https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Olivera&feedformat=atom AoPS Wiki - User contributions [en] 2020-01-23T15:03:35Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2007_AMC_12A_Problems/Problem_5&diff=114986 2007 AMC 12A Problems/Problem 5 2020-01-19T21:40:08Z <p>Olivera: /* Problem */ Fixed Problem</p> <hr /> <div>{{duplicate|[[2007 AMC 12A Problems|2007 AMC 12A #5]] and [[2007 AMC 10A Problems/Problem 7|2007 AMC 10A #7]]}}<br /> ==Problem==<br /> Last year Mr. Jon Q. Public received an inheritance. He paid &lt;math&gt;20\%&lt;/math&gt; in federal taxes on the inheritance, and paid &lt;math&gt;10\%&lt;/math&gt; of what he had left in state taxes. He paid a total of &lt;math&gt;\textdollar10500&lt;/math&gt; for both taxes. How many dollars was his inheritance?<br /> <br /> &lt;math&gt;(\mathrm {A})\ 30000 \qquad (\mathrm {B})\ 32500 \qquad(\mathrm {C})\ 35000 \qquad(\mathrm {D})\ 37500 \qquad(\mathrm {E})\ 40000&lt;/math&gt;<br /> <br /> ==Solution==<br /> After paying his taxes, he has &lt;math&gt;0.8*0.9=0.72&lt;/math&gt; of his earnings left. Since &lt;math&gt;10500&lt;/math&gt; is &lt;math&gt;0.28&lt;/math&gt; of his income, he got a total of &lt;math&gt;\frac{10500}{0.28}=37500\ \mathrm{(D)}&lt;/math&gt;.<br /> <br /> ==See also==<br /> {{AMC12 box|year=2007|ab=A|num-b=4|num-a=6}}<br /> {{AMC10 box|year=2007|ab=A|num-b=6|num-a=8}}<br /> <br /> [[Category:Introductory Algebra Problems]]<br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_13&diff=111963 2019 AMC 8 Problems/Problem 13 2019-11-21T18:02:20Z <p>Olivera: Fixed italics</p> <hr /> <div>==Problem 13==<br /> A ''palindrome'' is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> ==Solution 1==<br /> All the two digit palindromes are multiples of 11. The least 3 digit integer that is the sum of 2 two digit integers is a multiple of 11. The least 3 digit multiple of 11 is 110. The sum of the digits of 110 is 1 + 1 + 0 = &lt;math&gt;\boxed{\textbf{(A)}\ 2}&lt;/math&gt;.<br /> ~heeeeeeheeeee<br /> <br /> ==Solution 2==<br /> We let the two digit palindromes be &lt;math&gt;AA&lt;/math&gt;, &lt;math&gt;BB&lt;/math&gt;, and &lt;math&gt;CC&lt;/math&gt;, which sum to &lt;math&gt;11(A+B+C)&lt;/math&gt;. Now, we can let &lt;math&gt;A+B+C=k&lt;/math&gt;. This means we are looking for the smallest &lt;math&gt;k&lt;/math&gt; such that &lt;math&gt;11k&gt;100&lt;/math&gt; and &lt;math&gt;11k&lt;/math&gt; is not a palindrome. Thus, we test &lt;math&gt;10&lt;/math&gt; for &lt;math&gt;k&lt;/math&gt;, which works so &lt;math&gt;11k=110&lt;/math&gt;, meaning that the sum requested is &lt;math&gt;1+1+0=\boxed{\textbf{(A)}\ 2}&lt;/math&gt;. <br /> ~smartninja2000<br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=12|num-a=14}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_12&diff=111962 2019 AMC 8 Problems/Problem 12 2019-11-21T18:01:55Z <p>Olivera: Fixed image</p> <hr /> <div>==Problem==<br /> The faces of a cube are painted in six different colors: red (R), white (W), green (G), brown (B), aqua (A), and purple (P). Three views of the cube are shown below. What is the color of the face opposite the aqua face?<br /> <br /> [[File:2019AMC8Prob12.png]]<br /> &lt;math&gt;\textbf{(A) }Red\qquad\textbf{(B) }White\qquad\textbf{(C) }Green\qquad\textbf{(D) }Brown\qquad\textbf{(E) }Purple&lt;/math&gt;<br /> <br /> ==Solution 1==<br /> &lt;math&gt;B&lt;/math&gt; is on the top, and &lt;math&gt;R&lt;/math&gt; is on the side, and &lt;math&gt;G&lt;/math&gt; is on the right side. That means that (image 2)&lt;math&gt;W&lt;/math&gt; is on the left side. From the third image, you know that &lt;math&gt;P&lt;/math&gt; must be on the bottom since &lt;math&gt;G&lt;/math&gt; is sideways. That leaves us with the back, so the back must be &lt;math&gt;A&lt;/math&gt;. The front is opposite of the back, so the answer is &lt;math&gt;\boxed{\textbf{(A)}\ R}&lt;/math&gt;.~heeeeeeeheeeee<br /> <br /> ==Solution 2==<br /> {{AMC8 box|year=2019|num-b=11|num-a=13}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_11&diff=111961 2019 AMC 8 Problems/Problem 11 2019-11-21T18:00:16Z <p>Olivera: Fixed italics</p> <hr /> <div>==Problem 11==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eighth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eighth graders take ''only'' a math class and ''not'' a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> ==Solution 1==<br /> Let &lt;math&gt;x&lt;/math&gt; be the number of students taking both a math and a foreign language class.<br /> <br /> By P-I-E, we get &lt;math&gt;70 + 54 - x&lt;/math&gt; = &lt;math&gt;93&lt;/math&gt;. <br /> <br /> Solving gives us &lt;math&gt;x = 31&lt;/math&gt;.<br /> <br /> But we want the number of students taking only a math class.<br /> <br /> Which is &lt;math&gt;70 - 31 = 39&lt;/math&gt;.<br /> <br /> &lt;math&gt;\boxed{\textbf{(D)}\ 39}&lt;/math&gt;<br /> <br /> ~phoenixfire<br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=10|num-a=12}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_5&diff=111960 2019 AMC 8 Problems/Problem 5 2019-11-21T17:58:40Z <p>Olivera: Added problem</p> <hr /> <div>== Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> <br /> [[File:2019_AMC_8_-4_Image_1.png|900px]]<br /> <br /> [[File:2019_AMC_8_-4_Image_2.png|600px]]<br /> <br /> [[2019 AMC 8 Problems/Problem 5|Solution]]<br /> <br /> <br /> <br /> <br /> ==Solution 1==<br /> <br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=10|num-a=12}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111804 2019 AMC 8 Problems 2019-11-20T23:22:16Z <p>Olivera: Fixed typo</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually &lt;math&gt;21&lt;/math&gt; participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?<br /> &lt;asy&gt;<br /> unitsize(2mm);<br /> defaultpen(fontsize(8bp));<br /> real d = 5;<br /> real t = 0.7;<br /> real r;<br /> int[] num = {20,26,16,22,16};<br /> string[] days = {&quot;Monday&quot;,&quot;Tuesday&quot;,&quot;Wednesday&quot;,&quot;Thursday&quot;,&quot;Friday&quot;};<br /> for (int i=0; i&lt;30;<br /> i=i+2) { draw((i,0)--(i,-5*d),gray);<br /> }for (int i=0;<br /> i&lt;5;<br /> ++i) { r = -1*(i+0.5)*d;<br /> fill((0,r-t)--(0,r+t)--(num[i],r+t)--(num[i],r-t)--cycle,gray);<br /> label(days[i],(-1,r),W);<br /> }for(int i=0;<br /> i&lt;32;<br /> i=i+4) { label(string(i),(i,1));<br /> }label(&quot;Number of students at soccer practice&quot;,(14,3.5));<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median does not change.<br /> <br /> &lt;math&gt;\textbf{(B) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(C) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(D) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(E) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> [[2019 AMC 8 Problems/Problem 10|Solution]]<br /> <br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take ''only'' a math class and ''not'' a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A ''palindrome'' is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 18|Solution]]<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team that play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 19|Solution]]<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 20|Solution]]<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;y=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 21|Solution]]<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 22|Solution]]<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 23|Solution]]<br /> <br /> == Problem 24 ==<br /> In triangle &lt;math&gt;ABC&lt;/math&gt;, point &lt;math&gt;D&lt;/math&gt; divides side &lt;math&gt;\overline{AC}&lt;/math&gt; s that &lt;math&gt;AD:DC=1:2&lt;/math&gt;. Let &lt;math&gt;E&lt;/math&gt; be the midpoint of &lt;math&gt;\overline{BD}&lt;/math&gt; and left &lt;math&gt;F&lt;/math&gt; be the point of intersection of line &lt;math&gt;BC&lt;/math&gt; and line &lt;math&gt;AE&lt;/math&gt;. Given that the area of &lt;math&gt;\triangle ABC&lt;/math&gt; is &lt;math&gt;360&lt;/math&gt;, what is the area of &lt;math&gt;\triangle EBF&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> unitsize(2cm);<br /> pair A,B,C,DD,EE,FF;<br /> B = (0,0); C = (3,0); <br /> A = (1.2,1.7);<br /> DD = (2/3)*A+(1/3)*C;<br /> EE = (B+DD)/2;<br /> FF = intersectionpoint(B--C,A--A+2*(EE-A));<br /> draw(A--B--C--cycle);<br /> draw(A--FF); <br /> draw(B--DD);dot(A); <br /> label(&quot;$A$&quot;,A,N);<br /> dot(B); <br /> label(&quot;$B$&quot;,<br /> B,SW);dot(C); <br /> label(&quot;$C$&quot;,C,SE);<br /> dot(DD); <br /> label(&quot;$D$&quot;,DD,NE);<br /> dot(EE); <br /> label(&quot;$E$&quot;,EE,NW);<br /> dot(FF); <br /> label(&quot;$F$&quot;,FF,S);<br /> &lt;/asy&gt;<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }24\qquad\textbf{(B) }30\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 24|Solution]]<br /> <br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111803 2019 AMC 8 Problems 2019-11-20T23:21:38Z <p>Olivera: Changed italics</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually &lt;math&gt;21&lt;/math&gt; participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?<br /> &lt;asy&gt;<br /> unitsize(2mm);<br /> defaultpen(fontsize(8bp));<br /> real d = 5;<br /> real t = 0.7;<br /> real r;<br /> int[] num = {20,26,16,22,16};<br /> string[] days = {&quot;Monday&quot;,&quot;Tuesday&quot;,&quot;Wednesday&quot;,&quot;Thursday&quot;,&quot;Friday&quot;};<br /> for (int i=0; i&lt;30;<br /> i=i+2) { draw((i,0)--(i,-5*d),gray);<br /> }for (int i=0;<br /> i&lt;5;<br /> ++i) { r = -1*(i+0.5)*d;<br /> fill((0,r-t)--(0,r+t)--(num[i],r+t)--(num[i],r-t)--cycle,gray);<br /> label(days[i],(-1,r),W);<br /> }for(int i=0;<br /> i&lt;32;<br /> i=i+4) { label(string(i),(i,1));<br /> }label(&quot;Number of students at soccer practice&quot;,(14,3.5));<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median does not change.<br /> <br /> &lt;math&gt;\textbf{(B) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(C) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(D) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(E) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> [[2019 AMC 8 Problems/Problem 10|Solution]]<br /> <br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take ''only'' a math class and ''not'' a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A ''palindrome'' is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 18|Solution]]<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 19|Solution]]<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 20|Solution]]<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;y=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 21|Solution]]<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 22|Solution]]<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 23|Solution]]<br /> <br /> == Problem 24 ==<br /> In triangle &lt;math&gt;ABC&lt;/math&gt;, point &lt;math&gt;D&lt;/math&gt; divides side &lt;math&gt;\overline{AC}&lt;/math&gt; s that &lt;math&gt;AD:DC=1:2&lt;/math&gt;. Let &lt;math&gt;E&lt;/math&gt; be the midpoint of &lt;math&gt;\overline{BD}&lt;/math&gt; and left &lt;math&gt;F&lt;/math&gt; be the point of intersection of line &lt;math&gt;BC&lt;/math&gt; and line &lt;math&gt;AE&lt;/math&gt;. Given that the area of &lt;math&gt;\triangle ABC&lt;/math&gt; is &lt;math&gt;360&lt;/math&gt;, what is the area of &lt;math&gt;\triangle EBF&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> unitsize(2cm);<br /> pair A,B,C,DD,EE,FF;<br /> B = (0,0); C = (3,0); <br /> A = (1.2,1.7);<br /> DD = (2/3)*A+(1/3)*C;<br /> EE = (B+DD)/2;<br /> FF = intersectionpoint(B--C,A--A+2*(EE-A));<br /> draw(A--B--C--cycle);<br /> draw(A--FF); <br /> draw(B--DD);dot(A); <br /> label(&quot;$A$&quot;,A,N);<br /> dot(B); <br /> label(&quot;$B$&quot;,<br /> B,SW);dot(C); <br /> label(&quot;$C$&quot;,C,SE);<br /> dot(DD); <br /> label(&quot;$D$&quot;,DD,NE);<br /> dot(EE); <br /> label(&quot;$E$&quot;,EE,NW);<br /> dot(FF); <br /> label(&quot;$F$&quot;,FF,S);<br /> &lt;/asy&gt;<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }24\qquad\textbf{(B) }30\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 24|Solution]]<br /> <br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111802 2019 AMC 8 Problems 2019-11-20T23:21:20Z <p>Olivera: Changed italics</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually &lt;math&gt;21&lt;/math&gt; participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?<br /> &lt;asy&gt;<br /> unitsize(2mm);<br /> defaultpen(fontsize(8bp));<br /> real d = 5;<br /> real t = 0.7;<br /> real r;<br /> int[] num = {20,26,16,22,16};<br /> string[] days = {&quot;Monday&quot;,&quot;Tuesday&quot;,&quot;Wednesday&quot;,&quot;Thursday&quot;,&quot;Friday&quot;};<br /> for (int i=0; i&lt;30;<br /> i=i+2) { draw((i,0)--(i,-5*d),gray);<br /> }for (int i=0;<br /> i&lt;5;<br /> ++i) { r = -1*(i+0.5)*d;<br /> fill((0,r-t)--(0,r+t)--(num[i],r+t)--(num[i],r-t)--cycle,gray);<br /> label(days[i],(-1,r),W);<br /> }for(int i=0;<br /> i&lt;32;<br /> i=i+4) { label(string(i),(i,1));<br /> }label(&quot;Number of students at soccer practice&quot;,(14,3.5));<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median does not change.<br /> <br /> &lt;math&gt;\textbf{(B) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(C) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(D) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(E) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> [[2019 AMC 8 Problems/Problem 10|Solution]]<br /> <br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take ''only'' a math class and ''not'' a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 18|Solution]]<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 19|Solution]]<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 20|Solution]]<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;y=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 21|Solution]]<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 22|Solution]]<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 23|Solution]]<br /> <br /> == Problem 24 ==<br /> In triangle &lt;math&gt;ABC&lt;/math&gt;, point &lt;math&gt;D&lt;/math&gt; divides side &lt;math&gt;\overline{AC}&lt;/math&gt; s that &lt;math&gt;AD:DC=1:2&lt;/math&gt;. Let &lt;math&gt;E&lt;/math&gt; be the midpoint of &lt;math&gt;\overline{BD}&lt;/math&gt; and left &lt;math&gt;F&lt;/math&gt; be the point of intersection of line &lt;math&gt;BC&lt;/math&gt; and line &lt;math&gt;AE&lt;/math&gt;. Given that the area of &lt;math&gt;\triangle ABC&lt;/math&gt; is &lt;math&gt;360&lt;/math&gt;, what is the area of &lt;math&gt;\triangle EBF&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> unitsize(2cm);<br /> pair A,B,C,DD,EE,FF;<br /> B = (0,0); C = (3,0); <br /> A = (1.2,1.7);<br /> DD = (2/3)*A+(1/3)*C;<br /> EE = (B+DD)/2;<br /> FF = intersectionpoint(B--C,A--A+2*(EE-A));<br /> draw(A--B--C--cycle);<br /> draw(A--FF); <br /> draw(B--DD);dot(A); <br /> label(&quot;$A$&quot;,A,N);<br /> dot(B); <br /> label(&quot;$B$&quot;,<br /> B,SW);dot(C); <br /> label(&quot;$C$&quot;,C,SE);<br /> dot(DD); <br /> label(&quot;$D$&quot;,DD,NE);<br /> dot(EE); <br /> label(&quot;$E$&quot;,EE,NW);<br /> dot(FF); <br /> label(&quot;$F$&quot;,FF,S);<br /> &lt;/asy&gt;<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }24\qquad\textbf{(B) }30\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 24|Solution]]<br /> <br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111800 2019 AMC 8 Problems 2019-11-20T23:00:50Z <p>Olivera: Added link</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually &lt;math&gt;21&lt;/math&gt; participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?<br /> &lt;asy&gt;<br /> unitsize(2mm);<br /> defaultpen(fontsize(8bp));<br /> real d = 5;<br /> real t = 0.7;<br /> real r;<br /> int[] num = {20,26,16,22,16};<br /> string[] days = {&quot;Monday&quot;,&quot;Tuesday&quot;,&quot;Wednesday&quot;,&quot;Thursday&quot;,&quot;Friday&quot;};<br /> for (int i=0; i&lt;30;<br /> i=i+2) { draw((i,0)--(i,-5*d),gray);<br /> }for (int i=0;<br /> i&lt;5;<br /> ++i) { r = -1*(i+0.5)*d;<br /> fill((0,r-t)--(0,r+t)--(num[i],r+t)--(num[i],r-t)--cycle,gray);<br /> label(days[i],(-1,r),W);<br /> }for(int i=0;<br /> i&lt;32;<br /> i=i+4) { label(string(i),(i,1));<br /> }label(&quot;Number of students at soccer practice&quot;,(14,3.5));<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median does not change.<br /> <br /> &lt;math&gt;\textbf{(B) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(C) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(D) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(E) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> [[2019 AMC 8 Problems/Problem 10|Solution]]<br /> <br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 18|Solution]]<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 19|Solution]]<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 20|Solution]]<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;y=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 21|Solution]]<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 22|Solution]]<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 23|Solution]]<br /> <br /> == Problem 24 ==<br /> In triangle &lt;math&gt;ABC&lt;/math&gt;, point &lt;math&gt;D&lt;/math&gt; divides side &lt;math&gt;\overline{AC}&lt;/math&gt; s that &lt;math&gt;AD:DC=1:2&lt;/math&gt;. Let &lt;math&gt;E&lt;/math&gt; be the midpoint of &lt;math&gt;\overline{BD}&lt;/math&gt; and left &lt;math&gt;F&lt;/math&gt; be the point of intersection of line &lt;math&gt;BC&lt;/math&gt; and line &lt;math&gt;AE&lt;/math&gt;. Given that the area of &lt;math&gt;\triangle ABC&lt;/math&gt; is &lt;math&gt;360&lt;/math&gt;, what is the area of &lt;math&gt;\triangle EBF&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> unitsize(2cm);<br /> pair A,B,C,DD,EE,FF;<br /> B = (0,0); C = (3,0); <br /> A = (1.2,1.7);<br /> DD = (2/3)*A+(1/3)*C;<br /> EE = (B+DD)/2;<br /> FF = intersectionpoint(B--C,A--A+2*(EE-A));<br /> draw(A--B--C--cycle);<br /> draw(A--FF); <br /> draw(B--DD);dot(A); <br /> label(&quot;$A$&quot;,A,N);<br /> dot(B); <br /> label(&quot;$B$&quot;,<br /> B,SW);dot(C); <br /> label(&quot;$C$&quot;,C,SE);<br /> dot(DD); <br /> label(&quot;$D$&quot;,DD,NE);<br /> dot(EE); <br /> label(&quot;$E$&quot;,EE,NW);<br /> dot(FF); <br /> label(&quot;$F$&quot;,FF,S);<br /> &lt;/asy&gt;<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }24\qquad\textbf{(B) }30\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 24|Solution]]<br /> <br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111799 2019 AMC 8 Problems 2019-11-20T23:00:07Z <p>Olivera: Not needed; moved</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually &lt;math&gt;21&lt;/math&gt; participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?<br /> &lt;asy&gt;<br /> unitsize(2mm);<br /> defaultpen(fontsize(8bp));<br /> real d = 5;<br /> real t = 0.7;<br /> real r;<br /> int[] num = {20,26,16,22,16};<br /> string[] days = {&quot;Monday&quot;,&quot;Tuesday&quot;,&quot;Wednesday&quot;,&quot;Thursday&quot;,&quot;Friday&quot;};<br /> for (int i=0; i&lt;30;<br /> i=i+2) { draw((i,0)--(i,-5*d),gray);<br /> }for (int i=0;<br /> i&lt;5;<br /> ++i) { r = -1*(i+0.5)*d;<br /> fill((0,r-t)--(0,r+t)--(num[i],r+t)--(num[i],r-t)--cycle,gray);<br /> label(days[i],(-1,r),W);<br /> }for(int i=0;<br /> i&lt;32;<br /> i=i+4) { label(string(i),(i,1));<br /> }label(&quot;Number of students at soccer practice&quot;,(14,3.5));<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median does not change.<br /> <br /> &lt;math&gt;\textbf{(B) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(C) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(D) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(E) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 18|Solution]]<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 19|Solution]]<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 20|Solution]]<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;y=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 21|Solution]]<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 22|Solution]]<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 23|Solution]]<br /> <br /> == Problem 24 ==<br /> In triangle &lt;math&gt;ABC&lt;/math&gt;, point &lt;math&gt;D&lt;/math&gt; divides side &lt;math&gt;\overline{AC}&lt;/math&gt; s that &lt;math&gt;AD:DC=1:2&lt;/math&gt;. Let &lt;math&gt;E&lt;/math&gt; be the midpoint of &lt;math&gt;\overline{BD}&lt;/math&gt; and left &lt;math&gt;F&lt;/math&gt; be the point of intersection of line &lt;math&gt;BC&lt;/math&gt; and line &lt;math&gt;AE&lt;/math&gt;. Given that the area of &lt;math&gt;\triangle ABC&lt;/math&gt; is &lt;math&gt;360&lt;/math&gt;, what is the area of &lt;math&gt;\triangle EBF&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> unitsize(2cm);<br /> pair A,B,C,DD,EE,FF;<br /> B = (0,0); C = (3,0); <br /> A = (1.2,1.7);<br /> DD = (2/3)*A+(1/3)*C;<br /> EE = (B+DD)/2;<br /> FF = intersectionpoint(B--C,A--A+2*(EE-A));<br /> draw(A--B--C--cycle);<br /> draw(A--FF); <br /> draw(B--DD);dot(A); <br /> label(&quot;$A$&quot;,A,N);<br /> dot(B); <br /> label(&quot;$B$&quot;,<br /> B,SW);dot(C); <br /> label(&quot;$C$&quot;,C,SE);<br /> dot(DD); <br /> label(&quot;$D$&quot;,DD,NE);<br /> dot(EE); <br /> label(&quot;$E$&quot;,EE,NW);<br /> dot(FF); <br /> label(&quot;$F$&quot;,FF,S);<br /> &lt;/asy&gt;<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }24\qquad\textbf{(B) }30\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 24|Solution]]<br /> <br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_10&diff=111798 2019 AMC 8 Problems/Problem 10 2019-11-20T22:59:40Z <p>Olivera: Moved solution</p> <hr /> <div><br /> <br /> =Problem 10=<br /> The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually &lt;math&gt;21&lt;/math&gt; participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?<br /> &lt;asy&gt;<br /> unitsize(2mm);<br /> defaultpen(fontsize(8bp));<br /> real d = 5;<br /> real t = 0.7;<br /> real r;<br /> int[] num = {20,26,16,22,16};<br /> string[] days = {&quot;Monday&quot;,&quot;Tuesday&quot;,&quot;Wednesday&quot;,&quot;Thursday&quot;,&quot;Friday&quot;};<br /> for (int i=0; i&lt;30;<br /> i=i+2) { draw((i,0)--(i,-5*d),gray);<br /> }for (int i=0;<br /> i&lt;5;<br /> ++i) { r = -1*(i+0.5)*d;<br /> fill((0,r-t)--(0,r+t)--(num[i],r+t)--(num[i],r-t)--cycle,gray);<br /> label(days[i],(-1,r),W);<br /> }for(int i=0;<br /> i&lt;32;<br /> i=i+4) { label(string(i),(i,1));<br /> }label(&quot;Number of students at soccer practice&quot;,(14,3.5));<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median does not change.<br /> <br /> &lt;math&gt;\textbf{(B) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(C) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(D) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(E) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> <br /> <br /> ==Solution 1==<br /> On Monday, 20 people come. On Tuesday, 26 people come. On Wednesday, 16 people come. On Thursday, 22 people come. Finally, on Friday, 16 people come. 20+26+16+22+16=100, so the mean is 20. The median is (16,16,20,22,26) 20. The coach figures out that actually 21 people come on Wednesday. The new mean is 21, while the new median is (16,20,21,22,26)21. The median and mean both change, so the answer is &lt;math&gt;\boxed{\textbf{(B)}}&lt;/math&gt;~heeeeeeheeeee<br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=9|num-a=11}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111797 2019 AMC 8 Problems 2019-11-20T22:59:01Z <p>Olivera: Not needed</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually &lt;math&gt;21&lt;/math&gt; participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?<br /> &lt;asy&gt;<br /> unitsize(2mm);<br /> defaultpen(fontsize(8bp));<br /> real d = 5;<br /> real t = 0.7;<br /> real r;<br /> int[] num = {20,26,16,22,16};<br /> string[] days = {&quot;Monday&quot;,&quot;Tuesday&quot;,&quot;Wednesday&quot;,&quot;Thursday&quot;,&quot;Friday&quot;};<br /> for (int i=0; i&lt;30;<br /> i=i+2) { draw((i,0)--(i,-5*d),gray);<br /> }for (int i=0;<br /> i&lt;5;<br /> ++i) { r = -1*(i+0.5)*d;<br /> fill((0,r-t)--(0,r+t)--(num[i],r+t)--(num[i],r-t)--cycle,gray);<br /> label(days[i],(-1,r),W);<br /> }for(int i=0;<br /> i&lt;32;<br /> i=i+4) { label(string(i),(i,1));<br /> }label(&quot;Number of students at soccer practice&quot;,(14,3.5));<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median does not change.<br /> <br /> &lt;math&gt;\textbf{(B) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(C) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(D) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(E) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> ==Solution==<br /> On Monday, 20 people come. On Tuesday, 26 people come. On Wednesday, 16 people come. On Thursday, 22 people come. Finally, on Friday, 16 people come. 20+26+16+22+16=100, so the mean is 20. The median is (16,16,20,22,26) 20. The coach figures out that actually 21 people come on Wednesday. The new mean is 21, while the new median is (16,20,21,22,26)21. The median and mean both change, so the answer is &lt;math&gt;\boxed{\textbf{(B)}}&lt;/math&gt;~heeeeeeheeeee<br /> <br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 18|Solution]]<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 19|Solution]]<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 20|Solution]]<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;y=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 21|Solution]]<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 22|Solution]]<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 23|Solution]]<br /> <br /> == Problem 24 ==<br /> In triangle &lt;math&gt;ABC&lt;/math&gt;, point &lt;math&gt;D&lt;/math&gt; divides side &lt;math&gt;\overline{AC}&lt;/math&gt; s that &lt;math&gt;AD:DC=1:2&lt;/math&gt;. Let &lt;math&gt;E&lt;/math&gt; be the midpoint of &lt;math&gt;\overline{BD}&lt;/math&gt; and left &lt;math&gt;F&lt;/math&gt; be the point of intersection of line &lt;math&gt;BC&lt;/math&gt; and line &lt;math&gt;AE&lt;/math&gt;. Given that the area of &lt;math&gt;\triangle ABC&lt;/math&gt; is &lt;math&gt;360&lt;/math&gt;, what is the area of &lt;math&gt;\triangle EBF&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> unitsize(2cm);<br /> pair A,B,C,DD,EE,FF;<br /> B = (0,0); C = (3,0); <br /> A = (1.2,1.7);<br /> DD = (2/3)*A+(1/3)*C;<br /> EE = (B+DD)/2;<br /> FF = intersectionpoint(B--C,A--A+2*(EE-A));<br /> draw(A--B--C--cycle);<br /> draw(A--FF); <br /> draw(B--DD);dot(A); <br /> label(&quot;$A$&quot;,A,N);<br /> dot(B); <br /> label(&quot;$B$&quot;,<br /> B,SW);dot(C); <br /> label(&quot;$C$&quot;,C,SE);<br /> dot(DD); <br /> label(&quot;$D$&quot;,DD,NE);<br /> dot(EE); <br /> label(&quot;$E$&quot;,EE,NW);<br /> dot(FF); <br /> label(&quot;$F$&quot;,FF,S);<br /> &lt;/asy&gt;<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }24\qquad\textbf{(B) }30\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 24|Solution]]<br /> <br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_10&diff=111796 2019 AMC 8 Problems/Problem 10 2019-11-20T22:58:21Z <p>Olivera: Fixed typos</p> <hr /> <div><br /> <br /> =Problem 10=<br /> The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually &lt;math&gt;21&lt;/math&gt; participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?<br /> &lt;asy&gt;<br /> unitsize(2mm);<br /> defaultpen(fontsize(8bp));<br /> real d = 5;<br /> real t = 0.7;<br /> real r;<br /> int[] num = {20,26,16,22,16};<br /> string[] days = {&quot;Monday&quot;,&quot;Tuesday&quot;,&quot;Wednesday&quot;,&quot;Thursday&quot;,&quot;Friday&quot;};<br /> for (int i=0; i&lt;30;<br /> i=i+2) { draw((i,0)--(i,-5*d),gray);<br /> }for (int i=0;<br /> i&lt;5;<br /> ++i) { r = -1*(i+0.5)*d;<br /> fill((0,r-t)--(0,r+t)--(num[i],r+t)--(num[i],r-t)--cycle,gray);<br /> label(days[i],(-1,r),W);<br /> }for(int i=0;<br /> i&lt;32;<br /> i=i+4) { label(string(i),(i,1));<br /> }label(&quot;Number of students at soccer practice&quot;,(14,3.5));<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median does not change.<br /> <br /> &lt;math&gt;\textbf{(B) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(C) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(D) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(E) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> <br /> <br /> ==Solution 1==<br /> <br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=9|num-a=11}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111795 2019 AMC 8 Problems 2019-11-20T22:57:55Z <p>Olivera: Added problem</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually &lt;math&gt;21&lt;/math&gt; participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?<br /> &lt;asy&gt;<br /> unitsize(2mm);<br /> defaultpen(fontsize(8bp));<br /> real d = 5;<br /> real t = 0.7;<br /> real r;<br /> int[] num = {20,26,16,22,16};<br /> string[] days = {&quot;Monday&quot;,&quot;Tuesday&quot;,&quot;Wednesday&quot;,&quot;Thursday&quot;,&quot;Friday&quot;};<br /> for (int i=0; i&lt;30;<br /> i=i+2) { draw((i,0)--(i,-5*d),gray);<br /> }for (int i=0;<br /> i&lt;5;<br /> ++i) { r = -1*(i+0.5)*d;<br /> fill((0,r-t)--(0,r+t)--(num[i],r+t)--(num[i],r-t)--cycle,gray);<br /> label(days[i],(-1,r),W);<br /> }for(int i=0;<br /> i&lt;32;<br /> i=i+4) { label(string(i),(i,1));<br /> }label(&quot;Number of students at soccer practice&quot;,(14,3.5));<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median does not change.<br /> <br /> &lt;math&gt;\textbf{(B) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(C) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(D) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(E) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> ==Solution==<br /> <br /> ==Solution==<br /> On Monday, 20 people come. On Tuesday, 26 people come. On Wednesday, 16 people come. On Thursday, 22 people come. Finally, on Friday, 16 people come. 20+26+16+22+16=100, so the mean is 20. The median is (16,16,20,22,26) 20. The coach figures out that actually 21 people come on Wednesday. The new mean is 21, while the new median is (16,20,21,22,26)21. The median and mean both change, so the answer is &lt;math&gt;\boxed{\textbf{(B)}}&lt;/math&gt;~heeeeeeheeeee<br /> <br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 18|Solution]]<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 19|Solution]]<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 20|Solution]]<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;y=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 21|Solution]]<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 22|Solution]]<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 23|Solution]]<br /> <br /> == Problem 24 ==<br /> In triangle &lt;math&gt;ABC&lt;/math&gt;, point &lt;math&gt;D&lt;/math&gt; divides side &lt;math&gt;\overline{AC}&lt;/math&gt; s that &lt;math&gt;AD:DC=1:2&lt;/math&gt;. Let &lt;math&gt;E&lt;/math&gt; be the midpoint of &lt;math&gt;\overline{BD}&lt;/math&gt; and left &lt;math&gt;F&lt;/math&gt; be the point of intersection of line &lt;math&gt;BC&lt;/math&gt; and line &lt;math&gt;AE&lt;/math&gt;. Given that the area of &lt;math&gt;\triangle ABC&lt;/math&gt; is &lt;math&gt;360&lt;/math&gt;, what is the area of &lt;math&gt;\triangle EBF&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> unitsize(2cm);<br /> pair A,B,C,DD,EE,FF;<br /> B = (0,0); C = (3,0); <br /> A = (1.2,1.7);<br /> DD = (2/3)*A+(1/3)*C;<br /> EE = (B+DD)/2;<br /> FF = intersectionpoint(B--C,A--A+2*(EE-A));<br /> draw(A--B--C--cycle);<br /> draw(A--FF); <br /> draw(B--DD);dot(A); <br /> label(&quot;$A$&quot;,A,N);<br /> dot(B); <br /> label(&quot;$B$&quot;,<br /> B,SW);dot(C); <br /> label(&quot;$C$&quot;,C,SE);<br /> dot(DD); <br /> label(&quot;$D$&quot;,DD,NE);<br /> dot(EE); <br /> label(&quot;$E$&quot;,EE,NW);<br /> dot(FF); <br /> label(&quot;$F$&quot;,FF,S);<br /> &lt;/asy&gt;<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }24\qquad\textbf{(B) }30\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 24|Solution]]<br /> <br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_10&diff=111792 2019 AMC 8 Problems/Problem 10 2019-11-20T22:54:21Z <p>Olivera: Added space</p> <hr /> <div><br /> <br /> =Problem 10=<br /> The diagram shows the number of students at soccer practice each weekday during last wee. After computing the mean and median vlaues, Coach discovers that there were actually &lt;math&gt;21&lt;/math&gt; participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?<br /> &lt;asy&gt;<br /> unitsize(2mm);<br /> defaultpen(fontsize(8bp));<br /> real d = 5;<br /> real t = 0.7;<br /> real r;<br /> int[] num = {20,26,16,22,16};<br /> string[] days = {&quot;Monday&quot;,&quot;Tuesday&quot;,&quot;Wednesday&quot;,&quot;Thursday&quot;,&quot;Friday&quot;};<br /> for (int i=0; i&lt;30;<br /> i=i+2) { draw((i,0)--(i,-5*d),gray);<br /> }for (int i=0;<br /> i&lt;5;<br /> ++i) { r = -1*(i+0.5)*d;<br /> fill((0,r-t)--(0,r+t)--(num[i],r+t)--(num[i],r-t)--cycle,gray);<br /> label(days[i],(-1,r),W);<br /> }for(int i=0;<br /> i&lt;32;<br /> i=i+4) { label(string(i),(i,1));<br /> }label(&quot;Number of students at soccer practice&quot;,(14,3.5));<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median does not change.<br /> <br /> &lt;math&gt;\textbf{(B) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(C) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(D) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> <br /> &lt;math&gt;\textbf{(E) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> <br /> <br /> ==Solution 1==<br /> <br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=9|num-a=11}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_10&diff=111791 2019 AMC 8 Problems/Problem 10 2019-11-20T22:53:49Z <p>Olivera: Added problem</p> <hr /> <div><br /> <br /> =Problem 10=<br /> The diagram shows the number of students at soccer practice each weekday during last wee. After computing the mean and median vlaues, Coach discovers that there were actually &lt;math&gt;21&lt;/math&gt; participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?<br /> &lt;asy&gt;<br /> unitsize(2mm);<br /> defaultpen(fontsize(8bp));<br /> real d = 5;<br /> real t = 0.7;<br /> real r;<br /> int[] num = {20,26,16,22,16};<br /> string[] days = {&quot;Monday&quot;,&quot;Tuesday&quot;,&quot;Wednesday&quot;,&quot;Thursday&quot;,&quot;Friday&quot;};<br /> for (int i=0; i&lt;30;<br /> i=i+2) { draw((i,0)--(i,-5*d),gray);<br /> }for (int i=0;<br /> i&lt;5;<br /> ++i) { r = -1*(i+0.5)*d;<br /> fill((0,r-t)--(0,r+t)--(num[i],r+t)--(num[i],r-t)--cycle,gray);<br /> label(days[i],(-1,r),W);<br /> }for(int i=0;<br /> i&lt;32;<br /> i=i+4) { label(string(i),(i,1));<br /> }label(&quot;Number of students at soccer practice&quot;,(14,3.5));<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median does not change.<br /> &lt;math&gt;\textbf{(B) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> &lt;math&gt;\textbf{(C) }&lt;/math&gt;The mean increases by &lt;math&gt;1&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> &lt;math&gt;\textbf{(D) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;1&lt;/math&gt;.<br /> &lt;math&gt;\textbf{(E) }&lt;/math&gt;The mean increases by &lt;math&gt;5&lt;/math&gt; and the median increases by &lt;math&gt;5&lt;/math&gt;.<br /> <br /> <br /> <br /> ==Solution 1==<br /> <br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=9|num-a=11}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_10&diff=111778 2019 AMC 8 Problems/Problem 10 2019-11-20T22:36:21Z <p>Olivera: Added diagram</p> <hr /> <div>&lt;asy&gt;<br /> unitsize(2mm);<br /> defaultpen(fontsize(8bp));<br /> real d = 5;<br /> real t = 0.7;<br /> real r;<br /> int[] num = {20,26,16,22,16};<br /> string[] days = {&quot;Monday&quot;,&quot;Tuesday&quot;,&quot;Wednesday&quot;,&quot;Thursday&quot;,&quot;Friday&quot;};<br /> for (int i=0; i&lt;30;<br /> i=i+2) { draw((i,0)--(i,-5*d),gray);<br /> }for (int i=0;<br /> i&lt;5;<br /> ++i) { r = -1*(i+0.5)*d;<br /> fill((0,r-t)--(0,r+t)--(num[i],r+t)--(num[i],r-t)--cycle,gray);<br /> label(days[i],(-1,r),W);<br /> }for(int i=0;<br /> i&lt;32;<br /> i=i+4) { label(string(i),(i,1));<br /> }label(&quot;Number of students at soccer practice&quot;,(14,3.5));<br /> &lt;/asy&gt;</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111777 2019 AMC 8 Problems 2019-11-20T22:35:37Z <p>Olivera: Added diagram</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> &lt;asy&gt;<br /> unitsize(2mm);<br /> defaultpen(fontsize(8bp));<br /> real d = 5;<br /> real t = 0.7;<br /> real r;<br /> int[] num = {20,26,16,22,16};<br /> string[] days = {&quot;Monday&quot;,&quot;Tuesday&quot;,&quot;Wednesday&quot;,&quot;Thursday&quot;,&quot;Friday&quot;};<br /> for (int i=0; i&lt;30;<br /> i=i+2) { draw((i,0)--(i,-5*d),gray);<br /> }for (int i=0;<br /> i&lt;5;<br /> ++i) { r = -1*(i+0.5)*d;<br /> fill((0,r-t)--(0,r+t)--(num[i],r+t)--(num[i],r-t)--cycle,gray);<br /> label(days[i],(-1,r),W);<br /> }for(int i=0;<br /> i&lt;32;<br /> i=i+4) { label(string(i),(i,1));<br /> }label(&quot;Number of students at soccer practice&quot;,(14,3.5));<br /> &lt;/asy&gt;<br /> <br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 18|Solution]]<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 19|Solution]]<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 20|Solution]]<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;y=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 21|Solution]]<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 22|Solution]]<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 23|Solution]]<br /> <br /> == Problem 24 ==<br /> In triangle &lt;math&gt;ABC&lt;/math&gt;, point &lt;math&gt;D&lt;/math&gt; divides side &lt;math&gt;\overline{AC}&lt;/math&gt; s that &lt;math&gt;AD:DC=1:2&lt;/math&gt;. Let &lt;math&gt;E&lt;/math&gt; be the midpoint of &lt;math&gt;\overline{BD}&lt;/math&gt; and left &lt;math&gt;F&lt;/math&gt; be the point of intersection of line &lt;math&gt;BC&lt;/math&gt; and line &lt;math&gt;AE&lt;/math&gt;. Given that the area of &lt;math&gt;\triangle ABC&lt;/math&gt; is &lt;math&gt;360&lt;/math&gt;, what is the area of &lt;math&gt;\triangle EBF&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> unitsize(2cm);<br /> pair A,B,C,DD,EE,FF;<br /> B = (0,0); C = (3,0); <br /> A = (1.2,1.7);<br /> DD = (2/3)*A+(1/3)*C;<br /> EE = (B+DD)/2;<br /> FF = intersectionpoint(B--C,A--A+2*(EE-A));<br /> draw(A--B--C--cycle);<br /> draw(A--FF); <br /> draw(B--DD);dot(A); <br /> label(&quot;$A$&quot;,A,N);<br /> dot(B); <br /> label(&quot;$B$&quot;,<br /> B,SW);dot(C); <br /> label(&quot;$C$&quot;,C,SE);<br /> dot(DD); <br /> label(&quot;$D$&quot;,DD,NE);<br /> dot(EE); <br /> label(&quot;$E$&quot;,EE,NW);<br /> dot(FF); <br /> label(&quot;$F$&quot;,FF,S);<br /> &lt;/asy&gt;<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }24\qquad\textbf{(B) }30\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 24|Solution]]<br /> <br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111767 2019 AMC 8 Problems 2019-11-20T21:39:38Z <p>Olivera: Added link</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 18|Solution]]<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 19|Solution]]<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 20|Solution]]<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;7=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 21|Solution]]<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 22|Solution]]<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 23|Solution]]<br /> <br /> == Problem 24 ==<br /> In triangle &lt;math&gt;ABC&lt;/math&gt;, point &lt;math&gt;D&lt;/math&gt; divides side &lt;math&gt;\overline{AC}&lt;/math&gt; s that &lt;math&gt;AD:DC=1:2&lt;/math&gt;. Let &lt;math&gt;E&lt;/math&gt; be the midpoint of &lt;math&gt;\overline{BD}&lt;/math&gt; and left &lt;math&gt;F&lt;/math&gt; be the point of intersection of line &lt;math&gt;BC&lt;/math&gt; and line &lt;math&gt;AE&lt;/math&gt;. Given that the area of &lt;math&gt;\triangle ABC&lt;/math&gt; is &lt;math&gt;360&lt;/math&gt;, what is the area of &lt;math&gt;\triangle EBF&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> unitsize(2cm);<br /> pair A,B,C,DD,EE,FF;<br /> B = (0,0); C = (3,0); <br /> A = (1.2,1.7);<br /> DD = (2/3)*A+(1/3)*C;<br /> EE = (B+DD)/2;<br /> FF = intersectionpoint(B--C,A--A+2*(EE-A));<br /> draw(A--B--C--cycle);<br /> draw(A--FF); <br /> draw(B--DD);dot(A); <br /> label(&quot;$A$&quot;,A,N);<br /> dot(B); <br /> label(&quot;$B$&quot;,<br /> B,SW);dot(C); <br /> label(&quot;$C$&quot;,C,SE);<br /> dot(DD); <br /> label(&quot;$D$&quot;,DD,NE);<br /> dot(EE); <br /> label(&quot;$E$&quot;,EE,NW);<br /> dot(FF); <br /> label(&quot;$F$&quot;,FF,S);<br /> &lt;/asy&gt;<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }24\qquad\textbf{(B) }30\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 24|Solution]]<br /> <br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111766 2019 AMC 8 Problems 2019-11-20T21:39:12Z <p>Olivera: Added problem</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 18|Solution]]<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 19|Solution]]<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 20|Solution]]<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;7=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 21|Solution]]<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 22|Solution]]<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 23|Solution]]<br /> <br /> == Problem 24 ==<br /> In triangle &lt;math&gt;ABC&lt;/math&gt;, point &lt;math&gt;D&lt;/math&gt; divides side &lt;math&gt;\overline{AC}&lt;/math&gt; s that &lt;math&gt;AD:DC=1:2&lt;/math&gt;. Let &lt;math&gt;E&lt;/math&gt; be the midpoint of &lt;math&gt;\overline{BD}&lt;/math&gt; and left &lt;math&gt;F&lt;/math&gt; be the point of intersection of line &lt;math&gt;BC&lt;/math&gt; and line &lt;math&gt;AE&lt;/math&gt;. Given that the area of &lt;math&gt;\triangle ABC&lt;/math&gt; is &lt;math&gt;360&lt;/math&gt;, what is the area of &lt;math&gt;\triangle EBF&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> unitsize(2cm);<br /> pair A,B,C,DD,EE,FF;<br /> B = (0,0); C = (3,0); <br /> A = (1.2,1.7);<br /> DD = (2/3)*A+(1/3)*C;<br /> EE = (B+DD)/2;<br /> FF = intersectionpoint(B--C,A--A+2*(EE-A));<br /> draw(A--B--C--cycle);<br /> draw(A--FF); <br /> draw(B--DD);dot(A); <br /> label(&quot;$A$&quot;,A,N);<br /> dot(B); <br /> label(&quot;$B$&quot;,<br /> B,SW);dot(C); <br /> label(&quot;$C$&quot;,C,SE);<br /> dot(DD); <br /> label(&quot;$D$&quot;,DD,NE);<br /> dot(EE); <br /> label(&quot;$E$&quot;,EE,NW);<br /> dot(FF); <br /> label(&quot;$F$&quot;,FF,S);<br /> &lt;/asy&gt;<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }24\qquad\textbf{(B) }30\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_24&diff=111765 2019 AMC 8 Problems/Problem 24 2019-11-20T21:38:22Z <p>Olivera: Fixed LaTeX</p> <hr /> <div><br /> <br /> <br /> ==Problem 24==<br /> In triangle &lt;math&gt;ABC&lt;/math&gt;, point &lt;math&gt;D&lt;/math&gt; divides side &lt;math&gt;\overline{AC}&lt;/math&gt; s that &lt;math&gt;AD:DC=1:2&lt;/math&gt;. Let &lt;math&gt;E&lt;/math&gt; be the midpoint of &lt;math&gt;\overline{BD}&lt;/math&gt; and left &lt;math&gt;F&lt;/math&gt; be the point of intersection of line &lt;math&gt;BC&lt;/math&gt; and line &lt;math&gt;AE&lt;/math&gt;. Given that the area of &lt;math&gt;\triangle ABC&lt;/math&gt; is &lt;math&gt;360&lt;/math&gt;, what is the area of &lt;math&gt;\triangle EBF&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> unitsize(2cm);<br /> pair A,B,C,DD,EE,FF;<br /> B = (0,0); C = (3,0); <br /> A = (1.2,1.7);<br /> DD = (2/3)*A+(1/3)*C;<br /> EE = (B+DD)/2;<br /> FF = intersectionpoint(B--C,A--A+2*(EE-A));<br /> draw(A--B--C--cycle);<br /> draw(A--FF); <br /> draw(B--DD);dot(A); <br /> label(&quot;$A$&quot;,A,N);<br /> dot(B); <br /> label(&quot;$B$&quot;,<br /> B,SW);dot(C); <br /> label(&quot;$C$&quot;,C,SE);<br /> dot(DD); <br /> label(&quot;$D$&quot;,DD,NE);<br /> dot(EE); <br /> label(&quot;$E$&quot;,EE,NW);<br /> dot(FF); <br /> label(&quot;$F$&quot;,FF,S);<br /> &lt;/asy&gt;<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }24\qquad\textbf{(B) }30\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> ==Solution 1==<br /> <br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=19|num-a=21}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_24&diff=111764 2019 AMC 8 Problems/Problem 24 2019-11-20T21:37:49Z <p>Olivera: Added full problem</p> <hr /> <div><br /> <br /> <br /> ==Problem 24==<br /> In triangle &lt;math&gt;ABC&lt;/math&gt;, point &lt;math&gt;D&lt;/math&gt; divides side &lt;math&gt;\overline{AC}&lt;/math&gt; s that &lt;math&gt;AD&lt;/math&gt;:&lt;math&gt;DC=1&lt;/math&gt;:&lt;math&gt;2&lt;/math&gt;. Let &lt;math&gt;E&lt;/math&gt; be the midpoint of &lt;math&gt;\overline{BD}&lt;/math&gt; and left &lt;math&gt;F&lt;/math&gt; be the point of intersection of line &lt;math&gt;BC&lt;/math&gt; and line &lt;math&gt;AE&lt;/math&gt;. Given that the area of &lt;math&gt;\triangle ABC&lt;/math&gt; is &lt;math&gt;360&lt;/math&gt;, what is the area of &lt;math&gt;\triangle EBF&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> unitsize(2cm);<br /> pair A,B,C,DD,EE,FF;<br /> B = (0,0); C = (3,0); <br /> A = (1.2,1.7);<br /> DD = (2/3)*A+(1/3)*C;<br /> EE = (B+DD)/2;<br /> FF = intersectionpoint(B--C,A--A+2*(EE-A));<br /> draw(A--B--C--cycle);<br /> draw(A--FF); <br /> draw(B--DD);dot(A); <br /> label(&quot;$A$&quot;,A,N);<br /> dot(B); <br /> label(&quot;$B$&quot;,<br /> B,SW);dot(C); <br /> label(&quot;$C$&quot;,C,SE);<br /> dot(DD); <br /> label(&quot;$D$&quot;,DD,NE);<br /> dot(EE); <br /> label(&quot;$E$&quot;,EE,NW);<br /> dot(FF); <br /> label(&quot;$F$&quot;,FF,S);<br /> &lt;/asy&gt;<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }24\qquad\textbf{(B) }30\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> ==Solution 1==<br /> <br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=19|num-a=21}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_24&diff=111763 2019 AMC 8 Problems/Problem 24 2019-11-20T21:33:58Z <p>Olivera: Added diagram</p> <hr /> <div>==Problem==<br /> &lt;asy&gt;<br /> unitsize(2cm);<br /> pair A,B,C,DD,EE,FF;<br /> B = (0,0); C = (3,0); <br /> A = (1.2,1.7);<br /> DD = (2/3)*A+(1/3)*C;<br /> EE = (B+DD)/2;<br /> FF = intersectionpoint(B--C,A--A+2*(EE-A));<br /> draw(A--B--C--cycle);<br /> draw(A--FF); <br /> draw(B--DD);dot(A); <br /> label(&quot;$A$&quot;,A,N);<br /> dot(B); <br /> label(&quot;$B$&quot;,<br /> B,SW);dot(C); <br /> label(&quot;$C$&quot;,C,SE);<br /> dot(DD); <br /> label(&quot;$D$&quot;,DD,NE);<br /> dot(EE); <br /> label(&quot;$E$&quot;,EE,NW);<br /> dot(FF); <br /> label(&quot;$F$&quot;,FF,S);<br /> &lt;/asy&gt;<br /> <br /> ==Solution 1==<br /> ==Solution 2==<br /> {{AMC8 box|year=2019|num-b=23|num-a=25}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111735 2019 AMC 8 Problems 2019-11-20T18:27:33Z <p>Olivera: Added link</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 18|Solution]]<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 19|Solution]]<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 20|Solution]]<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;7=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 21|Solution]]<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 22|Solution]]<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 23|Solution]]<br /> <br /> == Problem 24 ==<br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111734 2019 AMC 8 Problems 2019-11-20T18:27:15Z <p>Olivera: Added link</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 18|Solution]]<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 19|Solution]]<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 20|Solution]]<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;7=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 21|Solution]]<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 22|Solution]]<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> == Problem 24 ==<br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111733 2019 AMC 8 Problems 2019-11-20T18:27:00Z <p>Olivera: Added link</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 18|Solution]]<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 19|Solution]]<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 20|Solution]]<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;7=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 21|Solution]]<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> == Problem 24 ==<br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111732 2019 AMC 8 Problems 2019-11-20T18:26:24Z <p>Olivera: Added link</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 18|Solution]]<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 19|Solution]]<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 20|Solution]]<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;7=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> == Problem 24 ==<br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111731 2019 AMC 8 Problems 2019-11-20T18:24:15Z <p>Olivera: Added link</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 18|Solution]]<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 19|Solution]]<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;7=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> == Problem 24 ==<br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111730 2019 AMC 8 Problems 2019-11-20T18:23:37Z <p>Olivera: Added link</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 18|Solution]]<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;7=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> == Problem 24 ==<br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111729 2019 AMC 8 Problems 2019-11-20T18:23:14Z <p>Olivera: Added link</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;7=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> == Problem 24 ==<br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111728 2019 AMC 8 Problems 2019-11-20T18:21:57Z <p>Olivera: Added link</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;7=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> == Problem 24 ==<br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111726 2019 AMC 8 Problems 2019-11-20T18:20:48Z <p>Olivera: Added problem</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;7=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> == Problem 23 ==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> == Problem 24 ==<br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111724 2019 AMC 8 Problems 2019-11-20T18:20:09Z <p>Olivera: Added problem</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;7=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> == Problem 22 ==<br /> A store increased the original price of a shirt by a certain percent and then decreased the new <br /> price by the same amount. Given that the resulting price was 84% of the original price, by what <br /> percent was the price increased and decreased? <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40&lt;/math&gt;<br /> <br /> == Problem 23 ==<br /> == Problem 24 ==<br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111723 2019 AMC 8 Problems 2019-11-20T18:19:42Z <p>Olivera: Added problem</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> == Problem 21 ==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;7=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> == Problem 22 ==<br /> == Problem 23 ==<br /> == Problem 24 ==<br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_20&diff=111722 2019 AMC 8 Problems/Problem 20 2019-11-20T18:19:18Z <p>Olivera: Fixed problem number</p> <hr /> <div>==Problem 20==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> ==Solution 1==<br /> <br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=19|num-a=21}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111721 2019 AMC 8 Problems 2019-11-20T18:17:29Z <p>Olivera: Added problem</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> == Problem 20 ==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> == Problem 21 ==<br /> == Problem 22 ==<br /> == Problem 23 ==<br /> == Problem 24 ==<br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111720 2019 AMC 8 Problems 2019-11-20T18:16:22Z <p>Olivera: Added problem</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> == Problem 19 ==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> == Problem 20 ==<br /> == Problem 21 ==<br /> == Problem 22 ==<br /> == Problem 23 ==<br /> == Problem 24 ==<br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111719 2019 AMC 8 Problems 2019-11-20T18:15:06Z <p>Olivera: Added problem</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> == Problem 18 ==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> == Problem 19 ==<br /> == Problem 20 ==<br /> == Problem 21 ==<br /> == Problem 22 ==<br /> == Problem 23 ==<br /> == Problem 24 ==<br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111718 2019 AMC 8 Problems 2019-11-20T18:14:09Z <p>Olivera: Added problem</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> == Problem 17 ==<br /> What is the value of the product <br /> &lt;cmath&gt;\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) }50&lt;/math&gt;<br /> <br /> == Problem 18 ==<br /> == Problem 19 ==<br /> == Problem 20 ==<br /> == Problem 21 ==<br /> == Problem 22 ==<br /> == Problem 23 ==<br /> == Problem 24 ==<br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems&diff=111716 2019 AMC 8 Problems 2019-11-20T18:12:50Z <p>Olivera: Added problem</p> <hr /> <div>== Problem 1 ==<br /> Ike and Mike go into a sandwich shop with a total of &lt;math&gt;\$30.00&lt;/math&gt; to spend. Sandwiches cost &lt;math&gt;\$4.50&lt;/math&gt; each and soft drinks cost &lt;math&gt;\$1.00&lt;/math&gt; each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?<br /> <br /> &lt;math&gt;\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> &lt;asy&gt;<br /> draw((0,0)--(3,0));<br /> draw((0,0)--(0,2));<br /> draw((0,2)--(3,2));<br /> draw((3,2)--(3,0));<br /> dot((0,0));<br /> dot((0,2));<br /> dot((3,0));<br /> dot((3,2));<br /> draw((2,0)--(2,2));<br /> draw((0,1)--(2,1));<br /> label(&quot;A&quot;,(0,0),S);<br /> label(&quot;B&quot;,(3,0),S);<br /> label(&quot;C&quot;,(3,2),N);<br /> label(&quot;D&quot;,(0,2),N);<br /> &lt;/asy&gt;<br /> Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is &lt;math&gt;5&lt;/math&gt; feet, what is the area in square feet of rectangle &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }75\qquad\textbf{(C) }100\qquad\textbf{(D) }125\qquad\textbf{(E) }150&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> 3. Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11}&lt;/math&gt;, &lt;math&gt;\frac{19}{15}&lt;/math&gt;, and &lt;math&gt;\frac{17}{13}&lt;/math&gt;, from least to greatest?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11} &lt; \frac{17}{13} &lt; \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} &lt; \frac{19}{15} &lt; \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} &lt; \frac{19}{15} &lt; \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} &lt; \frac{15}{11} &lt; \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} &lt; \frac{17}{13} &lt; \frac{15}{11}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance &lt;math&gt;d&lt;/math&gt; traveled by the two animals over time &lt;math&gt;t&lt;/math&gt; from start to finish?<br /> [img]https://latex.artofproblemsolving.com/e/f/9/ef92362133ad95b0788184ff802df2094337d377.png[/img]<br /> [img width=&quot;65&quot; height=&quot;5&quot;]https://latex.artofproblemsolving.com/6/8/2/682b63fdba98e33c13055463223d6c8ea409cf23.png[/img]<br /> <br /> == Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> [[2019 AMC 8 Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> Shauna takes five tests, each worth a maximum of &lt;math&gt;100&lt;/math&gt; points. Her scores on the first three tests are &lt;math&gt;76&lt;/math&gt;, &lt;math&gt;94&lt;/math&gt;, and &lt;math&gt;87&lt;/math&gt; . In order to average &lt;math&gt;81&lt;/math&gt; for all five tests, what is the lowest score she could earn on one of the other two tests?<br /> <br /> &lt;math&gt;\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?<br /> <br /> &lt;math&gt;\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are &lt;math&gt;6&lt;/math&gt; cm in diameter and &lt;math&gt;12&lt;/math&gt; cm high. Felicia buys cat food in cylindrical cans that are &lt;math&gt;12&lt;/math&gt; cm in diameter and &lt;math&gt;6&lt;/math&gt; cm high. What is the ratio of the volume one of Alex's cans to the volume one of Felicia's cans?<br /> <br /> &lt;math&gt;\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> == Problem 11 ==<br /> The eighth grade class at Lincoln Middle School has &lt;math&gt;93&lt;/math&gt; students. Each student takes a math class or a foreign language class or both. There are &lt;math&gt;70&lt;/math&gt; eigth graders taking a math class, and there are &lt;math&gt;54&lt;/math&gt; eight graders taking a foreign language class. How many eigth graders take &lt;math&gt;\textit{only}&lt;/math&gt; a math class and &lt;math&gt;\textit{not}&lt;/math&gt; a foreign language class?<br /> <br /> &lt;math&gt;\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> == Problem 13 ==<br /> A &lt;math&gt;\textit{palindrome}&lt;/math&gt; is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let &lt;math&gt;N&lt;/math&gt; be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of &lt;math&gt;N&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Isabella has &lt;math&gt;6&lt;/math&gt; coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every &lt;math&gt;10&lt;/math&gt; days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the &lt;math&gt;6&lt;/math&gt; dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?<br /> <br /> &lt;math&gt;\textbf{(A) }&lt;/math&gt;Monday&lt;math&gt;\qquad\textbf{(B) }&lt;/math&gt;Tuesday&lt;math&gt;\qquad\textbf{(C) }&lt;/math&gt;Wednesday&lt;math&gt;\qquad\textbf{(D) }&lt;/math&gt;Thursday&lt;math&gt;\qquad\textbf{(E) }&lt;/math&gt;Friday<br /> <br /> [[2019 AMC 8 Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> On a beach &lt;math&gt;50&lt;/math&gt; people are wearing sunglasses and &lt;math&gt;35&lt;/math&gt; people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is &lt;math&gt;\frac{2}{5}&lt;/math&gt;. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?<br /> &lt;math&gt;\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 15|Solution]]<br /> <br /> ==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> == Problem 17 ==<br /> == Problem 18 ==<br /> == Problem 19 ==<br /> == Problem 20 ==<br /> == Problem 21 ==<br /> == Problem 22 ==<br /> == Problem 23 ==<br /> == Problem 24 ==<br /> == Problem 25 ==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the people has at least &lt;math&gt;2&lt;/math&gt; apples?<br /> <br /> &lt;math&gt;\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380&lt;/math&gt;<br /> <br /> [[2019 AMC 8 Problems/Problem 25|Solution]]<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_4&diff=111714 2019 AMC 8 Problems/Problem 4 2019-11-20T18:09:35Z <p>Olivera: Added problem statement</p> <hr /> <div>== Problem 4 ==<br /> <br /> Quadrilateral &lt;math&gt;ABCD&lt;/math&gt; is a rhombus with perimeter &lt;math&gt;52&lt;/math&gt; meters. The length of diagonal &lt;math&gt;\overline{AC}&lt;/math&gt; is &lt;math&gt;24&lt;/math&gt; meters. What is the area in square meters of rhombus &lt;math&gt;ABCD&lt;/math&gt;?<br /> <br /> &lt;asy&gt;<br /> draw((-13,0)--(0,5));<br /> draw((0,5)--(13,0));<br /> draw((13,0)--(0,-5));<br /> draw((0,-5)--(-13,0));<br /> dot((-13,0));<br /> dot((0,5));<br /> dot((13,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-13,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(13,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }60\qquad\textbf{(B) }90\qquad\textbf{(C) }105\qquad\textbf{(D) }120\qquad\textbf{(E) }144&lt;/math&gt;<br /> <br /> <br /> == Solution 1 ==<br /> &lt;asy&gt;<br /> draw((-12,0)--(0,5));<br /> draw((0,5)--(12,0));<br /> draw((12,0)--(0,-5));<br /> draw((0,-5)--(-12,0));<br /> draw((0,0)--(12,0));<br /> draw((0,0)--(0,5));<br /> draw((0,0)--(-12,0));<br /> draw((0,0)--(0,-5));<br /> dot((-12,0));<br /> dot((0,5));<br /> dot((12,0));<br /> dot((0,-5));<br /> label(&quot;A&quot;,(-12,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;C&quot;,(12,0),E);<br /> label(&quot;D&quot;,(0,-5),S);<br /> label(&quot;E&quot;,(0,0),SW);<br /> &lt;/asy&gt;<br /> <br /> Because it is a rhombus all sides are equal. Implies all sides are 13. In a rhombus diagonals are perpendicular and bisect each other.<br /> Which means &lt;math&gt;\overline{AE}&lt;/math&gt; = &lt;math&gt;12&lt;/math&gt; = &lt;math&gt;\overline{EC}&lt;/math&gt;.<br /> <br /> Consider one of the right triangles.<br /> <br /> &lt;asy&gt;<br /> draw((-12,0)--(0,5));<br /> draw((0,0)--(-12,0));<br /> draw((0,0)--(0,5));<br /> dot((-12,0));<br /> dot((0,5));<br /> label(&quot;A&quot;,(-12,0),W);<br /> label(&quot;B&quot;,(0,5),N);<br /> label(&quot;E&quot;,(0,0),SE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\overline{AB}&lt;/math&gt; = &lt;math&gt;13&lt;/math&gt;.<br /> &lt;math&gt;\overline{AE}&lt;/math&gt; = &lt;math&gt;12&lt;/math&gt;.<br /> Which means &lt;math&gt;\overline{BE}&lt;/math&gt; = &lt;math&gt;5&lt;/math&gt;.<br /> <br /> Thus the values of the two diagonals are &lt;math&gt;\overline{AC}&lt;/math&gt; = &lt;math&gt;24&lt;/math&gt; and &lt;math&gt;\overline{BD}&lt;/math&gt; = &lt;math&gt;10&lt;/math&gt;.<br /> Which means area = &lt;math&gt;\frac{d_1*d_2}{2}&lt;/math&gt; = &lt;math&gt;\frac{24*10}{2}&lt;/math&gt; = &lt;math&gt;120&lt;/math&gt;<br /> <br /> &lt;math&gt;\boxed{\textbf{(D)}\ 120}&lt;/math&gt; ~phoenixfire<br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=1|num-a=3}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_6&diff=111711 2019 AMC 8 Problems/Problem 6 2019-11-20T17:59:57Z <p>Olivera: Fixed up problem</p> <hr /> <div>== Problem 6 ==<br /> <br /> There are &lt;math&gt;81&lt;/math&gt; grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point &lt;math&gt;P&lt;/math&gt; is in the center of the square. Given that point &lt;math&gt;Q&lt;/math&gt; is randomly chosen among the other 80 points, what is the probability that the line &lt;math&gt;PQ&lt;/math&gt; is a line of symmetry for the square?<br /> <br /> &lt;asy&gt;<br /> draw((0,0)--(0,8));<br /> draw((0,8)--(8,8));<br /> draw((8,8)--(8,0));<br /> draw((8,0)--(0,0));<br /> dot((0,0));<br /> dot((0,1));<br /> dot((0,2));<br /> dot((0,3));<br /> dot((0,4));<br /> dot((0,5));<br /> dot((0,6));<br /> dot((0,7));<br /> dot((0,8));<br /> <br /> dot((1,0));<br /> dot((1,1));<br /> dot((1,2));<br /> dot((1,3));<br /> dot((1,4));<br /> dot((1,5));<br /> dot((1,6));<br /> dot((1,7));<br /> dot((1,8));<br /> <br /> dot((2,0));<br /> dot((2,1));<br /> dot((2,2));<br /> dot((2,3));<br /> dot((2,4));<br /> dot((2,5));<br /> dot((2,6));<br /> dot((2,7));<br /> dot((2,8));<br /> <br /> dot((3,0));<br /> dot((3,1));<br /> dot((3,2));<br /> dot((3,3));<br /> dot((3,4));<br /> dot((3,5));<br /> dot((3,6));<br /> dot((3,7));<br /> dot((3,8));<br /> <br /> dot((4,0));<br /> dot((4,1));<br /> dot((4,2));<br /> dot((4,3));<br /> dot((4,4));<br /> dot((4,5));<br /> dot((4,6));<br /> dot((4,7));<br /> dot((4,8));<br /> <br /> dot((5,0));<br /> dot((5,1));<br /> dot((5,2));<br /> dot((5,3));<br /> dot((5,4));<br /> dot((5,5));<br /> dot((5,6));<br /> dot((5,7));<br /> dot((5,8));<br /> <br /> dot((6,0));<br /> dot((6,1));<br /> dot((6,2));<br /> dot((6,3));<br /> dot((6,4));<br /> dot((6,5));<br /> dot((6,6));<br /> dot((6,7));<br /> dot((6,8));<br /> <br /> dot((7,0));<br /> dot((7,1));<br /> dot((7,2));<br /> dot((7,3));<br /> dot((7,4));<br /> dot((7,5));<br /> dot((7,6));<br /> dot((7,7));<br /> dot((7,8));<br /> <br /> dot((8,0));<br /> dot((8,1));<br /> dot((8,2));<br /> dot((8,3));<br /> dot((8,4));<br /> dot((8,5));<br /> dot((8,6));<br /> dot((8,7));<br /> dot((8,8));<br /> label(&quot;P&quot;,(4,4),NE);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{1}{5}\qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{2}{5} \qquad\textbf{(D) }\frac{9}{20} \qquad\textbf{(E) }\frac{1}{2}&lt;/math&gt; <br /> <br /> ==Solution 1==<br /> <br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=14|num-a=16}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_16&diff=111702 2019 AMC 8 Problems/Problem 16 2019-11-20T17:40:31Z <p>Olivera: Added space</p> <hr /> <div>==Problem 16==<br /> Qiang drives &lt;math&gt;15&lt;/math&gt; miles at an average speed of &lt;math&gt;30&lt;/math&gt; miles per hour. How many additional miles will he have to drive at &lt;math&gt;55&lt;/math&gt; miles per hour to average &lt;math&gt;50&lt;/math&gt; miles per hour for the entire trip?<br /> <br /> &lt;math&gt;\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135&lt;/math&gt;<br /> <br /> ==Solution 1==<br /> <br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=15|num-a=17}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_21&diff=111700 2019 AMC 8 Problems/Problem 21 2019-11-20T17:39:47Z <p>Olivera: Fixed up problem</p> <hr /> <div><br /> ==Problem 21==<br /> What is the area of the triangle formed by the lines &lt;math&gt;y=5&lt;/math&gt;, &lt;math&gt;y=1+x&lt;/math&gt;, and &lt;math&gt;7=1-x&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16&lt;/math&gt;<br /> <br /> ==Solution 1==<br /> <br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=20|num-a=22}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_25&diff=111697 2019 AMC 8 Problems/Problem 25 2019-11-20T17:33:39Z <p>Olivera: Fixed Problem statement</p> <hr /> <div>==Problem 25==<br /> Alice has &lt;math&gt;24&lt;/math&gt; apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?<br /> <br /> ==Solution 1==<br /> Using [[Stars and bars]], and removing &lt;math&gt;6&lt;/math&gt; apples so each person can have &lt;math&gt;2&lt;/math&gt;, we get the total number of ways, which is &lt;math&gt;{20 \choose 2}&lt;/math&gt;, which is equal to &lt;math&gt;\boxed{\textbf{(C) }190}&lt;/math&gt;. ~~SmileKat32 <br /> <br /> ==Solution 2==<br /> Let's say you assume that Alice has 2 apples. There are 19 ways to split the rest of the apples with Becky and Chris. If Alice has 3 apples, there are 18 ways to split the rest of the apples with Becky and Chris. If Alice has 4 apples, there are 17 ways to split the rest. So the total number of ways to split 24 apples between the three friends is equal to 19+18+17...……+1=20(19/2)=&lt;math&gt;\boxed{\textbf{(C)}\ 190}&lt;/math&gt;~heeeeeeheeeeeee<br /> <br /> <br /> <br /> <br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=6|num-a=26}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_3&diff=111695 2019 AMC 8 Problems/Problem 3 2019-11-20T17:27:34Z <p>Olivera: Added copyright</p> <hr /> <div>===Problem 3===<br /> Which of the following is the correct order of the fractions &lt;math&gt;\frac{15}{11},\frac{19}{15},&lt;/math&gt; and &lt;math&gt;\frac{17}{13},&lt;/math&gt; from least to greatest? <br /> <br /> &lt;math&gt;\textbf{(A) }\frac{15}{11}&lt; \frac{17}{13}&lt; \frac{19}{15} \qquad\textbf{(B) }\frac{15}{11}&lt; \frac{19}{15}&lt;\frac{17}{13} \qquad\textbf{(C) }\frac{17}{13}&lt;\frac{19}{15}&lt;\frac{15}{11} \qquad\textbf{(D) } \frac{19}{15}&lt;\frac{15}{11}&lt;\frac{17}{13} \qquad\textbf{(E) } \frac{19}{15}&lt;\frac{17}{13}&lt;\frac{15}{11}&lt;/math&gt;<br /> <br /> ===Solution 1===<br /> Consider subtracting 1 from each of the fractions. Our new fractions would then be &lt;math&gt;\frac{4}{11}, \frac{4}{15}, and \frac{4}{13}&lt;/math&gt;. Since &lt;math&gt;\frac{4}{15}&lt;\frac{4}{13}&lt;\frac{4}{11}&lt;/math&gt;, it follows that the answer is &lt;math&gt;\boxed{\qquad\textbf{(E) }} \frac{19}{15}&lt;\frac{17}{13}&lt;\frac{15}{11}&lt;/math&gt;<br /> <br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=2|num-a=4}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_23&diff=111692 2019 AMC 8 Problems/Problem 23 2019-11-20T17:25:50Z <p>Olivera: Added Problem from pamphlet</p> <hr /> <div>==Problem 23==<br /> After Euclid High School's last basketball game, it was determined that &lt;math&gt;\frac{1}{4}&lt;/math&gt; of the team's points were scored by Alexa and &lt;math&gt;\frac{2}{7}&lt;/math&gt; were scored by Brittany. Chelsea scored &lt;math&gt;15&lt;/math&gt; points. None of the other &lt;math&gt;7&lt;/math&gt; team members scored more than &lt;math&gt;2&lt;/math&gt; points What was the total number of points scored by the other &lt;math&gt;7&lt;/math&gt; team members?<br /> <br /> &lt;math&gt;\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14&lt;/math&gt;<br /> <br /> ==Solution 1==<br /> <br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=22|num-a=24}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_20&diff=111690 2019 AMC 8 Problems/Problem 20 2019-11-20T17:23:47Z <p>Olivera: Added Problem from pamphlet</p> <hr /> <div>==Problem 16==<br /> How many different real numbers &lt;math&gt;x&lt;/math&gt; satisfy the equation &lt;cmath&gt;(x^{2}-5)^{2}=16?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8&lt;/math&gt;<br /> <br /> ==Solution 1==<br /> <br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=19|num-a=21}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_19&diff=111688 2019 AMC 8 Problems/Problem 19 2019-11-20T17:22:11Z <p>Olivera: Added Problem from pamphlet</p> <hr /> <div>==Problem 19==<br /> In a tournament there are six team taht play each other twice. A team earns &lt;math&gt;3&lt;/math&gt; points for a win, &lt;math&gt;1&lt;/math&gt; point for a draw, and &lt;math&gt;0&lt;/math&gt; points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?<br /> <br /> &lt;math&gt;\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30&lt;/math&gt;<br /> <br /> ==Solution 1==<br /> <br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=18|num-a=20}}<br /> <br /> {{MAA Notice}}</div> Olivera https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_18&diff=111685 2019 AMC 8 Problems/Problem 18 2019-11-20T17:19:33Z <p>Olivera: /* Problem 18 */</p> <hr /> <div>==Problem 18==<br /> The faces of each of two fair dice are numbered &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;8&lt;/math&gt;. When the two dice are tossed, what is the probability that their sum will be an even number?<br /> <br /> &lt;math&gt;\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}&lt;/math&gt;<br /> <br /> ==Solution 1==<br /> <br /> <br /> ==See Also==<br /> {{AMC8 box|year=2019|num-b=17|num-a=19}}<br /> <br /> {{MAA Notice}}</div> Olivera