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| − | ==Problem 1==
| + | '''2010 AMC 8''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution. |
| − | At Euclid High School, the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are <math>11</math> students in Mrs. Germain's class, 8 in Mr. Newton, and <math>9</math> in Mrs. Young's class are taking the AMC <math>8</math> this year. How many mathematics students at Euclid High School are taking the contest?
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| − | <math> \textbf{(A)}\ 26 \qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qquad\textbf{(E)}\ 30 </math>
| + | *[[2010 AMC 8 Problems]] |
| | + | *[[2010 AMC 8 Answer Key]] |
| | + | **[[2010 AMC 8 Problems/Problem 1|Problem 1]] |
| | + | **[[2010 AMC 8 Problems/Problem 2|Problem 2]] |
| | + | **[[2010 AMC 8 Problems/Problem 3|Problem 3]] |
| | + | **[[2010 AMC 8 Problems/Problem 4|Problem 4]] |
| | + | **[[2010 AMC 8 Problems/Problem 5|Problem 5]] |
| | + | **[[2010 AMC 8 Problems/Problem 6|Problem 6]] |
| | + | **[[2010 AMC 8 Problems/Problem 7|Problem 7]] |
| | + | **[[2010 AMC 8 Problems/Problem 8|Problem 8]] |
| | + | **[[2010 AMC 8 Problems/Problem 9|Problem 9]] |
| | + | **[[2010 AMC 8 Problems/Problem 10|Problem 10]] |
| | + | **[[2010 AMC 8 Problems/Problem 11|Problem 11]] |
| | + | **[[2010 AMC 8 Problems/Problem 12|Problem 12]] |
| | + | **[[2010 AMC 8 Problems/Problem 13|Problem 13]] |
| | + | **[[2010 AMC 8 Problems/Problem 14|Problem 14]] |
| | + | **[[2010 AMC 8 Problems/Problem 15|Problem 15]] |
| | + | **[[2010 AMC 8 Problems/Problem 16|Problem 16]] |
| | + | **[[2010 AMC 8 Problems/Problem 17|Problem 17]] |
| | + | **[[2010 AMC 8 Problems/Problem 18|Problem 18]] |
| | + | **[[2010 AMC 8 Problems/Problem 19|Problem 19]] |
| | + | **[[2010 AMC 8 Problems/Problem 20|Problem 20]] |
| | + | **[[2010 AMC 8 Problems/Problem 21|Problem 21]] |
| | + | **[[2010 AMC 8 Problems/Problem 22|Problem 22]] |
| | + | **[[2010 AMC 8 Problems/Problem 23|Problem 23]] |
| | + | **[[2010 AMC 8 Problems/Problem 24|Problem 24]] |
| | + | **[[2010 AMC 8 Problems/Problem 25|Problem 25]] |
| | | | |
| − | ==Problem 2==
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| − | If <math>a @ b = \frac{a\times b}{a+b}</math>, for <math>a,b</math> positive integers, then what is <math>5 @10</math>?
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| − | <math>\textbf{(A)}\ \frac{3}{10} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \frac{10}{3} \qquad\textbf{(E)}\ 50</math>
| + | == See also == |
| − | | + | {{Succession box| |
| − | ==Problem 3== | + | |header=2010 AMC 8 ([[2010 AMC 8 Problems|Problems]]) |
| − | The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price?
| + | |before=[[2009 AMC 8]] |
| − | | + | |title=[[AMC 8]] |
| − | <asy>
| + | |after=[[2011 AMC 8]]}} |
| − | import graph; size(16.38cm); real lsf=2; pathpen=linewidth(0.7); pointpen=black; pen fp = fontsize(10); pointfontpen=fp; real xmin=-1.33,xmax=11.05,ymin=-9.01,ymax=-0.44;
| + | * [[Mathematics competitions]] |
| − | pen ycycyc=rgb(0.55,0.55,0.55);
| + | * [[Mathematics competition resources]] |
| − | pair A=(1,-6), B=(1,-2), D=(1,-5.8), E=(1,-5.6), F=(1,-5.4), G=(1,-5.2), H=(1,-5), J=(1,-4.8), K=(1,-4.6), L=(1,-4.4), M=(1,-4.2), N=(1,-4), P=(1,-3.8), Q=(1,-3.6), R=(1,-3.4), S=(1,-3.2), T=(1,-3), U=(1,-2.8), V=(1,-2.6), W=(1,-2.4), Z=(1,-2.2), E_1=(1.4,-2.6), F_1=(1.8,-2.6), O_1=(14,-6), P_1=(14,-5), Q_1=(14,-4), R_1=(14,-3), S_1=(14,-2), C_1=(1.4,-6), D_1=(1.8,-6), G_1=(2.4,-6), H_1=(2.8,-6), I_1=(3.4,-6), J_1=(3.8,-6), K_1=(4.4,-6), L_1=(4.8,-6), M_1=(5.4,-6), N_1=(5.8,-6), T_1=(6.4,-6), U_1=(6.8,-6), V_1=(7.4,-6), W_1=(7.8,-6), Z_1=(8.4,-6), A_2=(8.8,-6), B_2=(9.4,-6), C_2=(9.8,-6), D_2=(10.4,-6), E_2=(10.8,-6), L_2=(2.4,-3.2), M_2=(2.8,-3.2), N_2=(3.4,-4), O_2=(3.8,-4), P_2=(4.4,-3.6), Q_2=(4.8,-3.6), R_2=(5.4,-3.6), S_2=(5.8,-3.6), T_2=(6.4,-3.4), U_2=(6.8,-3.4), V_2=(7.4,-3.8), W_2=(7.8,-3.8), Z_2=(8.4,-2.8), A_3=(8.8,-2.8), B_3=(9.4,-3.2), C_3=(9.8,-3.2), D_3=(10.4,-3.8), E_3=(10.8,-3.8);
| + | * [[Math books]] |
| − | filldraw(C_1--E_1--F_1--D_1--cycle,ycycyc); filldraw(G_1--L_2--M_2--H_1--cycle,ycycyc); filldraw(I_1--N_2--O_2--J_1--cycle,ycycyc); filldraw(K_1--P_2--Q_2--L_1--cycle,ycycyc); filldraw(M_1--R_2--S_2--N_1--cycle,ycycyc); filldraw(T_1--T_2--U_2--U_1--cycle,ycycyc); filldraw(V_1--V_2--W_2--W_1--cycle,ycycyc); filldraw(Z_1--Z_2--A_3--A_2--cycle,ycycyc); filldraw(B_2--B_3--C_3--C_2--cycle,ycycyc); filldraw(D_2--D_3--E_3--E_2--cycle,ycycyc); D(B--A,linewidth(0.4)); D(H--(8,-5),linewidth(0.4)); D(N--(8,-4),linewidth(0.4)); D(T--(8,-3),linewidth(0.4)); D(B--(8,-2),linewidth(0.4)); D(B--S_1); D(T--R_1); D(N--Q_1); D(H--P_1); D(A--O_1); D(C_1--E_1); D(E_1--F_1); D(F_1--D_1); D(D_1--C_1); D(G_1--L_2); D(L_2--M_2); D(M_2--H_1); D(H_1--G_1); D(I_1--N_2); D(N_2--O_2); D(O_2--J_1); D(J_1--I_1); D(K_1--P_2); D(P_2--Q_2); D(Q_2--L_1); D(L_1--K_1); D(M_1--R_2); D(R_2--S_2); D(S_2--N_1); D(N_1--M_1); D(T_1--T_2); D(T_2--U_2); D(U_2--U_1); D(U_1--T_1); D(V_1--V_2); D(V_2--W_2); D(W_2--W_1); D(W_1--V_1); D(Z_1--Z_2); D(Z_2--A_3); D(A_3--A_2); D(A_2--Z_1); D(B_2--B_3); D(B_3--C_3); D(C_3--C_2); D(C_2--B_2); D(D_2--D_3); D(D_3--E_3); D(E_3--E_2); D(E_2--D_2); label("0",(0.88,-5.91),SE*lsf,fp); label("\$ 5",(0.3,-4.84),SE*lsf,fp); label("\$ 10",(0.2,-3.84),SE*lsf,fp); label("\$ 15",(0.2,-2.85),SE*lsf,fp); label("\$ 20",(0.2,-1.85),SE*lsf,fp); label("$\mathrm{Price}$",(0.16,-3.45),SE*lsf,fp); label("$1$",(1.54,-5.97),SE*lsf,fp); label("$2$",(2.53,-5.95),SE*lsf,fp); label("$3$",(3.53,-5.94),SE*lsf,fp); label("$4$",(4.55,-5.94),SE*lsf,fp); label("$5$",(5.49,-5.95),SE*lsf,fp); label("$6$",(6.53,-5.95),SE*lsf,fp); label("$7$",(7.55,-5.95),SE*lsf,fp); label("$8$",(8.52,-5.95),SE*lsf,fp); label("$9$",(9.57,-5.97),SE*lsf,fp); label("$10$",(10.56,-5.94),SE*lsf,fp); label("Month",(7.14,-6.43),SE*lsf,fp);
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| − | D(A,linewidth(1pt)); D(B,linewidth(1pt)); D(D,linewidth(1pt)); D(E,linewidth(1pt)); D(F,linewidth(1pt)); D(G,linewidth(1pt)); D(H,linewidth(1pt)); D(J,linewidth(1pt)); D(K,linewidth(1pt)); D(L,linewidth(1pt)); D(M,linewidth(1pt)); D(N,linewidth(1pt)); D(P,linewidth(1pt)); D(Q,linewidth(1pt)); D(R,linewidth(1pt)); D(S,linewidth(1pt)); D(T,linewidth(1pt)); D(U,linewidth(1pt)); D(V,linewidth(1pt)); D(W,linewidth(1pt)); D(Z,linewidth(1pt)); D(E_1,linewidth(1pt)); D(F_1,linewidth(1pt)); D(O_1,linewidth(1pt)); D(P_1,linewidth(1pt)); D(Q_1,linewidth(1pt)); D(R_1,linewidth(1pt)); D(S_1,linewidth(1pt)); D(C_1,linewidth(1pt)); D(D_1,linewidth(1pt)); D(G_1,linewidth(1pt)); D(H_1,linewidth(1pt)); D(I_1,linewidth(1pt)); D(J_1,linewidth(1pt)); D(K_1,linewidth(1pt)); D(L_1,linewidth(1pt)); D(M_1,linewidth(1pt)); D(N_1,linewidth(1pt)); D(T_1,linewidth(1pt)); D(U_1,linewidth(1pt)); D(V_1,linewidth(1pt)); D(W_1,linewidth(1pt)); D(Z_1,linewidth(1pt)); D(A_2,linewidth(1pt)); D(B_2,linewidth(1pt)); D(C_2,linewidth(1pt)); D(D_2,linewidth(1pt)); D(E_2,linewidth(1pt)); D(L_2,linewidth(1pt)); D(M_2,linewidth(1pt)); D(N_2,linewidth(1pt)); D(O_2,linewidth(1pt)); D(P_2,linewidth(1pt)); D(Q_2,linewidth(1pt)); D(R_2,linewidth(1pt)); D(S_2,linewidth(1pt)); D(T_2,linewidth(1pt)); D(U_2,linewidth(1pt)); D(V_2,linewidth(1pt)); D(W_2,linewidth(1pt)); D(Z_2,linewidth(1pt)); D(A_3,linewidth(1pt)); D(B_3,linewidth(1pt)); D(C_3,linewidth(1pt)); D(D_3,linewidth(1pt)); D(E_3,linewidth(1pt));
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| − | clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);</asy>
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| − | <math>\textbf{(A)}\ 50 \qquad
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| − | \textbf{(B)}\ 62 \qquad
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| − | \textbf{(C)}\ 70 \qquad
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| − | \textbf{(D)}\ 89 \qquad
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| − | \textbf{(E)}\ 100</math>
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| − | | |
| − | ==Problem 4==
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| − | What is the sum of the mean, medium, and mode of the numbers, <math>2,3,0,3,1,4,0,3</math>?
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| − | <math> \textbf{(A)}\ 6.5 \qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 7.5\qquad\textbf{(D)}\ 8.5\qquad\textbf{(E)}\ 9 </math>
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| − | | |
| − | ==Problem 5==
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| − | Alice needs to replace a light bulb located <math>10</math> centimeters below the ceiling of her kitchen. The ceiling is <math>2.4</math> meters above the floor. Alice is <math>1.5</math> meters tall and can reach <math>46</math> centimeters above her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in centimeters?
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| − | <math> \textbf{(A)}\ 32 \qquad\textbf{(B)}\ 34\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 38\qquad\textbf{(E)}\ 40 </math>
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| − | | |
| − | ==Problem 6== | |
| − | Which of the following has the greatest number of line of symmetry?
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| − | | |
| − | <math> \textbf{(A)}\ \text{ Equilateral Triangle}</math>
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| − | <math>\textbf{(B)}\ \text{Non-square rhombus} </math>
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| − | <math>\textbf{(C)}\ \text{Non-square rectangle}</math>
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| − | <math>\textbf{(D)}\ \text{Isosceles Triangle}</math>
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| − | <math>\textbf{(E)}\ \text{Square} </math>
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| − | | |
| − | ==Problem 7==
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| − | Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than one dollar?
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| − | <math> \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 99 </math>
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| − | | |
| − | ==Problem 8==
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| − | As Emily is riding her bike on a long straight road, she spots Emerson skating in the same direction <math>1/2</math> mile in front of her. After she passes him, she can see him in her rear mirror until he is <math>1/2</math> mile behind her. Emily rides at a constant rate of <math>12</math> miles per hour. Emerson skates at a constant rate of <math>8</math> miles per hour. For how many minutes can Emily see Emerson?
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| − | <math> \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 16 </math>
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| − | | |
| − | ==Problem 9==
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| − | Ryan got <math>80\%</math> of the problems on a <math>25</math>-problem test, <math>90\%</math> on a <math>40</math>-problem test, and <math>70\%</math> on a <math>10</math>-problem test. What percent of all problems did Ryan answer correctly?
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| − | <math> \textbf{(A)}\ 64 \qquad\textbf{(B)}\ 75\qquad\textbf{(C)}\ 80\qquad\textbf{(D)}\ 84\qquad\textbf{(E)}\ 86 </math>
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| − | | |
| − | ==Problem 10==
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| − | <math>6</math> pepperoni circles will exactly fit across the diameter of a <math>12</math>-inch pizza when placed. If a total of <math>24</math> circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered with pepperoni?
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| − | <math> \textbf{(A)}\ \frac 12 \qquad\textbf{(B)}\ \frac 23 \qquad\textbf{(C)}\ \frac 34 \qquad\textbf{(D)}\ \frac 56 \qquad\textbf{(E)}\ \frac 78 </math>
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| − | | |
| − | ==Problem 11==
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| − | The top of one tree is <math>16</math> feet higher than the top of another tree. The height of the <math>2</math> trees are at a ratio of <math>3:4</math>. In feet, how tall is the taller tree?
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| − | <math> \textbf{(A)}\ 48 \qquad\textbf{(B)}\ 64 \qquad\textbf{(C)}\ 80 \qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 112 </math>
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| − | | |
| − | ==Problem 12==
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| − | Of the <math>500</math> balls in a large bag, <math>80\%</math> are red and the rest are blue. How many of the red balls must be removed so that <math>75\%</math> of the remaining balls are red?
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| − | <math> \textbf{(A)}\ 25 \qquad\textbf{(B)}\ 50\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 100\qquad\textbf{(E)}\ 150 </math>
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| − | | |
| − | ==Problem 13== | |
| − | The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shorter side is <math>30\%</math> of the perimeter. What is the length of the longest side?
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| − | | |
| − | <math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 </math>
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| − | | |
| − | ==Problem 14==
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| − | What is the sum of the prime factors of <math>2010</math>?
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| − | | |
| − | <math> \textbf{(A)}\ 67 \qquad\textbf{(B)}\ 75\qquad\textbf{(C)}\ 77\qquad\textbf{(D)}\ 201\qquad\textbf{(E)}\ 210 </math>
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| − | | |
| − | ==Problem 15==
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| − | A jar contains <math>5</math> different colors of gumdrops. <math>30\%</math> are blue, <math>20\%</math> are brown, <math>15\%</math> red, <math>10\%</math> yellow, and the other <math>30</math> gumdrops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown?
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| − | <math> \textbf{(A)}\ 35 \qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 64 </math>
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| − | | |
| − | ==Problem 16==
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| − | A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle?
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| − | | |
| − | <math> \textbf{(A)}\ \frac{\sqrt{\pi}}{2} \qquad\textbf{(B)}\ \sqrt{\pi} \qquad\textbf{(C)}\ \pi \qquad\textbf{(D)}\ 2\pi \qquad\textbf{(E)}\ \pi^{2}</math>
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| − | | |
| − | ==Problem 17==
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| − | The diagram shows an octagon consisting of <math>10</math> unit squares. The portion below <math>\overline{PQ}</math> is a unit square and a triangle with base <math>5</math>. If <math>\overline{PQ}</math> bisects the area of the octagon, what is the ratio <math>\frac{XQ}{QY}</math>?
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| − | | |
| − | <asy>
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| − | import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1);
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| − | draw((0,0)--(6,0),linewidth(1.2pt)); draw((0,0)--(0,1),linewidth(1.2pt)); draw((0,1)--(1,1),linewidth(1.2pt)); draw((1,1)--(1,2),linewidth(1.2pt)); draw((1,2)--(5,2),linewidth(1.2pt)); draw((5,2)--(5,1),linewidth(1.2pt)); draw((5,1)--(6,1),linewidth(1.2pt)); draw((6,1)--(6,0),linewidth(1.2pt)); draw((1,1)--(5,1),linewidth(1.2pt)+linetype("2pt 2pt")); draw((1,1)--(1,0),linewidth(1.2pt)+linetype("2pt 2pt")); draw((2,2)--(2,0),linewidth(1.2pt)+linetype("2pt 2pt")); draw((3,2)--(3,0),linewidth(1.2pt)+linetype("2pt 2pt")); draw((4,2)--(4,0),linewidth(1.2pt)+linetype("2pt 2pt")); draw((5,1)--(5,0),linewidth(1.2pt)+linetype("2pt 2pt")); draw((0,0)--(5,1.5),linewidth(1.2pt));
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| − | dot((0,0),ds); label("$P$", (-0.23,-0.26),NE*lsf); dot((0,1),ds); dot((1,1),ds); dot((1,2),ds); dot((5,2),ds); label("$X$", (5.14,2.02),NE*lsf); dot((5,1),ds); label("$Y$", (5.12,1.14),NE*lsf); dot((6,1),ds); dot((6,0),ds); dot((1,0),ds); dot((2,0),ds); dot((3,0),ds); dot((4,0),ds); dot((5,0),ds); dot((2,2),ds); dot((3,2),ds); dot((4,2),ds); dot((5,1.5),ds); label("$Q$", (5.14,1.51),NE*lsf); clip((-4.19,-5.52)--(-4.19,6.5)--(10.08,6.5)--(10.08,-5.52)--cycle);
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| − | </asy>
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| − | | |
| − | <math>\textbf{(A)}\ \frac 25 \qquad
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| − | \textbf{(B)}\ \frac 12 \qquad
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| − | \textbf{(C)}\ \frac 35 \qquad
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| − | \textbf{(D)}\ \frac 23 \qquad
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| − | \textbf{(E)}\ \frac 34</math>
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| − | | |
| − | ==Problem 18==
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| − | A decorative window is made up of a rectangle with semicircles at either end. The ratio of <math>AD</math> to <math>AB</math> is <math>3:2</math>. And <math>AB</math> is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircle.
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| − | | |
| − | <asy>
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| − | import graph; size(5cm); real lsf=0; pen dps=linewidth(0.7)+fontsize(8); defaultpen(dps); pen ds=black; real xmin=-4.27,xmax=14.73,ymin=-3.22,ymax=6.8; draw((0,4)--(0,0)); draw((0,0)--(2.5,0)); draw((2.5,0)--(2.5,4)); draw((2.5,4)--(0,4)); draw(shift((1.25,4))*xscale(1.25)*yscale(1.25)*arc((0,0),1,0,180)); draw(shift((1.25,0))*xscale(1.25)*yscale(1.25)*arc((0,0),1,-180,0));
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| − | dot((0,0),ds); label("$A$",(-0.26,-0.23),NE*lsf); dot((2.5,0),ds); label("$B$",(2.61,-0.26),NE*lsf); dot((0,4),ds); label("$D$",(-0.26,4.02),NE*lsf); dot((2.5,4),ds); label("$C$",(2.64,3.98),NE*lsf);
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| − | clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);</asy>
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| − | | |
| − | <math> \textbf{(A)}\ 2:3 \qquad\textbf{(B)}\ 3:2\qquad\textbf{(C)}\ 6:\pi \qquad\textbf{(D)}\ 9: \pi \qquad\textbf{(E)}\ 30 : \pi</math>
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| − | | |
| − | ==Problem 19==
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| − | The two circles pictured have the same center <math>C</math>. Chord <math>\overline{AD}</math> is tangent to the inner circle at <math>B</math>, <math>AC</math> is <math>10</math>, and chord <math>\overline{AD}</math> has length <math>16</math>. What is the area between the two circles?
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| − | | |
| − | <asy>
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| − | unitsize(45);
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| − | import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1);
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| − | draw((2,0.15)--(1.85,0.15)--(1.85,0)--(2,0)--cycle); draw(circle((2,1),2.24)); draw(circle((2,1),1)); draw((0,0)--(4,0)); draw((0,0)--(2,1)); draw((2,1)--(2,0)); draw((2,1)--(4,0));
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| − | dot((0,0),ds); label("$A$", (-0.19,-0.23),NE*lsf); dot((2,0),ds); label("$B$", (1.97,-0.31),NE*lsf); dot((2,1),ds); label("$C$", (1.96,1.09),NE*lsf); dot((4,0),ds); label("$D$", (4.07,-0.24),NE*lsf); clip((-3.1,-7.72)--(-3.1,4.77)--(11.74,4.77)--(11.74,-7.72)--cycle);
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| − | </asy>
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| − | | |
| − | <math> \textbf{(A)}\ 36 \pi \qquad\textbf{(B)}\ 49 \pi\qquad\textbf{(C)}\ 64 \pi\qquad\textbf{(D)}\ 81 \pi\qquad\textbf{(E)}\ 100 \pi </math>
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| − | | |
| − | ==Problem 20==
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| − | In a room, <math>2/5</math> of the people are wearing gloves, and <math>3/4</math> of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove?
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| − | | |
| − | <math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20 </math>
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| − | | |
| − | ==Problem 21==
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| − | Hui is an avid reader. She bought a copy of the best seller ''Math is Beautiful''. On the first day, she read <math>1/5</math> of the pages plus <math>12</math> more, and on the second day she read <math>1/4</math> of the remaining pages plus <math>15</math> more. On the third day she read <math>1/3</math> of the remaining pages plus <math>18</math> more. She then realizes she has <math>62</math> pages left, which she finishes the next day. How many pages are in this book?
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| − | <math> \textbf{(A)}\ 120 \qquad\textbf{(B)}\ 180\qquad\textbf{(C)}\ 240\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 360 </math>
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| − | ==Problem 22==
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| − | The hundreds digit of a three-digit number is <math>2</math> more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?
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| − | <math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 8 </math>
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| − | ==Problem 23==
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| − | Semicircles <math>POQ</math> and <math>ROS</math> pass through the center of circle <math>O</math>. What is the ratio of the combined areas of the two semicircles to the area of circle <math>O</math>?
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| − | <asy>
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| − | import graph; size(7.5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-6.27,xmax=10.01,ymin=-5.65,ymax=10.98; draw(circle((0,0),2)); draw((-3,0)--(3,0),EndArrow(6)); draw((0,-3)--(0,3),EndArrow(6)); draw(shift((0.01,1.42))*xscale(1.41)*yscale(1.41)*arc((0,0),1,179.76,359.76)); draw(shift((-0.01,-1.42))*xscale(1.41)*yscale(1.41)*arc((0,0),1,-0.38,179.62)); draw((-1.4,1.43)--(1.41,1.41)); draw((-1.42,-1.41)--(1.4,-1.42)); label("$ P(-1,1) $",(-2.57,2.17),SE*lsf); label("$ Q(1,1) $",(1.55,2.21),SE*lsf); label("$ R(-1,1) $",(-2.72,-1.45),SE*lsf); label("$S(1,-1)$",(1.59,-1.49),SE*lsf);
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| − | dot((0,0),ds); label("$O$",(-0.24,-0.35),NE*lsf); dot((1.41,1.41),ds); dot((-1.4,1.43),ds); dot((1.4,-1.42),ds); dot((-1.42,-1.41),ds);
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| − | clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
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| − | </asy>
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| − | <math> \textbf{(A)}\ \frac{\sqrt 2}4 \qquad\textbf{(B)}\ \frac 12 \qquad\textbf{(C)}\ \frac{2}{\pi} \qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{\sqrt 2}{2} </math>
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| − | ==Problem 24==
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| − | What is the correct ordering of the three numbers, <math>10^8</math>, <math>5^1^2</math>, and <math>2^2^4</math>?
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| − | <math> \textbf{(A)}\ 2^2^4<10^8<5^1^2 </math>
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| − | <math> \textbf{(B)}\ 2^2^4<5^1^2<10^8 </math>
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| − | <math> \textbf{(C)}\ 5^1^2<2^2^4<10^8 </math>
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| − | <math> \textbf{(D)}\ 10^8<5^1^2<2^2^4</math>
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| − | <math> \textbf{(E)}\ 10^8<2^2^4<5^1^2 </math>
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| − | | |
| − | ==Problem 25==
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| − | Everyday at school, Jo climbs a flight of <math>6</math> stairs. Joe can take the stairs <math>1,2</math>, or <math>3</math> at a time. For example, Jo could climb <math>3</math>, then <math>1</math>, then <math>2</math>. In how many ways can Jo climb the stairs?
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| − | <math> \textbf{(A)}\ 13 \qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24 </math>
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