2010 AMC 8

Part of AMC 8

Subcontests

(25)

1

2010 AMC 8 #22

The hundreds digit of a three-digit number is

2

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?

<span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 2\qquad<span class='latex-bold'>(C)</span>\ 4\qquad<span class='latex-bold'>(D)</span>\ 6\qquad<span class='latex-bold'>(E)</span>\ 8

1

2010 AMC 8 - Problem 23 - The ratio of areas

POQ

ROS

pass through the center of circle

O

. What is the ratio of the combined areas of the two semicircles to the area of circle

O

? [asy] 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); 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); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

<span class='latex-bold'>(A)</span>\ \frac{\sqrt 2}4 \qquad<span class='latex-bold'>(B)</span>\ \frac 12 \qquad<span class='latex-bold'>(C)</span>\ \frac{2}{\pi} \qquad<span class='latex-bold'>(D)</span>\ \frac{2}{3}\qquad<span class='latex-bold'>(E)</span>\ \frac{\sqrt 2}{2}

1

2010 AMC 8 - Problem 19 - Area between the two circles

The two circles pictured have the same center

C

\overline{AD}

is tangent to the inner circle at

B

AC

10

\overline{AD}

16

. What is the area between the two circles?
[asy] unitsize(45); 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); 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)); 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); [/asy]

<span class='latex-bold'>(A)</span>\ 36 \pi \qquad<span class='latex-bold'>(B)</span>\ 49 \pi\qquad<span class='latex-bold'>(C)</span>\ 64 \pi\qquad<span class='latex-bold'>(D)</span>\ 81 \pi\qquad<span class='latex-bold'>(E)</span>\ 100 \pi

1

2010 AMC 8 - Problem 17 - Find the ratio XQ/QY

The diagram shows an octagon consisting of

10

unit squares. The portion below

\overline{PQ}

is a unit square and a triangle with base

5

\overline{PQ}

bisects the area of the octagon, what is the ratio

\frac{XQ}{QY}

?
[asy] 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); 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)); draw((1,1)--(1,0),linewidth(1.2pt));draw((2,2)--(2,0),linewidth(1.2pt)); draw((3,2)--(3,0),linewidth(1.2pt)); draw((4,2)--(4,0),linewidth(1.2pt)); draw((5,1)--(5,0),linewidth(1.2pt)); draw((0,0)--(5,1.5),linewidth(1.2pt)); 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); [/asy]

<span class='latex-bold'>(A)</span>\ \frac 25 \qquad <span class='latex-bold'>(B)</span>\ \frac 12 \qquad <span class='latex-bold'>(C)</span>\ \frac 35 \qquad <span class='latex-bold'>(D)</span>\ \frac 23 \qquad <span class='latex-bold'>(E)</span>\ \frac 34

1

2010 AMC 8 - Problem 3 - difference of highest and lowest

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?
[asy] import graph; size(12.5cm); real lsf=2; pathpen=linewidth(0.5); pointpen=black; pen fp = fontsize(10); pointfontpen=fp; real xmin=-1.33,xmax=11.05,ymin=-9.01,ymax=-0.44; pen ycycyc=rgb(0.55,0.55,0.55); 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); 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",(0.52,-5.77),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("

",(-.65,-3.84),SE*lsf,fp); label("

",(1.45,-5.95),SE*lsf,fp); label("

",(2.44,-5.95),SE*lsf,fp); label("

",(3.44,-5.95),SE*lsf,fp); label("

",(4.46,-5.95),SE*lsf,fp); label("

",(5.43,-5.95),SE*lsf,fp); label("

",(6.42,-5.95),SE*lsf,fp); label("

",(7.44,-5.95),SE*lsf,fp); label("

",(8.43,-5.95),SE*lsf,fp); label("

",(9.44,-5.95),SE*lsf,fp); label("

",(10.37,-5.95),SE*lsf,fp); label("Month",(5.67,-6.43),SE*lsf,fp); 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)); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]</br>

(A)\ 50 \qquad (B)\ 62 \qquad (C)\ 70 \qquad (D)\ 89 \qquad (E)\ 100$

1

2010 AMC 8 #21

Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, she read

1/5

of the pages plus

12

more, and on the second day she read

1/4

of the remaining pages plus

15

more. On the third day she read

1/3

of the remaining pages plus

18

more. She then realizes she has

62

pages left, which she finishes the next day. How many pages are in this book?

<span class='latex-bold'>(A)</span>\ 120 \qquad<span class='latex-bold'>(B)</span>\ 180\qquad<span class='latex-bold'>(C)</span>\ 240\qquad<span class='latex-bold'>(D)</span>\ 300\qquad<span class='latex-bold'>(E)</span>\ 360

1

2010 AMC 8 #24

What is the correct ordering of the three numbers,

10^8

5{}^1{}^2

2{}^2{}^4

?

<span class='latex-bold'>(A)</span>\ 2{}^2{}^4<10^8<5{}^1{}^2

<span class='latex-bold'>(B)</span>\ 2{}^2{}^4<5{}^1{}^2<10^8

<span class='latex-bold'>(C)</span>\ 5{}^1{}^2<2{}^2{}^4<10^8

<span class='latex-bold'>(D)</span>\ 10^8<5{}^1{}^2<2{}^2{}^4

<span class='latex-bold'>(E)</span>\ 10^8<2{}^2{}^4<5{}^1{}^2

1

2010 AMC 8 #18

A decorative window is made up of a rectangle with semicircles at either end. The ratio of

AD

AB

3:2

AB

is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircle.
[asy] 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)); 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); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]

<span class='latex-bold'>(A)</span>\ 2:3 \qquad<span class='latex-bold'>(B)</span>\ 3:2\qquad<span class='latex-bold'>(C)</span>\ 6:\pi \qquad<span class='latex-bold'>(D)</span>\ 9: \pi \qquad<span class='latex-bold'>(E)</span>\ 30 : \pi

1

2010 AMC 8 #20

2/5

of the people are wearing gloves, and

3/4

of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove?

<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 5\qquad<span class='latex-bold'>(C)</span>\ 8\qquad<span class='latex-bold'>(D)</span>\ 15\qquad<span class='latex-bold'>(E)</span>\ 20

1

2010 AMC 8 #25

Everyday at school, Jo climbs a flight of

6

stairs. Joe can take the stairs

1,2

3

at a time. For example, Jo could climb

3

1

2

. In how many ways can Jo climb the stairs?

<span class='latex-bold'>(A)</span>\ 13 \qquad<span class='latex-bold'>(B)</span>\ 18\qquad<span class='latex-bold'>(C)</span>\ 20\qquad<span class='latex-bold'>(D)</span>\ 22\qquad<span class='latex-bold'>(E)</span>\ 24

1

2010 AMC 8 #16

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?

<span class='latex-bold'>(A)</span>\ \frac{\sqrt{\pi}}{2} \qquad<span class='latex-bold'>(B)</span>\ \sqrt{\pi} \qquad<span class='latex-bold'>(C)</span>\ \pi \qquad<span class='latex-bold'>(D)</span>\ 2\pi \qquad<span class='latex-bold'>(E)</span>\ \pi^{2}

1

2010 AMC 8 #15

5

different colors of gumdrops.

30\%

20\%

15\%

10\%

yellow, and the other

30

gumdrops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown?

<span class='latex-bold'>(A)</span>\ 35 \qquad<span class='latex-bold'>(B)</span>\ 36\qquad<span class='latex-bold'>(C)</span>\ 42\qquad<span class='latex-bold'>(D)</span>\ 48\qquad<span class='latex-bold'>(E)</span>\ 64

1

2010 AMC 8 #14

What is the sum of the prime factors of

2010

<span class='latex-bold'>(A)</span>\ 67 \qquad<span class='latex-bold'>(B)</span>\ 75\qquad<span class='latex-bold'>(C)</span>\ 77\qquad<span class='latex-bold'>(D)</span>\ 201\qquad<span class='latex-bold'>(E)</span>\ 210

1

2010 AMC 8 #13

The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shorter side is

30\%

of the perimeter. What is the length of the longest side?

<span class='latex-bold'>(A)</span>\ 7 \qquad<span class='latex-bold'>(B)</span>\ 8\qquad<span class='latex-bold'>(C)</span>\ 9\qquad<span class='latex-bold'>(D)</span>\ 10\qquad<span class='latex-bold'>(E)</span>\ 11

1

2010 AMC 8 #12

500

balls in a large bag,

80\%

are red and the rest are blue. How many of the red balls must be removed so that

75\%

of the remaining balls are red?

<span class='latex-bold'>(A)</span>\ 25 \qquad<span class='latex-bold'>(B)</span>\ 50\qquad<span class='latex-bold'>(C)</span>\ 75\qquad<span class='latex-bold'>(D)</span>\ 100\qquad<span class='latex-bold'>(E)</span>\ 150

1

2010 AMC 8 #11

The top of one tree is

16

feet higher than the top of another tree. The height of the

2

trees are at a ratio of

3:4

. In feet, how tall is the taller tree?

<span class='latex-bold'>(A)</span>\ 48 \qquad<span class='latex-bold'>(B)</span>\ 64 \qquad<span class='latex-bold'>(C)</span>\ 80 \qquad<span class='latex-bold'>(D)</span>\ 96\qquad<span class='latex-bold'>(E)</span>\ 112

1

2010 AMC 8 #10

6

pepperoni circles will exactly fit across the diameter of a

12

-inch pizza when placed. If a total of

24

circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered with pepperoni?

<span class='latex-bold'>(A)</span>\ \frac 12 \qquad<span class='latex-bold'>(B)</span>\ \frac 23 \qquad<span class='latex-bold'>(C)</span>\ \frac 34 \qquad<span class='latex-bold'>(D)</span>\ \frac 56 \qquad<span class='latex-bold'>(E)</span>\ \frac 78

1

2010 AMC 8 #9

80\%

of the problems on a

25

90\%

40

-problem test, and

70\%

10

-problem test. What percent of all problems did Ryan answer correctly?

<span class='latex-bold'>(A)</span>\ 64 \qquad<span class='latex-bold'>(B)</span>\ 75\qquad<span class='latex-bold'>(C)</span>\ 80\qquad<span class='latex-bold'>(D)</span>\ 84\qquad<span class='latex-bold'>(E)</span>\ 86

1

2010 AMC 8 #8

As Emily is riding her bike on a long straight road, she spots Ermenson skating in the same direction

1/2

mile in front of her. After she passes him, she can see him in her rear mirror until he is

1/2

mile behind her. Emily rides at a constant rate of

12

miles per hour. Ermenson skates at a constant rate of

8

miles per hour. For how many minutes can Emily see Ermenson?

<span class='latex-bold'>(A)</span>\ 6 \qquad<span class='latex-bold'>(B)</span>\ 8\qquad<span class='latex-bold'>(C)</span>\ 12\qquad<span class='latex-bold'>(D)</span>\ 15\qquad<span class='latex-bold'>(E)</span>\ 16

1

2010 AMC 8 #7

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?

<span class='latex-bold'>(A)</span>\ 6 \qquad<span class='latex-bold'>(B)</span>\ 10\qquad<span class='latex-bold'>(C)</span>\ 15\qquad<span class='latex-bold'>(D)</span>\ 25\qquad<span class='latex-bold'>(E)</span>\ 99

1

2010 AMC 8 #5

Alice needs to replace a light bulb located

10

centimeters below the ceiling of her kitchen. The ceiling is

2.4

meters above the floor. Alice is

1.5

meters tall and can reach

46

centimeters above her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in centimeters?

<span class='latex-bold'>(A)</span>\ 32 \qquad<span class='latex-bold'>(B)</span>\ 34\qquad<span class='latex-bold'>(C)</span>\ 36\qquad<span class='latex-bold'>(D)</span>\ 38\qquad<span class='latex-bold'>(E)</span>\ 40

1

2010 AMC 8 #4

What is the sum of the mean, median, and mode of the numbers,

2,3,0,3,1,4,0,3

<span class='latex-bold'>(A)</span>\ 6.5 \qquad<span class='latex-bold'>(B)</span>\ 7\qquad<span class='latex-bold'>(C)</span>\ 7.5\qquad<span class='latex-bold'>(D)</span>\ 8.5\qquad<span class='latex-bold'>(E)</span>\ 9

1

2010 AMC 8 #2

a @ b = \frac{a\times b}{a+b}

a,b

positive integers, then what is

5 @10

<span class='latex-bold'>(A)</span>\ \frac{3}{10} \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ \frac{10}{3} \qquad<span class='latex-bold'>(E)</span>\ 50

1

2010 AMC 8 #1

At Euclid High School, the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are

11

students in Mrs. Germain's class, 8 in Mr. Newton, and

9

in Mrs. Young's class are taking the AMC

8

this year. How many mathematics students at Euclid High School are taking the contest?

<span class='latex-bold'>(A)</span>\ 26 \qquad<span class='latex-bold'>(B)</span>\ 27\qquad<span class='latex-bold'>(C)</span>\ 28\qquad<span class='latex-bold'>(D)</span>\ 29\qquad<span class='latex-bold'>(E)</span>\ 30

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2011 AMC 8 #6

Which of the following has the greatest number of line of symmetry?

<span class='latex-bold'>(A)</span>\ \text{ Equilateral Triangle}

<span class='latex-bold'>(B)</span>\ \text{Non-square rhombus}

<span class='latex-bold'>(C)</span>\ \text{Non-square rectangle}

<span class='latex-bold'>(D)</span>\ \text{Isosceles Triangle}

<span class='latex-bold'>(E)</span>\ \text{Square}