Difference between revisions of "2013 AMC 8 Problems"
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+ | {{AMC8 Problems|year=2013|}} | ||
==Problem 1== | ==Problem 1== | ||
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==Problem 2== | ==Problem 2== | ||
− | A sign at the fish market says, "50% off, today only: half-pound packages for just | + | A sign at the fish market says, "50% off, today only: half-pound packages for just \$3 per package." What is the regular price for a full pound of fish, in dollars? (Assume that there are no deals for bulk) |
<math>\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 15</math> | <math>\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 15</math> | ||
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==Problem 4== | ==Problem 4== | ||
− | Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra | + | Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra \$2.50 to cover her portion of the total bill. What was the total bill? |
− | <math> \textbf{(A)}\ | + | <math> \textbf{(A)}\ \text{\textdollar}120\qquad\textbf{(B)}\ \text{\textdollar}128\qquad\textbf{(C)}\ \text{\textdollar}140\qquad\textbf{(D)}\ \text{\textdollar}144\qquad\textbf{(E)}\ \text{\textdollar}160 </math> |
[[2013 AMC 8 Problems/Problem 4|Solution]] | [[2013 AMC 8 Problems/Problem 4|Solution]] | ||
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− | Hammie is in the <math>6^\text{th}</math> grade and weighs 106 pounds. | + | Hammie is in the <math>6^\text{th}</math> grade and weighs 106 pounds. Her quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds? |
<math>\textbf{(A)}\ \text{median, by 60} \qquad \textbf{(B)}\ \text{median, by 20} \qquad \textbf{(C)}\ \text{average, by 5} \qquad \textbf{(D)}\ \text{average, by 15} \qquad \textbf{(E)}\ \text{average, by 20}</math> | <math>\textbf{(A)}\ \text{median, by 60} \qquad \textbf{(B)}\ \text{median, by 20} \qquad \textbf{(C)}\ \text{average, by 5} \qquad \textbf{(D)}\ \text{average, by 15} \qquad \textbf{(E)}\ \text{average, by 20}</math> | ||
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==Problem 9== | ==Problem 9== | ||
+ | The Incredible Hulk can double the distance it jumps with each succeeding jump. If its first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will it first be able to jump more than 1 kilometer? | ||
− | <math> \textbf{(A)}\ | + | <math>\textbf{(A)}\ 9^\text{th} \qquad \textbf{(B)}\ 10^\text{th} \qquad \textbf{(C)}\ 11^\text{th} \qquad \textbf{(D)}\ 12^\text{th} \qquad \textbf{(E)}\ 13^\text{th}</math> |
[[2013 AMC 8 Problems/Problem 9|Solution]] | [[2013 AMC 8 Problems/Problem 9|Solution]] | ||
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− | <math> \textbf{(A)}\ | + | What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594? |
+ | |||
+ | <math>\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 165 \qquad \textbf{(C)}\ 330 \qquad \textbf{(D)}\ 625 \qquad \textbf{(E)}\ 660</math> | ||
[[2013 AMC 8 Problems/Problem 10|Solution]] | [[2013 AMC 8 Problems/Problem 10|Solution]] | ||
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==Problem 11== | ==Problem 11== | ||
+ | Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less? | ||
− | <math> \textbf{(A)}\ | + | <math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5</math> |
[[2013 AMC 8 Problems/Problem 11|Solution]] | [[2013 AMC 8 Problems/Problem 11|Solution]] | ||
==Problem 12== | ==Problem 12== | ||
+ | At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of <math>50</math>, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the 150 dollar regular price did he save? | ||
− | + | <math>\textbf{(A)}\ 25\% \qquad \textbf{(B)}\ 30\% \qquad \textbf{(C)}\ 33\% \qquad \textbf{(D)}\ 40\% \qquad \textbf{(E)}\ 45\%</math> | |
− | <math> \textbf{(A)}\ | ||
[[2013 AMC 8 Problems/Problem 12|Solution]] | [[2013 AMC 8 Problems/Problem 12|Solution]] | ||
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==Problem 14== | ==Problem 14== | ||
− | Abe holds 1 green and 1 red jelly bean in his hand. | + | Abe holds 1 green and 1 red jelly bean in his hand. Bob holds 1 green, 1 yellow, and 2 red jelly beans in his hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match? |
<math>\textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac13 \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac23</math> | <math>\textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac13 \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac23</math> | ||
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==Problem 15== | ==Problem 15== | ||
+ | If <math>3^p + 3^4 = 90</math>, <math>2^r + 44 = 76</math>, and <math>5^3 + 6^s = 1421</math>, what is the product of <math>p</math>, <math>r</math>, and <math>s</math>? | ||
− | + | <math>\textbf{(A)}\ 27 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 70 \qquad \textbf{(E)}\ 90</math> | |
− | <math> \textbf{(A)}\ | ||
[[2013 AMC 8 Problems/Problem 15|Solution]] | [[2013 AMC 8 Problems/Problem 15|Solution]] | ||
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− | <math> \textbf{(A)}\ | + | A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of <math>8^\text{th}</math>-graders to <math>6^\text{th}</math>-graders is <math>5:3</math>, and the ratio of <math>8^\text{th}</math>-graders to <math>7^\text{th}</math>-graders is <math>8:5</math>. What is the smallest number of students that could be participating in the project? |
+ | |||
+ | <math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 55 \qquad \textbf{(D)}\ 79 \qquad \textbf{(E)}\ 89</math> | ||
+ | |||
[[2013 AMC 8 Problems/Problem 16|Solution]] | [[2013 AMC 8 Problems/Problem 16|Solution]] | ||
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==Problem 17== | ==Problem 17== | ||
+ | The sum of six consecutive positive integers is 2013. What is the largest of these six integers? | ||
− | <math> \textbf{(A)}\ | + | <math>\textbf{(A)}\ 335 \qquad \textbf{(B)}\ 338 \qquad \textbf{(C)}\ 340 \qquad \textbf{(D)}\ 345 \qquad \textbf{(E)}\ 350</math> |
[[2013 AMC 8 Problems/Problem 17|Solution]] | [[2013 AMC 8 Problems/Problem 17|Solution]] | ||
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==Problem 18== | ==Problem 18== | ||
+ | Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain? | ||
− | <math> \textbf{(A)}\ | + | <asy> |
+ | import three; | ||
+ | size(3inch); | ||
+ | currentprojection=orthographic(-8,15,15); | ||
+ | triple A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P; | ||
+ | A = (0,0,0); | ||
+ | B = (0,10,0); | ||
+ | C = (12,10,0); | ||
+ | D = (12,0,0); | ||
+ | E = (0,0,5); | ||
+ | F = (0,10,5); | ||
+ | G = (12,10,5); | ||
+ | H = (12,0,5); | ||
+ | I = (1,1,1); | ||
+ | J = (1,9,1); | ||
+ | K = (11,9,1); | ||
+ | L = (11,1,1); | ||
+ | M = (1,1,5); | ||
+ | N = (1,9,5); | ||
+ | O = (11,9,5); | ||
+ | P = (11,1,5); | ||
+ | //outside box far | ||
+ | draw(surface(A--B--C--D--cycle),white,nolight); | ||
+ | draw(A--B--C--D--cycle); | ||
+ | draw(surface(E--A--D--H--cycle),white,nolight); | ||
+ | draw(E--A--D--H--cycle); | ||
+ | draw(surface(D--C--G--H--cycle),white,nolight); | ||
+ | draw(D--C--G--H--cycle); | ||
+ | //inside box far | ||
+ | draw(surface(I--J--K--L--cycle),white,nolight); | ||
+ | draw(I--J--K--L--cycle); | ||
+ | draw(surface(I--L--P--M--cycle),white,nolight); | ||
+ | draw(I--L--P--M--cycle); | ||
+ | draw(surface(L--K--O--P--cycle),white,nolight); | ||
+ | draw(L--K--O--P--cycle); | ||
+ | //inside box near | ||
+ | draw(surface(I--J--N--M--cycle),white,nolight); | ||
+ | draw(I--J--N--M--cycle); | ||
+ | draw(surface(J--K--O--N--cycle),white,nolight); | ||
+ | draw(J--K--O--N--cycle); | ||
+ | //outside box near | ||
+ | draw(surface(A--B--F--E--cycle),white,nolight); | ||
+ | draw(A--B--F--E--cycle); | ||
+ | draw(surface(B--C--G--F--cycle),white,nolight); | ||
+ | draw(B--C--G--F--cycle); | ||
+ | //top | ||
+ | draw(surface(E--H--P--M--cycle),white,nolight); | ||
+ | draw(surface(E--M--N--F--cycle),white,nolight); | ||
+ | draw(surface(F--N--O--G--cycle),white,nolight); | ||
+ | draw(surface(O--G--H--P--cycle),white,nolight); | ||
+ | draw(M--N--O--P--cycle); | ||
+ | draw(E--F--G--H--cycle); | ||
+ | label("10",(A--B),SE); | ||
+ | label("12",(C--B),SW); | ||
+ | label("5",(F--B),W);</asy> | ||
+ | |||
+ | <math>\textbf{(A)}\ 204 \qquad \textbf{(B)}\ 280 \qquad \textbf{(C)}\ 320 \qquad \textbf{(D)}\ 340 \qquad \textbf{(E)}\ 600</math> | ||
[[2013 AMC 8 Problems/Problem 18|Solution]] | [[2013 AMC 8 Problems/Problem 18|Solution]] | ||
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==Problem 19== | ==Problem 19== | ||
+ | Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I didn't get the lowest score in our class,' and Bridget adds, 'I didn't get the highest score.' What is the ranking of the three girls from the highest score to the lowest score? | ||
− | <math> \textbf{(A)}\ | + | <math>\textbf{(A)}\ \text{Hannah, Cassie, Bridget} \qquad \textbf{(B)}\ \text{Hannah, Bridget, Cassie} \\ \qquad \textbf{(C)}\ \text{Cassie, Bridget, Hannah} \qquad \textbf{(D)}\ \text{Cassie, Hannah, Bridget} \\ \qquad \textbf{(E)}\ \text{Bridget, Cassie, Hannah}</math> |
[[2013 AMC 8 Problems/Problem 19|Solution]] | [[2013 AMC 8 Problems/Problem 19|Solution]] | ||
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==Problem 20== | ==Problem 20== | ||
+ | A <math>1\times 2</math> rectangle is inscribed in a semicircle with longer side on the diameter. What is the area of the semicircle? | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac\pi2 \qquad \textbf{(B)}\ \frac{2\pi}3 \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}3 \qquad \textbf{(E)}\ \frac{5\pi}3</math> | ||
− | |||
[[2013 AMC 8 Problems/Problem 20|Solution]] | [[2013 AMC 8 Problems/Problem 20|Solution]] | ||
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==Problem 21== | ==Problem 21== | ||
+ | Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take? | ||
− | <math> \textbf{(A)}\ | + | <math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 18</math> |
[[2013 AMC 8 Problems/Problem 21|Solution]] | [[2013 AMC 8 Problems/Problem 21|Solution]] | ||
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Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether? | Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether? | ||
− | + | <asy> | |
picture corner; | picture corner; | ||
draw(corner,(5,0)--(35,0)); | draw(corner,(5,0)--(35,0)); | ||
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for (int i=0; i<3; ++i) | for (int i=0; i<3; ++i) | ||
{ | { | ||
− | + | for (int j=0; j>-2; --j) | |
− | + | { | |
− | + | if ((i-j)<3) | |
− | + | { | |
− | + | add(corner,(50i,50j)); | |
− | + | } | |
− | + | } | |
} | } | ||
draw((5,-100)--(45,-100)); | draw((5,-100)--(45,-100)); | ||
Line 199: | Line 269: | ||
draw((50,-105)--(50,-135),dotted); | draw((50,-105)--(50,-135),dotted); | ||
draw((0,-105)--(0,-135),dotted); | draw((0,-105)--(0,-135),dotted); | ||
− | + | </asy> | |
− | <math>\textbf{(A)}\ 1920 \qquad \textbf{(B)}\ 1952 \qquad \textbf{(C)}\ 1980 \qquad \textbf{(D)}\ 2013 \qquad \textbf{(E)}\ | + | <math>\textbf{(A)}\ 1920 \qquad \textbf{(B)}\ 1952 \qquad \textbf{(C)}\ 1980 \qquad \textbf{(D)}\ 2013 \qquad \textbf{(E)}\ 3932</math> |
[[2013 AMC 8 Problems/Problem 22|Solution]] | [[2013 AMC 8 Problems/Problem 22|Solution]] | ||
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==Problem 23== | ==Problem 23== | ||
− | <math> \textbf{(A)}\ | + | Angle <math>ABC</math> of <math>\triangle ABC</math> is a right angle. The sides of <math>\triangle ABC</math> are the diameters of semicircles as shown. The area of the semicircle on <math>\overline{AB}</math> equals <math>8\pi</math>, and the arc of the semicircle on <math>\overline{AC}</math> has length <math>8.5\pi</math>. What is the radius of the semicircle on <math>\overline{BC}</math>? |
+ | |||
+ | <asy> | ||
+ | import graph; | ||
+ | pair A,B,C; | ||
+ | A=(0,8); | ||
+ | B=(0,0); | ||
+ | C=(15,0); | ||
+ | draw((0,8)..(-4,4)..(0,0)--(0,8)); | ||
+ | draw((0,0)..(7.5,-7.5)..(15,0)--(0,0)); | ||
+ | real theta = aTan(8/15); | ||
+ | draw(arc((15/2,4),17/2,-theta,180-theta)); | ||
+ | draw((0,8)--(15,0)); | ||
+ | dot(A); | ||
+ | dot(B); | ||
+ | dot(C); | ||
+ | label("$A$", A, NW); | ||
+ | label("$B$", B, SW); | ||
+ | label("$C$", C, SE);</asy> | ||
+ | |||
+ | <math>\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 7.5 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 8.5 \qquad \textbf{(E)}\ 9</math> | ||
[[2013 AMC 8 Problems/Problem 23|Solution]] | [[2013 AMC 8 Problems/Problem 23|Solution]] | ||
==Problem 24== | ==Problem 24== | ||
+ | |||
+ | Squares <math>ABCD</math>, <math>EFGH</math>, and <math>GHIJ</math> are equal in area. Points <math>C</math> and <math>D</math> are the midpoints of sides <math>IH</math> and <math>HE</math>, respectively. What is the ratio of the area of the shaded pentagon <math>AJICB</math> to the sum of the areas of the three squares? | ||
+ | |||
+ | <asy> | ||
+ | pair A,B,C,D,E,F,G,H,I,J; | ||
+ | |||
+ | A = (0.5,2); | ||
+ | B = (1.5,2); | ||
+ | C = (1.5,1); | ||
+ | D = (0.5,1); | ||
+ | E = (0,1); | ||
+ | F = (0,0); | ||
+ | G = (1,0); | ||
+ | H = (1,1); | ||
+ | I = (2,1); | ||
+ | J = (2,0); | ||
+ | draw(A--B); | ||
+ | draw(C--B); | ||
+ | draw(D--A); | ||
+ | draw(F--E); | ||
+ | draw(I--J); | ||
+ | draw(J--F); | ||
+ | draw(G--H); | ||
+ | draw(A--J); | ||
+ | filldraw(A--B--C--I--J--cycle,grey); | ||
+ | draw(E--I); | ||
+ | label("$A$", A, NW); | ||
+ | label("$B$", B, NE); | ||
+ | label("$C$", C, NE); | ||
+ | label("$D$", D, NW); | ||
+ | label("$E$", E, NW); | ||
+ | label("$F$", F, SW); | ||
+ | label("$G$", G, S); | ||
+ | label("$H$", H, N); | ||
+ | label("$I$", I, NE); | ||
+ | label("$J$", J, SE); | ||
+ | </asy> | ||
− | <math> \textbf{(A)}\hspace{.05in}\qquad\textbf{(B)}\hspace{.05in}\qquad\textbf{(C)}\hspace{.05in}\qquad\textbf{(D)}\hspace{.05in}\qquad\textbf{(E)}\hspace{.05in} </math> | + | <math> \textbf{(A)}\hspace{.05in}\frac{1}{4}\qquad\textbf{(B)}\hspace{.05in}\frac{7}{24}\qquad\textbf{(C)}\hspace{.05in}\frac{1}{3}\qquad\textbf{(D)}\hspace{.05in}\frac{3}{8}\qquad\textbf{(E)}\hspace{.05in}\frac{5}{12}</math> |
[[2013 AMC 8 Problems/Problem 24|Solution]] | [[2013 AMC 8 Problems/Problem 24|Solution]] | ||
==Problem 25== | ==Problem 25== | ||
+ | A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are <math>R_1 = 100</math> inches, <math>R_2 = 60</math> inches, and <math>R_3 = 80</math> inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B? | ||
+ | <asy> | ||
+ | pair A,B; | ||
+ | size(8cm); | ||
+ | A=(0,0); | ||
+ | B=(480,0); | ||
+ | draw((0,0)--(480,0),linetype("3 4")); | ||
+ | filldraw(circle((8,0),8),black); | ||
+ | draw((0,0)..(100,-100)..(200,0)); | ||
+ | draw((200,0)..(260,60)..(320,0)); | ||
+ | draw((320,0)..(400,-80)..(480,0)); | ||
+ | draw((100,0)--(150,-50sqrt(3)),Arrow(size=4)); | ||
+ | draw((260,0)--(290,30sqrt(3)),Arrow(size=4)); | ||
+ | draw((400,0)--(440,-40sqrt(3)),Arrow(size=4)); | ||
+ | label("$A$", A, SW); | ||
+ | label("$B$", B, SE); | ||
+ | label("$R_1$", (100,-40), W); | ||
+ | label("$R_2$", (260,40), SW); | ||
+ | label("$R_3$", (400,-40), W);</asy> | ||
− | <math> \textbf{(A)}\ | + | <math> \textbf{(A)}\ 238\pi\qquad\textbf{(B)}\ 240\pi\qquad\textbf{(C)}\ 260\pi\qquad\textbf{(D)}\ 280\pi\qquad\textbf{(E)}\ 500\pi </math> |
[[2013 AMC 8 Problems/Problem 25|Solution]] | [[2013 AMC 8 Problems/Problem 25|Solution]] | ||
+ | |||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 20:05, 8 May 2023
2013 AMC 8 (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
Problem 1
Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way?
Problem 2
A sign at the fish market says, "50% off, today only: half-pound packages for just $3 per package." What is the regular price for a full pound of fish, in dollars? (Assume that there are no deals for bulk)
Problem 3
What is the value of ?
Problem 4
Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $2.50 to cover her portion of the total bill. What was the total bill?
Problem 5
Hammie is in the grade and weighs 106 pounds. Her quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?
Problem 6
The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, . What is the missing number in the top row?
Problem 7
Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely number of cars in the train?
Problem 8
A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?
Problem 9
The Incredible Hulk can double the distance it jumps with each succeeding jump. If its first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will it first be able to jump more than 1 kilometer?
Problem 10
What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?
Problem 11
Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?
Problem 12
At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of , you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the 150 dollar regular price did he save?
Problem 13
When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?
Problem 14
Abe holds 1 green and 1 red jelly bean in his hand. Bob holds 1 green, 1 yellow, and 2 red jelly beans in his hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?
Problem 15
If ,
, and
, what is the product of
,
, and
?
Problem 16
A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of -graders to
-graders is
, and the ratio of
-graders to
-graders is
. What is the smallest number of students that could be participating in the project?
Problem 17
The sum of six consecutive positive integers is 2013. What is the largest of these six integers?
Problem 18
Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?
Problem 19
Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I didn't get the lowest score in our class,' and Bridget adds, 'I didn't get the highest score.' What is the ranking of the three girls from the highest score to the lowest score?
Problem 20
A rectangle is inscribed in a semicircle with longer side on the diameter. What is the area of the semicircle?
Problem 21
Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?
Problem 22
Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?
Problem 23
Angle of
is a right angle. The sides of
are the diameters of semicircles as shown. The area of the semicircle on
equals
, and the arc of the semicircle on
has length
. What is the radius of the semicircle on
?
Problem 24
Squares ,
, and
are equal in area. Points
and
are the midpoints of sides
and
, respectively. What is the ratio of the area of the shaded pentagon
to the sum of the areas of the three squares?
Problem 25
A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are inches,
inches, and
inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B?
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.