https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Drakodin&feedformat=atom AoPS Wiki - User contributions [en] 2021-07-30T04:58:38Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=1994_AHSME_Problems/Problem_26&diff=83568 1994 AHSME Problems/Problem 26 2017-02-15T00:04:56Z <p>Drakodin: /* Solution */</p> <hr /> <div>==Problem==<br /> A regular polygon of &lt;math&gt;m&lt;/math&gt; sides is exactly enclosed (no overlaps, no gaps) by &lt;math&gt;m&lt;/math&gt; regular polygons of &lt;math&gt;n&lt;/math&gt; sides each. (Shown here for &lt;math&gt;m=4, n=8&lt;/math&gt;.) If &lt;math&gt;m=10&lt;/math&gt;, what is the value of &lt;math&gt;n&lt;/math&gt;?<br /> &lt;asy&gt;<br /> size(200);<br /> defaultpen(linewidth(0.8));<br /> draw(unitsquare);<br /> path p=(0,1)--(1,1)--(1+sqrt(2)/2,1+sqrt(2)/2)--(1+sqrt(2)/2,2+sqrt(2)/2)--(1,2+sqrt(2))--(0,2+sqrt(2))--(-sqrt(2)/2,2+sqrt(2)/2)--(-sqrt(2)/2,1+sqrt(2)/2)--cycle;<br /> draw(p);<br /> draw(shift((1+sqrt(2)/2,-sqrt(2)/2-1))*p);<br /> draw(shift((0,-2-sqrt(2)))*p);<br /> draw(shift((-1-sqrt(2)/2,-sqrt(2)/2-1))*p);&lt;/asy&gt;<br /> &lt;math&gt; \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 26 &lt;/math&gt;<br /> ==Solution==<br /> Note: might be a horrible solution but:<br /> <br /> To find the number of sides on the regular polygons that surround the decagon, we can find the interior angles and work from there. Knowing that the measure of the interior angle of any regular polygon is &lt;math&gt;\frac{(n-2)*180}{n}&lt;/math&gt;, the measure of the decagon's interior angle is &lt;math&gt;\frac{8*180}{10} = 144&lt;/math&gt; degrees. <br /> <br /> <br /> The regular polygons meet at every vertex such that the angle outside of the decagon is divided evenly in two. With this, we know that the angle of the regular polygon is &lt;math&gt;216/2=108&lt;/math&gt; degrees. Using the previous formula, &lt;math&gt;n=5&lt;/math&gt; &lt;math&gt;\boxed{\textbf{(A) }5}&lt;/math&gt;</div> Drakodin https://artofproblemsolving.com/wiki/index.php?title=1994_AHSME_Problems/Problem_27&diff=83567 1994 AHSME Problems/Problem 27 2017-02-15T00:04:36Z <p>Drakodin: /* Solution */</p> <hr /> <div>==Problem==<br /> A bag of popping corn contains &lt;math&gt;\frac{2}{3}&lt;/math&gt; white kernels and &lt;math&gt;\frac{1}{3}&lt;/math&gt; yellow kernels. Only &lt;math&gt;\frac{1}{2}&lt;/math&gt; of the white kernels will pop, whereas &lt;math&gt;\frac{2}{3}&lt;/math&gt; of the yellow ones will pop. A kernel is selected at random from the bag, and pops when placed in the popper. What is the probability that the kernel selected was white?<br /> <br /> &lt;math&gt; \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{5}{9} \qquad\textbf{(C)}\ \frac{4}{7} \qquad\textbf{(D)}\ \frac{3}{5} \qquad\textbf{(E)}\ \frac{2}{3} &lt;/math&gt;<br /> ==Solution==<br /> To find the probability that the kernel is white, the probability of &lt;math&gt;P(white|popped) = \frac{P(white, popped)}{P(popped)}&lt;/math&gt;<br /> <br /> Running a bit of calculations &lt;math&gt;P(white, popped) = \frac{1}{3}&lt;/math&gt; while &lt;math&gt;P(popped) = \frac{1}{3} + \frac{2}{9} = \frac{5}{9}&lt;/math&gt; Plugging this into the earlier equation, &lt;math&gt;P(white|popped) = \frac{\frac{1}{3}}{\frac{5}{9}}&lt;/math&gt;. Meaning that the answer is &lt;math&gt;\boxed{\textbf{(D)}\ \frac{3}{5}}&lt;/math&gt;.</div> Drakodin https://artofproblemsolving.com/wiki/index.php?title=1994_AHSME_Problems/Problem_26&diff=83565 1994 AHSME Problems/Problem 26 2017-02-14T23:24:41Z <p>Drakodin: /* Solution */</p> <hr /> <div>==Problem==<br /> A regular polygon of &lt;math&gt;m&lt;/math&gt; sides is exactly enclosed (no overlaps, no gaps) by &lt;math&gt;m&lt;/math&gt; regular polygons of &lt;math&gt;n&lt;/math&gt; sides each. (Shown here for &lt;math&gt;m=4, n=8&lt;/math&gt;.) If &lt;math&gt;m=10&lt;/math&gt;, what is the value of &lt;math&gt;n&lt;/math&gt;?<br /> &lt;asy&gt;<br /> size(200);<br /> defaultpen(linewidth(0.8));<br /> draw(unitsquare);<br /> path p=(0,1)--(1,1)--(1+sqrt(2)/2,1+sqrt(2)/2)--(1+sqrt(2)/2,2+sqrt(2)/2)--(1,2+sqrt(2))--(0,2+sqrt(2))--(-sqrt(2)/2,2+sqrt(2)/2)--(-sqrt(2)/2,1+sqrt(2)/2)--cycle;<br /> draw(p);<br /> draw(shift((1+sqrt(2)/2,-sqrt(2)/2-1))*p);<br /> draw(shift((0,-2-sqrt(2)))*p);<br /> draw(shift((-1-sqrt(2)/2,-sqrt(2)/2-1))*p);&lt;/asy&gt;<br /> &lt;math&gt; \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 26 &lt;/math&gt;<br /> ==Solution==<br /> Note: might be a horrible solution but:<br /> <br /> To find the number of sides on the regular polygons that surround the decagon, we can find the interior angles and work from there. Knowing that the measure of the interior angle of any regular polygon is &lt;math&gt;\frac{(n-2)*180}{n}&lt;/math&gt;, the measure of the decagon's interior angle is &lt;math&gt;\frac{8*180}{10} = 144&lt;/math&gt; degrees. <br /> <br /> <br /> The regular polygons meet at every vertex such that the angle outside of the decagon is divided evenly in two. With this, we know that the angle of the regular polygon is &lt;math&gt;216/2=108&lt;/math&gt; degrees. Using the previous formula, &lt;math&gt;n=5&lt;/math&gt; &lt;math&gt;\boxed{\textbf{(A) }5}&lt;/math&gt;<br /> <br /> <br /> - proto-solution by Drakodin -</div> Drakodin https://artofproblemsolving.com/wiki/index.php?title=1994_AHSME_Problems/Problem_26&diff=83564 1994 AHSME Problems/Problem 26 2017-02-14T23:23:28Z <p>Drakodin: /* Solution */</p> <hr /> <div>==Problem==<br /> A regular polygon of &lt;math&gt;m&lt;/math&gt; sides is exactly enclosed (no overlaps, no gaps) by &lt;math&gt;m&lt;/math&gt; regular polygons of &lt;math&gt;n&lt;/math&gt; sides each. (Shown here for &lt;math&gt;m=4, n=8&lt;/math&gt;.) If &lt;math&gt;m=10&lt;/math&gt;, what is the value of &lt;math&gt;n&lt;/math&gt;?<br /> &lt;asy&gt;<br /> size(200);<br /> defaultpen(linewidth(0.8));<br /> draw(unitsquare);<br /> path p=(0,1)--(1,1)--(1+sqrt(2)/2,1+sqrt(2)/2)--(1+sqrt(2)/2,2+sqrt(2)/2)--(1,2+sqrt(2))--(0,2+sqrt(2))--(-sqrt(2)/2,2+sqrt(2)/2)--(-sqrt(2)/2,1+sqrt(2)/2)--cycle;<br /> draw(p);<br /> draw(shift((1+sqrt(2)/2,-sqrt(2)/2-1))*p);<br /> draw(shift((0,-2-sqrt(2)))*p);<br /> draw(shift((-1-sqrt(2)/2,-sqrt(2)/2-1))*p);&lt;/asy&gt;<br /> &lt;math&gt; \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 26 &lt;/math&gt;<br /> ==Solution==<br /> Note: might be a horrible solution but<br /> <br /> To find the number of sides on the regular polygons that surround the decagon, we can find the interior angles and work from there. Knowing that the measure of the interior angle of any regular polygon is &lt;math&gt;\frac{(n-2)*180}{n}&lt;/math&gt;, the measure of the decagon's interior angle is &lt;math&gt;\frac{8*180}{10} = 144&lt;/math&gt; degrees. <br /> <br /> The regular polygons meet at every vertex such that the angle outside of the decagon is divided evenly in two. With this, we know that the angle of the regular polygon is &lt;math&gt;216/2=108&lt;/math&gt; degrees. Using the previous formula, &lt;math&gt;n=5&lt;/math&gt; &lt;math&gt;\boxed{\textbf{(A) }5}&lt;/math&gt;</div> Drakodin https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_10A_Problems&diff=82987 2017 AMC 10A Problems 2017-02-08T22:18:25Z <p>Drakodin: /* Problem 18 */</p> <hr /> <div>==Problem 1==<br /> What is the value of &lt;math&gt;(2(2(2(2(2(2+1)+1)+1)+1)+1)+1)&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 70\qquad\textbf{(B)}\ 97\qquad\textbf{(C)}\ 127\qquad\textbf{(D)}\ 159\qquad\textbf{(E)}\ 729&lt;/math&gt;<br /> <br /> [[2017 AMC 10A Problems/Problem 1|Solution]]<br /> <br /> ==Problem 2==<br /> Pablo buys popsicles for his friends. The store sells single popsicles for \$1 each, 3-popsicle boxes for \$2 each, and 5-popsicle boxes for \$3. What is the greatest number of popsicles that Pablo can buy with \$8?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 8\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 13\qquad\textbf{(E)}\ 15&lt;/math&gt;<br /> <br /> [[2017 AMC 10A Problems/Problem 2|Solution]]<br /> <br /> ==Problem 3==<br /> Tamara has three rows of two 6-feet by 2-feet flower beds in her garden. The beds are separated and also surrounded by 1-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?<br /> <br /> [[2017 AMC 10A Problems/Problem 3|Solution]]<br /> <br /> ==Problem 4==<br /> Mia is “helping” her mom pick up &lt;math&gt;30&lt;/math&gt; toys that are strewn on the floor. Mia’s mom manages to put &lt;math&gt;3&lt;/math&gt; toys into the toy box every &lt;math&gt;30&lt;/math&gt; seconds, but each time immediately after those &lt;math&gt;30&lt;/math&gt; seconds have elapsed, Mia takes &lt;math&gt;2&lt;/math&gt; toys out of the box. How much time, in minutes, will it take Mia and her mom to put all &lt;math&gt;30&lt;/math&gt; toys into the box for the first time?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 13.5\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 14.5\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 15.5&lt;/math&gt;<br /> <br /> ==Problem 5==<br /> The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12&lt;/math&gt;<br /> <br /> ==Problem 6==<br /> Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which on of these statements necessarily follows logically?<br /> <br /> ==Problem 7==<br /> Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 30\%\qquad\textbf{(B)}\ 40\%\qquad\textbf{(C)}\ 50\%\qquad\textbf{(D)}\ 60\%\qquad\textbf{(E)}\ 70\%&lt;/math&gt;<br /> <br /> ==Problem 8==<br /> At a gathering of 30 people, there are 20 people who all know each other and 10 people who know no one. People who know each other a hug, and people who do not know each other shake hands. How many handshakes occur?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 240\qquad\textbf{(B)}\ 245\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 490&lt;/math&gt;<br /> <br /> ==Problem 9==<br /> Minnie rides on a flat road at &lt;math&gt;20&lt;/math&gt; kilometers per hour (kph), downhill at &lt;math&gt;30&lt;/math&gt; kph, and uphill at &lt;math&gt;5&lt;/math&gt; kph. Penny rides on a flat road at &lt;math&gt;30&lt;/math&gt; kph, downhill at &lt;math&gt;40&lt;/math&gt; kph, and uphill at &lt;math&gt;10&lt;/math&gt; kph. Minnie goes from town &lt;math&gt;A&lt;/math&gt; to town &lt;math&gt;B&lt;/math&gt;, a distance of &lt;math&gt;10&lt;/math&gt; km all uphill, then from town &lt;math&gt;B&lt;/math&gt; to town &lt;math&gt;C&lt;/math&gt;, a distance of &lt;math&gt;10&lt;/math&gt; km all uphill, then from town &lt;math&gt;B&lt;/math&gt; to town &lt;math&gt;C&lt;/math&gt;, a distance of &lt;math&gt;15&lt;/math&gt; km all downhill, and then back to town &lt;math&gt;A&lt;/math&gt;, a distance of &lt;math&gt;20&lt;/math&gt; km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the &lt;math&gt;45&lt;/math&gt;-km ride than it takes Penny?<br /> <br /> ==Problem 10==<br /> Joy has &lt;math&gt;30&lt;/math&gt; thin rods, one each of every integer length from &lt;math&gt;1&lt;/math&gt; cm through &lt;math&gt;30&lt;/math&gt; cm. She places the rods with lengths &lt;math&gt;3&lt;/math&gt; cm, &lt;math&gt;7&lt;/math&gt; cm, and &lt;math&gt;15&lt;/math&gt; cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?<br /> <br /> &lt;math&gt;\text{(A) 16}\qquad\text{(B) 17}\qquad\text{(C) 18}\qquad\text{(D) 19}\qquad\text{(E) 20}&lt;/math&gt;<br /> <br /> ==Problem 11==<br /> The region consisting of all point in three-dimensional space within 3 units of line segment &lt;math&gt;\overline{AB}&lt;/math&gt; has volume 216&lt;math&gt;\pi&lt;/math&gt;. What is the length &lt;math&gt;\textit{AB}&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24&lt;/math&gt;<br /> <br /> ==Problem 12==<br /> Let &lt;math&gt;S&lt;/math&gt; be a set of points &lt;math&gt;(x,y)&lt;/math&gt; in the coordinate plane such that two of the three quantities &lt;math&gt;3,~x+2,&lt;/math&gt; and &lt;math&gt;y-4&lt;/math&gt; are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description for &lt;math&gt;S?&lt;/math&gt;<br /> <br /> &lt;math&gt;\textbf{(A)}\ \text{a single point} \qquad\textbf{(B)}\ \text{two intersecting lines} \\\qquad\textbf{(C)}\ \text{ three lines whose pairwise intersections are three distinct points} \\\qquad\textbf{(D)}\ \text{a triangle} \qquad\textbf{(E)}\ \text{three rays with a common endpoint}&lt;/math&gt;<br /> <br /> ==Problem 13==<br /> Define a sequence recursively by &lt;math&gt;F_{0}=0,~F_{1}=1,&lt;/math&gt; and &lt;math&gt;F_{n}=&lt;/math&gt; the remainder when &lt;math&gt;F_{n-1}+F_{n-2}&lt;/math&gt; is divided by &lt;math&gt;3,&lt;/math&gt; for all &lt;math&gt;n\geq 2.&lt;/math&gt; Thus the sequence starts &lt;math&gt;0,1,1,2,0,2,\ldots&lt;/math&gt; What is &lt;math&gt;F_{2017}+F_{2018}+F_{2019}+F_{2020}+F_{2021}+F_{2022}+F_{2023}+F_{2024}?&lt;/math&gt;<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 /> ==Problem 14==<br /> Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was &lt;math&gt;A&lt;/math&gt; dollars. The cost of his movie ticket was &lt;math&gt;20\%&lt;/math&gt; of the difference between &lt;math&gt;A&lt;/math&gt; and the cost of his soda, while the cost of his soda was &lt;math&gt;5\%&lt;/math&gt; of the difference between &lt;math&gt;A&lt;/math&gt; and the cost of his movie ticket. To the nearest whole percent, what fraction of &lt;math&gt;A&lt;/math&gt; did Roger pay for his movie ticket and soda?<br /> <br /> &lt;math&gt; \mathrm{(A) \ }9\%\qquad \mathrm{(B) \ } 19\%\qquad \mathrm{(C) \ } 22\%\qquad \mathrm{(D) \ } 23\%\qquad \mathrm{(E) \ }25\%&lt;/math&gt;<br /> <br /> ==Problem 15==<br /> Chloé chooses a real number uniformly at random from the interval &lt;math&gt;[0, 2017]&lt;/math&gt;. Independently, Laurent cooses a real number uniformly at random from the interval &lt;math&gt;[0, 4034]&lt;/math&gt;. What is the probability that Laurent's number is greater than Chloé's number?<br /> <br /> &lt;math&gt; \mathrm{(A) \ }\frac{1}{2}\qquad \mathrm{(B) \ } \frac{2}{3}\qquad \mathrm{(C) \ } \frac{3}{4}\qquad \mathrm{(D) \ } \frac{5}{6}\qquad \mathrm{(E) \ }\frac{7}{8}&lt;/math&gt;<br /> <br /> ==Problem 16==<br /> There are 10 horses, named Horse 1, Horse 2, &lt;math&gt;\ldots&lt;/math&gt;, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse &lt;math&gt;k&lt;/math&gt; runs one lap in exactly &lt;math&gt;k&lt;/math&gt; minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time &lt;math&gt;S&gt;0&lt;/math&gt;, in minutes, at which all 10 horses will again simultaneously be at the starting point is &lt;math&gt;S=2520&lt;/math&gt;. Let &lt;math&gt;T&gt;0&lt;/math&gt; be the least time, in minutes, such that at least 5 of the horses are again at the starting point. What is the sum of the digits of &lt;math&gt;T&lt;/math&gt;?<br /> <br /> &lt;math&gt; \mathrm{(A) \ }2\qquad \mathrm{(B) \ }3\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ }5\qquad \mathrm{(E) \ }6&lt;/math&gt;<br /> <br /> ==Problem 17==<br /> Distinct points &lt;math&gt;P&lt;/math&gt;, &lt;math&gt;Q&lt;/math&gt;, &lt;math&gt;R&lt;/math&gt;, &lt;math&gt;S&lt;/math&gt; lie on the circle &lt;math&gt;x^2+y^2=25&lt;/math&gt; and have integer coordinates. The distances &lt;math&gt;PQ&lt;/math&gt; and &lt;math&gt;RS&lt;/math&gt; are irrational numbers. What is the greatest possible value of the ratio &lt;math&gt;\frac{PQ}{RS}&lt;/math&gt;?<br /> <br /> &lt;math&gt;\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ 5\qquad\mathrm{(C)}\ 3\sqrt{5}\qquad\mathrm{(D)}\ 7\qquad\mathrm{(E)}\ 5\sqrt{2}&lt;/math&gt;<br /> <br /> ==Problem 18==<br /> Amelia has a coin that lands heads with probability &lt;math&gt;\frac{1}{3}&lt;/math&gt;, and Blaine has a coin that lands on heads with probability &lt;math&gt;\frac{2}{5}&lt;/math&gt;. Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is &lt;math&gt;\frac{p}{q}&lt;/math&gt;, where &lt;math&gt;p&lt;/math&gt; and &lt;math&gt;q&lt;/math&gt; are relatively prime positive integers. What is &lt;math&gt;q-p&lt;/math&gt;?<br /> <br /> &lt;math&gt;\mathrm{(A)}\ 1\qquad\mathrm{(B)}\ 2\qquad\mathrm{(C)}\ 3\qquad\mathrm{(D)}\ 4\qquad\mathrm{(E)}\ 5&lt;/math&gt;<br /> <br /> ==Problem 19==<br /> <br /> ==Problem 20==<br /> <br /> ==Problem 21==<br /> <br /> ==Problem 22==<br /> Sides &lt;math&gt;\overline{AB}&lt;/math&gt; and &lt;math&gt;\overline{AC}&lt;/math&gt; of equilateral triangle &lt;math&gt;ABC&lt;/math&gt; are tangent to a circle as points &lt;math&gt;B&lt;/math&gt; and &lt;math&gt;C&lt;/math&gt; respectively. What fraction of the area of &lt;math&gt;\triangle ABC&lt;/math&gt; lies outside the circle?<br /> <br /> &lt;math&gt; \mathrm{(A) \ }\dfrac{4\sqrt{3}\pi}{27}-\frac{1}{3}\qquad \mathrm{(B) \ } \frac{\sqrt{3}}{2}-\frac{\pi}{8}\qquad \mathrm{(C) \ } \frac{1}{2} \qquad \mathrm{(D) \ }\sqrt{3}-\frac{2\sqrt{3}\pi}{9}\qquad \mathrm{(E) \ } \frac{4}{3}-\dfrac{4\sqrt{3}\pi}{27}&lt;/math&gt;<br /> <br /> ==Problem 23==<br /> How many triangles with positive area have all their vertices at points &lt;math&gt;(i,j)&lt;/math&gt; in the coordinate plane, where &lt;math&gt;i&lt;/math&gt; and &lt;math&gt;j&lt;/math&gt; are integers between &lt;math&gt;1&lt;/math&gt; and &lt;math&gt;5&lt;/math&gt;, inclusive?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 2128 \qquad\textbf{(B)}\ 2148 \qquad\textbf{(C)}\ 2160 \qquad\textbf{(D)}\ 2200 \qquad\textbf{(E)}\ 2300&lt;/math&gt;<br /> <br /> ==Problem 24==<br /> For certain real numbers &lt;math&gt;a&lt;/math&gt;, &lt;math&gt;b&lt;/math&gt;, and &lt;math&gt;c&lt;/math&gt;, the polynomial &lt;cmath&gt;g(x) = x^3 + ax^2 + x + 10&lt;/cmath&gt;has three distinct roots, and each root of &lt;math&gt;g(x)&lt;/math&gt; is also a root of the polynomial &lt;cmath&gt;f(x) = x^4 + x^3 + bx^2 + 100x + c.&lt;/cmath&gt;What is &lt;math&gt;f(1)&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{(D)}\ -6006 \qquad\textbf{(E)}\ -5005&lt;/math&gt;<br /> <br /> ==Problem 25==<br /> How many integers between 100 and 999, inclusive, have the property that some permutation of its digits is a multiple of 11 between 100 and 999? For example, both 121 and 211 have this property.<br /> <br /> &lt;math&gt;\mathrm{(A)}\ 226\qquad\mathrm{(B)}\ 243\qquad\mathrm{(C)}\ 270\qquad\mathrm{(D)}\ 469\qquad\mathrm{(E)}\ 486&lt;/math&gt;</div> Drakodin https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_10A_Problems&diff=82974 2017 AMC 10A Problems 2017-02-08T22:14:50Z <p>Drakodin: /* Problem 25 */</p> <hr /> <div>==Problem 1==<br /> What is the value of &lt;math&gt;(2(2(2(2(2(2+1)+1)+1)+1)+1)+1)&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 70\qquad\textbf{(B)}\ 97\qquad\textbf{(C)}\ 127\qquad\textbf{(D)}\ 159\qquad\textbf{(E)}\ 729&lt;/math&gt;<br /> <br /> [[2017 AMC 10A Problems/Problem 1|Solution]]<br /> <br /> ==Problem 2==<br /> Pablo buys popsicles for his friends. The store sells single popsicles for \$1 each, 3-popsicle boxes for \$2 each, and 5-popsicle boxes for \$3. What is the greatest number of popsicles that Pablo can buy with \$8?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 8\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 13\qquad\textbf{(E)}\ 15&lt;/math&gt;<br /> <br /> [[2017 AMC 10A Problems/Problem 2|Solution]]<br /> <br /> ==Problem 3==<br /> Tamara has three rows of two 6-feet by 2-feet flower beds in her garden. The beds are separated and also surrounded by 1-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?<br /> <br /> [[2017 AMC 10A Problems/Problem 3|Solution]]<br /> <br /> ==Problem 4==<br /> Mia is “helping” her mom pick up &lt;math&gt;30&lt;/math&gt; toys that are strewn on the floor. Mia’s mom manages to put &lt;math&gt;3&lt;/math&gt; toys into the toy box every &lt;math&gt;30&lt;/math&gt; seconds, but each time immediately after those &lt;math&gt;30&lt;/math&gt; seconds have elapsed, Mia takes &lt;math&gt;2&lt;/math&gt; toys out of the box. How much time, in minutes, will it take Mia and her mom to put all &lt;math&gt;30&lt;/math&gt; toys into the box for the first time?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 13.5\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 14.5\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 15.5&lt;/math&gt;<br /> <br /> ==Problem 5==<br /> The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12&lt;/math&gt;<br /> <br /> ==Problem 6==<br /> Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which on of these statements necessarily follows logically?<br /> <br /> ==Problem 7==<br /> Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 30\%\qquad\textbf{(B)}\ 40\%\qquad\textbf{(C)}\ 50\%\qquad\textbf{(D)}\ 60\%\qquad\textbf{(E)}\ 70\%&lt;/math&gt;<br /> <br /> ==Problem 8==<br /> At a gathering of 30 people, there are 20 people who all know each other and 10 people who know no one. People who know each other a hug, and people who do not know each other shake hands. How many handshakes occur?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 240\qquad\textbf{(B)}\ 245\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 490&lt;/math&gt;<br /> <br /> ==Problem 9==<br /> Minnie rides on a flat road at &lt;math&gt;20&lt;/math&gt; kilometers per hour (kph), downhill at &lt;math&gt;30&lt;/math&gt; kph, and uphill at &lt;math&gt;5&lt;/math&gt; kph. Penny rides on a flat road at &lt;math&gt;30&lt;/math&gt; kph, downhill at &lt;math&gt;40&lt;/math&gt; kph, and uphill at &lt;math&gt;10&lt;/math&gt; kph. Minnie goes from town &lt;math&gt;A&lt;/math&gt; to town &lt;math&gt;B&lt;/math&gt;, a distance of &lt;math&gt;10&lt;/math&gt; km all uphill, then from town &lt;math&gt;B&lt;/math&gt; to town &lt;math&gt;C&lt;/math&gt;, a distance of &lt;math&gt;10&lt;/math&gt; km all uphill, then from town &lt;math&gt;B&lt;/math&gt; to town &lt;math&gt;C&lt;/math&gt;, a distance of &lt;math&gt;15&lt;/math&gt; km all downhill, and then back to town &lt;math&gt;A&lt;/math&gt;, a distance of &lt;math&gt;20&lt;/math&gt; km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the &lt;math&gt;45&lt;/math&gt;-km ride than it takes Penny?<br /> <br /> ==Problem 10==<br /> Joy has &lt;math&gt;30&lt;/math&gt; thin rods, one each of every integer length from &lt;math&gt;1&lt;/math&gt; cm through &lt;math&gt;30&lt;/math&gt; cm. She places the rods with lengths &lt;math&gt;3&lt;/math&gt; cm, &lt;math&gt;7&lt;/math&gt; cm, and &lt;math&gt;15&lt;/math&gt; cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?<br /> <br /> &lt;math&gt;\text{(A) 16}\qquad\text{(B) 17}\qquad\text{(C) 18}\qquad\text{(D) 19}\qquad\text{(E) 20}&lt;/math&gt;<br /> <br /> ==Problem 11==<br /> The region consisting of all point in three-dimensional space within 3 units of line segment &lt;math&gt;\overline{AB}&lt;/math&gt; has volume 216&lt;math&gt;\pi&lt;/math&gt;. What is the length &lt;math&gt;\textit{AB}&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24&lt;/math&gt;<br /> <br /> ==Problem 12==<br /> Let &lt;math&gt;S&lt;/math&gt; be a set of points &lt;math&gt;(x,y)&lt;/math&gt; in the coordinate plane such that two of the three quantities &lt;math&gt;3,~x+2,&lt;/math&gt; and &lt;math&gt;y-4&lt;/math&gt; are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description for &lt;math&gt;S?&lt;/math&gt;<br /> <br /> &lt;math&gt;\textbf{(A)}\ \text{a single point} \qquad\textbf{(B)}\ \text{two intersecting lines} \\\qquad\textbf{(C)}\ \text{ three lines whose pairwise intersections are three distinct points} \\\qquad\textbf{(D)}\ \text{a triangle} \qquad\textbf{(E)}\ \text{three rays with a common endpoint}&lt;/math&gt;<br /> <br /> ==Problem 13==<br /> Define a sequence recursively by &lt;math&gt;F_{0}=0,~F_{1}=1,&lt;/math&gt; and &lt;math&gt;F_{n}=&lt;/math&gt; the remainder when &lt;math&gt;F_{n-1}+F_{n-2}&lt;/math&gt; is divided by &lt;math&gt;3,&lt;/math&gt; for all &lt;math&gt;n\geq 2.&lt;/math&gt; Thus the sequence starts &lt;math&gt;0,1,1,2,0,2,\ldots&lt;/math&gt; What is &lt;math&gt;F_{2017}+F_{2018}+F_{2019}+F_{2020}+F_{2021}+F_{2022}+F_{2023}+F_{2024}?&lt;/math&gt;<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 /> ==Problem 14==<br /> Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was &lt;math&gt;A&lt;/math&gt; dollars. The cost of his movie ticket was &lt;math&gt;20\%&lt;/math&gt; of the difference between &lt;math&gt;A&lt;/math&gt; and the cost of his soda, while the cost of his soda was &lt;math&gt;5\%&lt;/math&gt; of the difference between &lt;math&gt;A&lt;/math&gt; and the cost of his movie ticket. To the nearest whole percent, what fraction of &lt;math&gt;A&lt;/math&gt; did Roger pay for his movie ticket and soda?<br /> <br /> &lt;math&gt; \mathrm{(A) \ }9\%\qquad \mathrm{(B) \ } 19\%\qquad \mathrm{(C) \ } 22\%\qquad \mathrm{(D) \ } 23\%\qquad \mathrm{(E) \ }25\%&lt;/math&gt;<br /> <br /> ==Problem 15==<br /> Chloé chooses a real number uniformly at random from the interval &lt;math&gt;[0, 2017]&lt;/math&gt;. Independently, Laurent cooses a real number uniformly at random from the interval &lt;math&gt;[0, 4034]&lt;/math&gt;. What is the probability that Laurent's number is greater than Chloé's number?<br /> <br /> &lt;math&gt; \mathrm{(A) \ }\frac{1}{2}\qquad \mathrm{(B) \ } \frac{2}{3}\qquad \mathrm{(C) \ } \frac{3}{4}\qquad \mathrm{(D) \ } \frac{5}{6}\qquad \mathrm{(E) \ }\frac{7}{8}&lt;/math&gt;<br /> <br /> ==Problem 16==<br /> There are 10 horses, named Horse 1, Horse 2, &lt;math&gt;\ldots&lt;/math&gt;, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse &lt;math&gt;k&lt;/math&gt; runs one lap in exactly &lt;math&gt;k&lt;/math&gt; minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time &lt;math&gt;S&gt;0&lt;/math&gt;, in minutes, at which all 10 horses will again simultaneously be at the starting point is &lt;math&gt;S=2520&lt;/math&gt;. Let &lt;math&gt;T&gt;0&lt;/math&gt; be the least time, in minutes, such that at least 5 of the horses are again at the starting point. What is the sum of the digits of &lt;math&gt;T&lt;/math&gt;?<br /> <br /> &lt;math&gt; \mathrm{(A) \ }2\qquad \mathrm{(B) \ }3\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ }5\qquad \mathrm{(E) \ }6&lt;/math&gt;<br /> <br /> ==Problem 17==<br /> Distinct points &lt;math&gt;P&lt;/math&gt;, &lt;math&gt;Q&lt;/math&gt;, &lt;math&gt;R&lt;/math&gt;, &lt;math&gt;S&lt;/math&gt; lie on the circle &lt;math&gt;x^2+y^2=25&lt;/math&gt; and have integer coordinates. The distances &lt;math&gt;PQ&lt;/math&gt; and &lt;math&gt;RS&lt;/math&gt; are irrational numbers. What is the greatest possible value of the ratio &lt;math&gt;\frac{PQ}{RS}&lt;/math&gt;?<br /> <br /> &lt;math&gt;\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ 5\qquad\mathrm{(C)}\ 3\sqrt{5}\qquad\mathrm{(D)}\ 7\qquad\mathrm{(E)}\ 5\sqrt{2}&lt;/math&gt;<br /> <br /> ==Problem 18==<br /> <br /> ==Problem 19==<br /> <br /> ==Problem 20==<br /> <br /> ==Problem 21==<br /> <br /> ==Problem 22==<br /> Sides &lt;math&gt;\overline{AB}&lt;/math&gt; and &lt;math&gt;\overline{AC}&lt;/math&gt; of equilateral triangle &lt;math&gt;ABC&lt;/math&gt; are tangent to a circle as points &lt;math&gt;B&lt;/math&gt; and &lt;math&gt;C&lt;/math&gt; respectively. What fraction of the area of &lt;math&gt;\triangle ABC&lt;/math&gt; lies outside the circle?<br /> <br /> &lt;math&gt; \mathrm{(A) \ }\dfrac{4\sqrt{3}\pi}{27}-\frac{1}{3}\qquad \mathrm{(B) \ } \frac{\sqrt{3}}{2}-\frac{\pi}{8}\qquad \mathrm{(C) \ } \frac{1}{2} \qquad \mathrm{(D) \ }\sqrt{3}-\frac{2\sqrt{3}\pi}{9}\qquad \mathrm{(E) \ } \frac{4}{3}-\dfrac{4\sqrt{3}\pi}{27}&lt;/math&gt;<br /> <br /> ==Problem 23==<br /> How many triangles with positive area have all their vertices at points &lt;math&gt;(i,j)&lt;/math&gt; in the coordinate plane, where &lt;math&gt;i&lt;/math&gt; and &lt;math&gt;j&lt;/math&gt; are integers between &lt;math&gt;1&lt;/math&gt; and &lt;math&gt;5&lt;/math&gt;, inclusive?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 2128 \qquad\textbf{(B)}\ 2148 \qquad\textbf{(C)}\ 2160 \qquad\textbf{(D)}\ 2200 \qquad\textbf{(E)}\ 2300&lt;/math&gt;<br /> <br /> ==Problem 24==<br /> For certain real numbers &lt;math&gt;a&lt;/math&gt;, &lt;math&gt;b&lt;/math&gt;, and &lt;math&gt;c&lt;/math&gt;, the polynomial &lt;cmath&gt;g(x) = x^3 + ax^2 + x + 10&lt;/cmath&gt;has three distinct roots, and each root of &lt;math&gt;g(x)&lt;/math&gt; is also a root of the polynomial &lt;cmath&gt;f(x) = x^4 + x^3 + bx^2 + 100x + c.&lt;/cmath&gt;What is &lt;math&gt;f(1)&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{(D)}\ -6006 \qquad\textbf{(E)}\ -5005&lt;/math&gt;<br /> <br /> ==Problem 25==<br /> How many integers between 100 and 999, inclusive, have the property that some permutation of its digits is a multiple of 11 between 100 and 999? For example, both 121 and 211 have this property.<br /> <br /> &lt;math&gt;\mathrm{(A)}\ 226\qquad\mathrm{(B)}\ 243\qquad\mathrm{(C)}\ 270\qquad\mathrm{(D)}\ 469\qquad\mathrm{(E)}\ 486&lt;/math&gt;</div> Drakodin https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_10A_Problems&diff=82965 2017 AMC 10A Problems 2017-02-08T22:12:08Z <p>Drakodin: /* Problem 17 */</p> <hr /> <div>==Problem 1==<br /> What is the value of &lt;math&gt;(2(2(2(2(2(2+1)+1)+1)+1)+1)+1)&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 70\qquad\textbf{(B)}\ 97\qquad\textbf{(C)}\ 127\qquad\textbf{(D)}\ 159\qquad\textbf{(E)}\ 729&lt;/math&gt;<br /> <br /> [[2017 AMC 10A Problems/Problem 1|Solution]]<br /> <br /> ==Problem 2==<br /> Pablo buys popsicles for his friends. The store sells single popsicles for \$1 each, 3-popsicle boxes for \$2 each, and 5-popsicle boxes for \$3. What is the greatest number of popsicles that Pablo can buy with \$8?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 8\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 13\qquad\textbf{(E)}\ 15&lt;/math&gt;<br /> <br /> [[2017 AMC 10A Problems/Problem 2|Solution]]<br /> <br /> ==Problem 3==<br /> Tamara has three rows of two 6-feet by 2-feet flower beds in her garden. The beds are separated and also surrounded by 1-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?<br /> <br /> [[2017 AMC 10A Problems/Problem 3|Solution]]<br /> <br /> ==Problem 4==<br /> Mia is “helping” her mom pick up &lt;math&gt;30&lt;/math&gt; toys that are strewn on the floor. Mia’s mom manages to put &lt;math&gt;3&lt;/math&gt; toys into the toy box every &lt;math&gt;30&lt;/math&gt; seconds, but each time immediately after those &lt;math&gt;30&lt;/math&gt; seconds have elapsed, Mia takes &lt;math&gt;2&lt;/math&gt; toys out of the box. How much time, in minutes, will it take Mia and her mom to put all &lt;math&gt;30&lt;/math&gt; toys into the box for the first time?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 13.5\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 14.5\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 15.5&lt;/math&gt;<br /> <br /> ==Problem 5==<br /> The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12&lt;/math&gt;<br /> <br /> ==Problem 6==<br /> Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which on of these statements necessarily follows logically?<br /> <br /> ==Problem 7==<br /> Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 30\%\qquad\textbf{(B)}\ 40\%\qquad\textbf{(C)}\ 50\%\qquad\textbf{(D)}\ 60\%\qquad\textbf{(E)}\ 70\%&lt;/math&gt;<br /> <br /> ==Problem 8==<br /> At a gathering of 30 people, there are 20 people who all know each other and 10 people who know no one. People who know each other a hug, and people who do not know each other shake hands. How many handshakes occur?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 240\qquad\textbf{(B)}\ 245\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 490&lt;/math&gt;<br /> <br /> ==Problem 9==<br /> Minnie rides on a flat road at &lt;math&gt;20&lt;/math&gt; kilometers per hour (kph), downhill at &lt;math&gt;30&lt;/math&gt; kph, and uphill at &lt;math&gt;5&lt;/math&gt; kph. Penny rides on a flat road at &lt;math&gt;30&lt;/math&gt; kph, downhill at &lt;math&gt;40&lt;/math&gt; kph, and uphill at &lt;math&gt;10&lt;/math&gt; kph. Minnie goes from town &lt;math&gt;A&lt;/math&gt; to town &lt;math&gt;B&lt;/math&gt;, a distance of &lt;math&gt;10&lt;/math&gt; km all uphill, then from town &lt;math&gt;B&lt;/math&gt; to town &lt;math&gt;C&lt;/math&gt;, a distance of &lt;math&gt;10&lt;/math&gt; km all uphill, then from town &lt;math&gt;B&lt;/math&gt; to town &lt;math&gt;C&lt;/math&gt;, a distance of &lt;math&gt;15&lt;/math&gt; km all downhill, and then back to town &lt;math&gt;A&lt;/math&gt;, a distance of &lt;math&gt;20&lt;/math&gt; km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the &lt;math&gt;45&lt;/math&gt;-km ride than it takes Penny?<br /> <br /> ==Problem 10==<br /> Joy has &lt;math&gt;30&lt;/math&gt; thin rods, one each of every integer length from &lt;math&gt;1&lt;/math&gt; cm through &lt;math&gt;30&lt;/math&gt; cm. She places the rods with lengths &lt;math&gt;3&lt;/math&gt; cm, &lt;math&gt;7&lt;/math&gt; cm, and &lt;math&gt;15&lt;/math&gt; cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?<br /> <br /> &lt;math&gt;\text{(A) 16}\qquad\text{(B) 17}\qquad\text{(C) 18}\qquad\text{(D) 19}\qquad\text{(E) 20}&lt;/math&gt;<br /> <br /> ==Problem 11==<br /> The region consisting of all point in three-dimensional space within 3 units of line segment &lt;math&gt;\overline{AB}&lt;/math&gt; has volume 216&lt;math&gt;\pi&lt;/math&gt;. What is the length &lt;math&gt;\textit{AB}&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24&lt;/math&gt;<br /> <br /> ==Problem 12==<br /> Let &lt;math&gt;S&lt;/math&gt; be a set of points &lt;math&gt;(x,y)&lt;/math&gt; in the coordinate plane such that two of the three quantities &lt;math&gt;3,~x+2,&lt;/math&gt; and &lt;math&gt;y-4&lt;/math&gt; are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description for &lt;math&gt;S?&lt;/math&gt;<br /> <br /> &lt;math&gt;\textbf{(A)}\ \text{a single point} \qquad\textbf{(B)}\ \text{two intersecting lines} \\\qquad\textbf{(C)}\ \text{ three lines whose pairwise intersections are three distinct points} \\\qquad\textbf{(D)}\ \text{a triangle} \qquad\textbf{(E)}\ \text{three rays with a common endpoint}&lt;/math&gt;<br /> <br /> ==Problem 13==<br /> Define a sequence recursively by &lt;math&gt;F_{0}=0,~F_{1}=1,&lt;/math&gt; and &lt;math&gt;F_{n}=&lt;/math&gt; the remainder when &lt;math&gt;F_{n-1}+F_{n-2}&lt;/math&gt; is divided by &lt;math&gt;3,&lt;/math&gt; for all &lt;math&gt;n\geq 2.&lt;/math&gt; Thus the sequence starts &lt;math&gt;0,1,1,2,0,2,\ldots&lt;/math&gt; What is &lt;math&gt;F_{2017}+F_{2018}+F_{2019}+F_{2020}+F_{2021}+F_{2022}+F_{2023}+F_{2024}?&lt;/math&gt;<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 /> ==Problem 14==<br /> Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was &lt;math&gt;A&lt;/math&gt; dollars. The cost of his movie ticket was &lt;math&gt;20\%&lt;/math&gt; of the difference between &lt;math&gt;A&lt;/math&gt; and the cost of his soda, while the cost of his soda was &lt;math&gt;5\%&lt;/math&gt; of the difference between &lt;math&gt;A&lt;/math&gt; and the cost of his movie ticket. To the nearest whole percent, what fraction of &lt;math&gt;A&lt;/math&gt; did Roger pay for his movie ticket and soda?<br /> <br /> &lt;math&gt; \mathrm{(A) \ }9\%\qquad \mathrm{(B) \ } 19\%\qquad \mathrm{(C) \ } 22\%\qquad \mathrm{(D) \ } 23\%\qquad \mathrm{(E) \ }25\%&lt;/math&gt;<br /> <br /> ==Problem 15==<br /> Chloé chooses a real number uniformly at random from the interval &lt;math&gt;[0, 2017]&lt;/math&gt;. Independently, Laurent cooses a real number uniformly at random from the interval &lt;math&gt;[0, 4034]&lt;/math&gt;. What is the probability that Laurent's number is greater than Chloé's number?<br /> <br /> &lt;math&gt; \mathrm{(A) \ }\frac{1}{2}\qquad \mathrm{(B) \ } \frac{2}{3}\qquad \mathrm{(C) \ } \frac{3}{4}\qquad \mathrm{(D) \ } \frac{5}{6}\qquad \mathrm{(E) \ }\frac{7}{8}&lt;/math&gt;<br /> <br /> ==Problem 16==<br /> There are 10 horses, named Horse 1, Horse 2, &lt;math&gt;\ldots&lt;/math&gt;, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse &lt;math&gt;k&lt;/math&gt; runs one lap in exactly &lt;math&gt;k&lt;/math&gt; minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time &lt;math&gt;S&gt;0&lt;/math&gt;, in minutes, at which all 10 horses will again simultaneously be at the starting point is &lt;math&gt;S=2520&lt;/math&gt;. Let &lt;math&gt;T&gt;0&lt;/math&gt; be the least time, in minutes, such that at least 5 of the horses are again at the starting point. What is the sum of the digits of &lt;math&gt;T&lt;/math&gt;?<br /> <br /> &lt;math&gt; \mathrm{(A) \ }2\qquad \mathrm{(B) \ }3\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ }5\qquad \mathrm{(E) \ }6&lt;/math&gt;<br /> <br /> ==Problem 17==<br /> Distinct points &lt;math&gt;P&lt;/math&gt;, &lt;math&gt;Q&lt;/math&gt;, &lt;math&gt;R&lt;/math&gt;, &lt;math&gt;S&lt;/math&gt; lie on the circle &lt;math&gt;x^2+y^2=25&lt;/math&gt; and have integer coordinates. The distances &lt;math&gt;PQ&lt;/math&gt; and &lt;math&gt;RS&lt;/math&gt; are irrational numbers. What is the greatest possible value of the ratio &lt;math&gt;\frac{PQ}{RS}&lt;/math&gt;?<br /> <br /> &lt;math&gt;\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ 5\qquad\mathrm{(C)}\ 3\sqrt{5}\qquad\mathrm{(D)}\ 7\qquad\mathrm{(E)}\ 5\sqrt{2}&lt;/math&gt;<br /> <br /> ==Problem 18==<br /> <br /> ==Problem 19==<br /> <br /> ==Problem 20==<br /> <br /> ==Problem 21==<br /> <br /> ==Problem 22==<br /> Sides &lt;math&gt;\overline{AB}&lt;/math&gt; and &lt;math&gt;\overline{AC}&lt;/math&gt; of equilateral triangle &lt;math&gt;ABC&lt;/math&gt; are tangent to a circle as points &lt;math&gt;B&lt;/math&gt; and &lt;math&gt;C&lt;/math&gt; respectively. What fraction of the area of &lt;math&gt;\triangle ABC&lt;/math&gt; lies outside the circle?<br /> <br /> &lt;math&gt; \mathrm{(A) \ }\dfrac{4\sqrt{3}\pi}{27}-\frac{1}{3}\qquad \mathrm{(B) \ } \frac{\sqrt{3}}{2}-\frac{\pi}{8}\qquad \mathrm{(C) \ } \frac{1}{2} \qquad \mathrm{(D) \ }\sqrt{3}-\frac{2\sqrt{3}\pi}{9}\qquad \mathrm{(E) \ } \frac{4}{3}-\dfrac{4\sqrt{3}\pi}{27}&lt;/math&gt;<br /> <br /> ==Problem 23==<br /> How many triangles with positive area have all their vertices at points &lt;math&gt;(i,j)&lt;/math&gt; in the coordinate plane, where &lt;math&gt;i&lt;/math&gt; and &lt;math&gt;j&lt;/math&gt; are integers between &lt;math&gt;1&lt;/math&gt; and &lt;math&gt;5&lt;/math&gt;, inclusive?<br /> <br /> &lt;math&gt;\textbf{(A)}\ 2128 \qquad\textbf{(B)}\ 2148 \qquad\textbf{(C)}\ 2160 \qquad\textbf{(D)}\ 2200 \qquad\textbf{(E)}\ 2300&lt;/math&gt;<br /> <br /> ==Problem 24==<br /> For certain real numbers &lt;math&gt;a&lt;/math&gt;, &lt;math&gt;b&lt;/math&gt;, and &lt;math&gt;c&lt;/math&gt;, the polynomial &lt;cmath&gt;g(x) = x^3 + ax^2 + x + 10&lt;/cmath&gt;has three distinct roots, and each root of &lt;math&gt;g(x)&lt;/math&gt; is also a root of the polynomial &lt;cmath&gt;f(x) = x^4 + x^3 + bx^2 + 100x + c.&lt;/cmath&gt;What is &lt;math&gt;f(1)&lt;/math&gt;?<br /> <br /> &lt;math&gt;\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{(D)}\ -6006 \qquad\textbf{(E)}\ -5005&lt;/math&gt;<br /> <br /> ==Problem 25==</div> Drakodin https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_10A_Problems/Problem_17&diff=82964 2017 AMC 10A Problems/Problem 17 2017-02-08T22:10:34Z <p>Drakodin: /* Problem */</p> <hr /> <div>==Problem==<br /> Distinct points &lt;math&gt;P&lt;/math&gt;, &lt;math&gt;Q&lt;/math&gt;, &lt;math&gt;R&lt;/math&gt;, &lt;math&gt;S&lt;/math&gt; lie on the circle &lt;math&gt;x^2+y^2=25&lt;/math&gt; and have integer coordinates. The distances &lt;math&gt;PQ&lt;/math&gt; and &lt;math&gt;RS&lt;/math&gt; are irrational numbers. What is the greatest possible value of the ratio &lt;math&gt;\frac{PQ}{RS}&lt;/math&gt;?<br /> <br /> &lt;math&gt;\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ 5\qquad\mathrm{(C)}\ 3\sqrt{5}\qquad\mathrm{(D)}\ 7\qquad\mathrm{(E)}\ 5\sqrt{2}&lt;/math&gt;<br /> <br /> ==See Also==<br /> {{AMC10 box|year=2017|ab=A|num-b=16|num-a=18}}<br /> {{MAA Notice}}</div> Drakodin https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_10A_Problems/Problem_18&diff=82958 2017 AMC 10A Problems/Problem 18 2017-02-08T22:08:08Z <p>Drakodin: Created page with &quot;==Problem== Amelia has a coin that lands heads with probability &lt;math&gt;\frac{1}{3}&lt;/math&gt;, and Blaine has a coin that lands on heads with probability &lt;math&gt;\frac{2}{5}&lt;/math&gt;....&quot;</p> <hr /> <div>==Problem==<br /> Amelia has a coin that lands heads with probability &lt;math&gt;\frac{1}{3}&lt;/math&gt;, and Blaine has a coin that lands on heads with probability &lt;math&gt;\frac{2}{5}&lt;/math&gt;. Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is &lt;math&gt;\frac{p}{q}&lt;/math&gt;, where &lt;math&gt;p&lt;/math&gt; and &lt;math&gt;q&lt;/math&gt; are relatively prime positive integers. What is &lt;math&gt;q-p&lt;/math&gt;?<br /> <br /> &lt;math&gt;\mathrm{(A)}\ 1\qquad\mathrm{(B)}\ 2\qquad\mathrm{(C)}\ 3\qquad\mathrm{(D)}\ 4\qquad\mathrm{(E)}\ 5&lt;/math&gt;</div> Drakodin https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_10A_Problems/Problem_17&diff=82940 2017 AMC 10A Problems/Problem 17 2017-02-08T22:00:03Z <p>Drakodin: /* Problem 17 */</p> <hr /> <div>Distinct points &lt;math&gt;P&lt;/math&gt;, &lt;math&gt;Q&lt;/math&gt;, &lt;math&gt;R&lt;/math&gt;, &lt;math&gt;S&lt;/math&gt; lie on the circle &lt;math&gt;x^2+y^2=25&lt;/math&gt; and have integer coordinates. The distances &lt;math&gt;PQ&lt;/math&gt; and &lt;math&gt;RS&lt;/math&gt; are irrational numbers. What is the greatest possible value of the ratio &lt;math&gt;\frac{PQ}{RS}&lt;/math&gt;?</div> Drakodin https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_10A_Problems/Problem_17&diff=82935 2017 AMC 10A Problems/Problem 17 2017-02-08T21:58:22Z <p>Drakodin: Created page with &quot;==Problem 17== Distinct points &lt;math&gt;P&lt;/math&gt;, &lt;math&gt;Q&lt;/math&gt;, &lt;math&gt;R&lt;/math&gt;, &lt;math&gt;S&lt;/math&gt; lie on the circle &lt;math&gt;x^2+y^2=25&lt;/math&gt; and have integer coordinates. The dista...&quot;</p> <hr /> <div>==Problem 17==<br /> Distinct points &lt;math&gt;P&lt;/math&gt;, &lt;math&gt;Q&lt;/math&gt;, &lt;math&gt;R&lt;/math&gt;, &lt;math&gt;S&lt;/math&gt; lie on the circle &lt;math&gt;x^2+y^2=25&lt;/math&gt; and have integer coordinates. The distances &lt;math&gt;PQ&lt;/math&gt; and &lt;math&gt;RS&lt;/math&gt; are irrational numbers. What is the greatest possible value of the ratio &lt;math&gt;\frac{PQ}{RS}&lt;/math&gt;?</div> Drakodin