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| − | ==Problem 1==
| + | '''2008 AMC 8''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution. |
| − | Susan had 50 dollars to spend at the carnival. She spent 12 dollars on food and twice as much on rides. How many dollars did she have left to spend?
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| | | | |
| − | <math>\textbf{(A)}\ 12 \qquad
| + | * [[2008 AMC 8 Answer Key]] |
| − | \textbf{(B)}\ 14 \qquad
| + | * [[2008 AMC 8 Problems]] |
| − | \textbf{(C)}\ 26 \qquad
| + | ** [[2008 AMC 8 Problems/Problem 1]] |
| − | \textbf{(D)}\ 38 \qquad</math>
| + | ** [[2008 AMC 8 Problems/Problem 2]] |
| − | | + | ** [[2008 AMC 8 Problems/Problem 3]] |
| − | ==Problem 2==
| + | ** [[2008 AMC 8 Problems/Problem 4]] |
| − | The ten-letter code <math>\text{BEST OF LUCK}</math> represents the ten digits <math>0-9</math>, in order. What 4-digit number is represented by the code word <math>\text{CLUE}</math>?
| + | ** [[2008 AMC 8 Problems/Problem 5]] |
| − | | + | ** [[2008 AMC 8 Problems/Problem 6]] |
| − | <math>\textbf{(A)}\ 8671 \qquad
| + | ** [[2008 AMC 8 Problems/Problem 7]] |
| − | \textbf{(B)}\ 8672 \qquad
| + | ** [[2008 AMC 8 Problems/Problem 8]] |
| − | \textbf{(C)}\ 9781 \qquad
| + | ** [[2008 AMC 8 Problems/Problem 9]] |
| − | \textbf{(D)}\ 9782 \qquad
| + | ** [[2008 AMC 8 Problems/Problem 10]] |
| − | \textbf{(E)}\ 9872</math>
| + | ** [[2008 AMC 8 Problems/Problem 11]] |
| − | | + | ** [[2008 AMC 8 Problems/Problem 12]] |
| − | ==Problem 3==
| + | ** [[2008 AMC 8 Problems/Problem 13]] |
| − | If February is a month that contains Friday the <math>13^{\text{th}}</math>, what day of the week is February 1?
| + | ** [[2008 AMC 8 Problems/Problem 14]] |
| − | | + | ** [[2008 AMC 8 Problems/Problem 15]] |
| − | <math>\textbf{(A)}\ \text{Sunday} \qquad
| + | ** [[2008 AMC 8 Problems/Problem 16]] |
| − | \textbf{(B)}\ \text{Monday} \qquad
| + | ** [[2008 AMC 8 Problems/Problem 17]] |
| − | \textbf{(C)}\ \text{Wednesday} \qquad
| + | ** [[2008 AMC 8 Problems/Problem 18]] |
| − | \textbf{(D)}\ \text{Thursday}\qquad
| + | ** [[2008 AMC 8 Problems/Problem 19]] |
| − | \textbf{(E)}\ \text{Saturday} </math>
| + | ** [[2008 AMC 8 Problems/Problem 20]] |
| − | | + | ** [[2008 AMC 8 Problems/Problem 21]] |
| − | ==Problem 4==
| + | ** [[2008 AMC 8 Problems/Problem 22]] |
| − | In the figure, the outer equilateral triangle has area <math>16</math>, the inner equilateral triangle has area <math>1</math>, and the three trapezoids are congruent. What is the area of one of the trapezoids?
| + | ** [[2008 AMC 8 Problems/Problem 23]] |
| − | <asy>
| + | ** [[2008 AMC 8 Problems/Problem 24]] |
| − | size((70));
| + | ** [[2008 AMC 8 Problems/Problem 25]] |
| − | draw((0,0)--(7.5,13)--(15,0)--(0,0));
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| − | draw((1.88,3.25)--(9.45,3.25));
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| − | draw((11.2,0)--(7.5,6.5));
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| − | draw((9.4,9.7)--(5.6,3.25));
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| − | </asy>
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| − | <math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7</math>
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| − | | |
| − | ==Problem 5==
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| − | Barney Schwinn notices that the odometer on his bicycle reads <math>1441</math>, a palindrome, because it reads the same forward and backward. After riding <math>4</math> more hours that day and <math>6</math> the next, he notices that the odometer shows another palindrome, <math>1661</math>. What was his average speed in miles per hour?
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| − | | |
| − | <math>\textbf{(A)}\ 15\qquad
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| − | \textbf{(B)}\ 16\qquad
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| − | \textbf{(C)}\ 18\qquad
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| − | \textbf{(D)}\ 20\qquad
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| − | \textbf{(E)}\ 22</math>
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| − | | |
| − | ==Problem 6==
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| − | In the figure, what is the ratio of the area of the gray squares to the area of the white squares?
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| − | <asy>
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| − | size((70));
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| − | draw((10,0)--(0,10)--(-10,0)--(0,-10)--(10,0));
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| − | draw((-2.5,-7.5)--(7.5,2.5));
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| − | draw((-5,-5)--(5,5));
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| − | draw((-7.5,-2.5)--(2.5,7.5));
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| − | draw((-7.5,2.5)--(2.5,-7.5));
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| − | draw((-5,5)--(5,-5));
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| − | draw((-2.5,7.5)--(7.5,-2.5));
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| − | fill((-10,0)--(-7.5,2.5)--(-5,0)--(-7.5,-2.5)--cycle, gray);
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| − | fill((-5,0)--(0,5)--(5,0)--(0,-5)--cycle, gray);
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| − | fill((5,0)--(7.5,2.5)--(10,0)--(7.5,-2.5)--cycle, gray);
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| − | </asy>
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| − | <math> \textbf{(A)}\ 3:10 \qquad\textbf{(B)}\ 3:8 \qquad\textbf{(C)}\ 3:7 \qquad\textbf{(D)}\ 3:5 \qquad\textbf{(E)}\ 1:1 </math>
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| − | | |
| − | ==Problem 7==
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| − | If <math>\frac{3}{5}=\frac{M}{45}=\frac{60}{N}</math>, what is <math>M+N</math>?
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| − | | |
| − | <math>\textbf{(A)}\ 27\qquad
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| − | \textbf{(B)}\ 29 \qquad
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| − | \textbf{(C)}\ 45 \qquad
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| − | \textbf{(D)}\ 105\qquad
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| − | \textbf{(E)}\ 127</math>
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| − | | |
| − | ==Problem 8==
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| − | Candy sales from the Boosters Club from January through April are shown. What were the average sales per month in dollars?
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| − | <asy>
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| − | draw((0,0)--(36,0)--(36,24)--(0,24)--cycle);
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| − | draw((0,4)--(36,4));
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| − | draw((0,8)--(36,8));
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| − | draw((0,12)--(36,12));
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| − | draw((0,16)--(36,16));
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| − | draw((0,20)--(36,20));
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| − | fill((4,0)--(8,0)--(8,20)--(4,20)--cycle, black);
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| − | fill((12,0)--(16,0)--(16,12)--(12,12)--cycle, black);
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| − | fill((20,0)--(24,0)--(24,8)--(20,8)--cycle, black);
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| − | fill((28,0)--(32,0)--(32,24)--(28,24)--cycle, black);
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| − | label("\$120", (0,24), W);
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| − | label("\$80", (0,16), W);
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| − | label("\$40", (0,8), W);
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| − | label("Jan", (6,0), S);
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| − | label("Feb", (14,0), S);
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| − | label("Mar", (22,0), S);
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| − | label("Apr", (30,0), S);
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| − | </asy>
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| − | <math> \textbf{(A)}\ 60\qquad\textbf{(B)}\ 70\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 80\qquad\textbf{(E)}\ 85 </math>
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| − | | |
| − | ==Problem 9==
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| − | In <math>2005</math> Tycoon Tammy invested <math>100</math> dollars for two years. During the the first year
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| − | her investment suffered a <math>15\%</math> loss, but during the second year the remaining
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| − | investment showed a <math>20\%</math> gain. Over the two-year period, what was the change
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| − | in Tammy's investment?
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| − | | |
| − | <math>\textbf{(A)}\ 5\%\text{ loss}\qquad
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| − | \textbf{(B)}\ 2\%\text{ loss}\qquad
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| − | \textbf{(C)}\ 1\%\text{ gain}\qquad
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| − | \textbf{(D)}\ 2\% \text{ gain} \qquad
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| − | \textbf{(E)}\ 5\%\text{ gain}</math>
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| − | | |
| − | ==Problem 10==
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| − | The average age of the <math>6</math> people in Room A is <math>40</math>. The average age of the <math>4</math> people in Room B is <math>25</math>. If the two groups are combined, what is the average age of all the people?
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| − | | |
| − | <math>\textbf{(A)}\ 32.5 \qquad
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| − | \textbf{(B)}\ 33 \qquad
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| − | \textbf{(C)}\ 33.5 \qquad
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| − | \textbf{(D)}\ 34\qquad
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| − | \textbf{(E)}\ 35</math>
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| − | | |
| − | ==Problem 11==
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| − | Each of the <math>39</math> students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and <math>26</math> students have a cat. How many students have both a dog and a cat?
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| − | | |
| − | <math>\textbf{(A)}\ 7\qquad
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| − | \textbf{(B)}\ 13\qquad
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| − | \textbf{(C)}\ 19\qquad
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| − | \textbf{(D)}\ 39\qquad
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| − | \textbf{(E)}\ 46</math>
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| − | | |
| − | ==Problem 12==
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| − | A ball is dropped from a height of <math>3</math> meters. On its first bounce it rises to a height of <math>2</math> meters. It keeps falling and bouncing to <math>\frac{2}{3}</math> of the height it reached in the previous bounce. On which bounce will it not rise to a height of <math>0.5</math> meters?
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| − | | |
| − | <math>\textbf{(A)}\ 3 \qquad
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| − | \textbf{(B)}\ 4 \qquad
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| − | \textbf{(C)}\ 5 \qquad
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| − | \textbf{(D)}\ 6 \qquad
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| − | \textbf{(E)}\ 7</math>
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| − | | |
| − | ==Problem 13==
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| − | Mr. Harman needs to know the combined weight in pounds of three boxes he wants to mail. However, the only available scale is not accurate for weights less than <math>100</math> pounds or more than <math>150</math> pounds. So the boxes are weighed in pairs in every possible way. The results are <math>122</math>, <math>125</math> and <math>127</math> pounds. What is the combined weight in pounds of the three boxes?
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| − | | |
| − | <math>\textbf{(A)}\ 160\qquad
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| − | \textbf{(B)}\ 170\qquad
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| − | \textbf{(C)}\ 187\qquad
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| − | \textbf{(D)}\ 195\qquad
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| − | \textbf{(E)}\ 354</math>
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| − | | |
| − | ==Problem 14==
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| − | Three <math>\text{A's}</math>, three <math>\text{B's}</math>, and three <math>\text{C's}</math> are placed in the nine spaces so that each row and column contain one of each letter. If <math>\text{A}</math> is placed in the upper left corner, how many arrangements are possible?
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| − | <asy>
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| − | size((80));
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| − | draw((0,0)--(9,0)--(9,9)--(0,9)--(0,0));
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| − | draw((3,0)--(3,9));
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| − | draw((6,0)--(6,9));
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| − | draw((0,3)--(9,3));
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| − | draw((0,6)--(9,6));
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| − | label("A", (1.5,7.5));
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| − | </asy>
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| − | <math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6 </math>
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| − | | |
| − | ==Problem 15==
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| − | In Theresa's first <math>8</math> basketball games, she scored <math>7, 4, 3, 6, 8, 3, 1</math> and <math>5</math> points. In her ninth game, she scored fewer than <math>10</math> points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than <math>10</math> points and her points-per-game average for the <math>10</math> games was also an integer. What is the product of the number of points she scored in the ninth and tenth games?
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| − | | |
| − | <math>\textbf{(A)}\ 35\qquad
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| − | \textbf{(B)}\ 40\qquad
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| − | \textbf{(C)}\ 48\qquad
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| − | \textbf{(D)}\ 56\qquad
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| − | \textbf{(E)}\ 72</math>
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| − | | |
| − | ==Problem 16==
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| − | A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?
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| − | | |
| − | <asy>
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| − | import three;
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| − | defaultpen(linewidth(0.8));
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| − | real r=0.5;
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| − | currentprojection=orthographic(1,1/2,1/4);
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| − | draw(unitcube, white, thick(), nolight);
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| − | draw(shift(1,0,0)*unitcube, white, thick(), nolight);
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| − | draw(shift(1,-1,0)*unitcube, white, thick(), nolight);
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| − | draw(shift(1,0,-1)*unitcube, white, thick(), nolight);
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| − | draw(shift(2,0,0)*unitcube, white, thick(), nolight);
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| − | draw(shift(1,1,0)*unitcube, white, thick(), nolight);
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| − | draw(shift(1,0,1)*unitcube, white, thick(), nolight);</asy>
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| − | | |
| − | <math>\textbf{(A)} \:1 : 6 \qquad\textbf{ (B)}\: 7 : 36 \qquad\textbf{(C)}\: 1 : 5 \qquad\textbf{(D)}\: 7 : 30\qquad\textbf{ (E)}\: 6 : 25</math>
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| − | | |
| − | ==Problem 17==
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| − | Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of <math>50</math> units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles?
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| − | | |
| − | <math>\textbf{(A)}\ 76\qquad
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| − | \textbf{(B)}\ 120\qquad
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| − | \textbf{(C)}\ 128\qquad
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| − | \textbf{(D)}\ 132\qquad
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| − | \textbf{(E)}\ 136</math>
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| − | | |
| − | ==Problem 18==
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| − | Two circles that share the same center have radii <math>10</math> meters and <math>20</math> meters. An aardvark runs along the path shown, starting at <math>A</math> and ending at <math>K</math>. How many meters does the aardvark run?
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| − | <asy>
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| − | size((150));
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| − | draw((10,0)..(0,10)..(-10,0)..(0,-10)..cycle);
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| − | draw((20,0)..(0,20)..(-20,0)..(0,-20)..cycle);
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| − | draw((20,0)--(-20,0));
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| − | draw((0,20)--(0,-20));
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| − | draw((-2,21.5)..(-15.4, 15.4)..(-22,0), EndArrow);
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| − | draw((-18,1)--(-12, 1), EndArrow);
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| − | draw((-12,0)..(-8.3,-8.3)..(0,-12), EndArrow);
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| − | draw((1,-9)--(1,9), EndArrow);
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| − | draw((0,12)..(8.3, 8.3)..(12,0), EndArrow);
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| − | draw((12,-1)--(18,-1), EndArrow);
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| − | label("$A$", (0,20), N);
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| − | label("$K$", (20,0), E);
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| − | </asy>
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| − | <math> \textbf{(A)}\ 10\pi+20\qquad\textbf{(B)}\ 10\pi+30\qquad\textbf{(C)}\ 10\pi+40\qquad\textbf{(D)}\ 20\pi+20\qquad \\ \textbf{(E)}\ 20\pi+40</math>
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| − | | |
| − | ==Problem 19==
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| − | Eight points are spaced around at intervals of one unit around a <math>2 \times 2</math> square, as shown. Two of the <math>8</math> points are chosen at random. What is the probability that the two points are one unit apart?
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| − | <asy>
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| − | size((50));
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| − | dot((5,0));
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| − | dot((5,5));
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| − | dot((0,5));
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| − | dot((-5,5));
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| − | dot((-5,0));
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| − | dot((-5,-5));
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| − | dot((0,-5));
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| − | dot((5,-5));
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| − | </asy>
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| − | <math> \textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{2}{7}\qquad\textbf{(C)}\ \frac{4}{11}\qquad\textbf{(D)}\ \frac{1}{2}\qquad\textbf{(E)}\ \frac{4}{7} </math>
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| − | | |
| − | ==Problem 20==
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| − | The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and <math>\frac{3}{4}</math> of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class?
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| − | | |
| − | <math>\textbf{(A)}\ 12\qquad
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| − | \textbf{(B)}\ 17\qquad
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| − | \textbf{(C)}\ 24\qquad
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| − | \textbf{(D)}\ 27\qquad
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| − | \textbf{(E)}\ 36</math>
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| − | | |
| − | ==Problem 21==
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| − | Jerry cuts a wedge from a <math>6</math>-cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters?
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| − | <asy>
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| − | defaultpen(linewidth(0.65));
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| − | real d=90-63.43494882;
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| − | draw(ellipse((origin), 2, 4));
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| − | fill((0,4)--(0,-4)--(-8,-4)--(-8,4)--cycle, white);
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| − | draw(ellipse((-4,0), 2, 4));
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| − | draw((0,4)--(-4,4));
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| − | draw((0,-4)--(-4,-4));
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| − | draw(shift(-2,0)*rotate(-d-5)*ellipse(origin, 1.82, 4.56), linetype("10 10"));
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| − | draw((-4,4)--(-8,4), dashed);
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| − | draw((-4,-4)--(-8,-4), dashed);
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| − | draw((-4,4.3)--(-4,5));
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| − | draw((0,4.3)--(0,5));
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| − | draw((-7,4)--(-7,-4), Arrows(5));
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| − | draw((-4,4.7)--(0,4.7), Arrows(5));
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| − | label("$8$ cm", (-7,0), W);
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| − | label("$6$ cm", (-2,4.7), N);</asy>
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| − | | |
| − | <math>\textbf{(A)} 48 \qquad
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| − | \textbf{(B)} 75 \qquad
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| − | \textbf{(C)}151\qquad
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| − | \textbf{(D)}192 \qquad
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| − | \textbf{(E)}603</math>
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| − | | |
| − | ==Problem 22==
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| − | For how many positive integer values of <math>n</math> are both <math>\frac{n}{3}</math> and <math>3n</math> three-digit whole numbers?
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| − | | |
| − | <math>\textbf{(A)}\ 12\qquad
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| − | \textbf{(B)}\ 21\qquad
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| − | \textbf{(C)}\ 27\qquad
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| − | \textbf{(D)}\ 33\qquad
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| − | \textbf{(E)}\ 34</math>
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| − | | |
| − | ==Problem 23==
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| − | In square <math>ABCE</math>, <math>AF=2FE</math> and <math>CD=2DE</math>. What is the ratio of the area of <math>\triangle BFD</math> to the area of square <math>ABCE</math>?
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| − | <asy>
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| − | size((100));
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| − | draw((0,0)--(9,0)--(9,9)--(0,9)--cycle);
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| − | draw((3,0)--(9,9)--(0,3)--cycle);
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| − | dot((3,0));
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| − | dot((0,3));
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| − | dot((9,9));
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| − | dot((0,0));
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| − | dot((9,0));
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| − | dot((0,9));
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| − | label("$A$", (0,9), NW);
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| − | label("$B$", (9,9), NE);
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| − | label("$C$", (9,0), SE);
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| − | label("$D$", (3,0), S);
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| − | label("$E$", (0,0), SW);
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| − | label("$F$", (0,3), W);
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| − | </asy>
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| − | <math> \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{2}{9}\qquad\textbf{(C)}\ \frac{5}{18}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{7}{20} </math>
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| − | | |
| − | ==Problem 24==
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| − | Ten tiles numbered <math>1</math> through <math>10</math> are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?
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| − | | |
| − | <math>\textbf{(A)}\ \frac{1}{10}\qquad
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| − | \textbf{(B)}\ \frac{1}{6}\qquad
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| − | \textbf{(C)}\ \frac{11}{60}\qquad
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| − | \textbf{(D)}\ \frac{1}{5}\qquad
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| − | \textbf{(E)}\ \frac{7}{30}</math>
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| − | | |
| − | ==Problem 25==
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| − | Margie's winning art design is shown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Approximately what percent of the design is black?
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| − | | |
| − | <asy>
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| − | real d=320;
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| − | pair O=origin;
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| − | pair P=O+8*dir(d);
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| − | pair A0 = origin;
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| − | pair A1 = O+1*dir(d);
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| − | pair A2 = O+2*dir(d);
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| − | pair A3 = O+3*dir(d);
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| − | pair A4 = O+4*dir(d);
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| − | pair A5 = O+5*dir(d);
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| − | filldraw(Circle(A0, 6), white, black);
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| − | filldraw(circle(A1, 5), black, black);
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| − | filldraw(circle(A2, 4), white, black);
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| − | filldraw(circle(A3, 3), black, black);
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| − | filldraw(circle(A4, 2), white, black);
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| − | filldraw(circle(A5, 1), black, black);
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| − | </asy>
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| − | <math> \textbf{(A)}\ 42\qquad \textbf{(B)}\ 44\qquad \textbf{(C)}\ 45\qquad \textbf{(D)}\ 46\qquad \textbf{(E)}\ 48\qquad</math>
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