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Difference between revisions of "2015 AMC 12A Problems"
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{{AMC12 Problems|year=2015|ab=A}}
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==Problem 1==
 
==Problem 1==
  
 
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math>
 
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math>
  
−
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}}\ \frac{5}{24}\qquad\textbf{(E)}\ 25</math>
+
<math> \textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25 </math>
  
 
[[2015 AMC 12A Problems/Problem 1|Solution]]
 
[[2015 AMC 12A Problems/Problem 1|Solution]]
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Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?
 
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?
  
−
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72</math>
+
<math> \textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72 </math>
  
 
[[2015 AMC 12A Problems/Problem 2|Solution]]
 
[[2015 AMC 12A Problems/Problem 2|Solution]]
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==Problem 3==
 
==Problem 3==
  
−
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?
+
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. After he graded Payton's test, the class average became 81. What was Payton's score on the test?
  
−
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95</math>
+
<math> \textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95 </math>
  
 
[[2015 AMC 12A Problems/Problem 3|Solution]]
 
[[2015 AMC 12A Problems/Problem 3|Solution]]
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The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?
 
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?
  
−
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}}\ 2 \qquad\textbf{(E)}\ \frac52</math>
+
<math> \textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52 </math>
  
 
[[2015 AMC 12A Problems/Problem 4|Solution]]
 
[[2015 AMC 12A Problems/Problem 4|Solution]]
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<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\
 
<math> \textbf{(A)}\ \text{She rounds all three numbers up.}\\
−
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{down.}\\
+
\qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\
−
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{down.} \\
+
\qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\
−
\qquad\textbf{(D)}}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{down.}\\
+
\qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\
−
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{down.}</math>
+
\qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.} </math>
  
 
[[2015 AMC 12A Problems/Problem 5|Solution]]
 
[[2015 AMC 12A Problems/Problem 5|Solution]]
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Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?
 
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be <math>2 : 1</math>?
  
−
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math>
+
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math>
  
 
[[2015 AMC 12A Problems/Problem 6|Solution]]
 
[[2015 AMC 12A Problems/Problem 6|Solution]]
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Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?
 
Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders?
  
−
<math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math>
+
<math> \textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\
 +
\qquad\textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.} \\
 +
\qquad\textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\
 +
\qquad\textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.} \\
 +
\qquad\textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.} </math>
  
 
[[2015 AMC 12A Problems/Problem 7|Solution]]
 
[[2015 AMC 12A Problems/Problem 7|Solution]]
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The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?
 
The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>?
  
−
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}}\ \frac{16}{26} \qquad\textbf{(E)}\ \frac34</math>
+
<math> \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{25} \qquad\textbf{(E)}\ \frac34</math>
  
 
[[2015 AMC 12A Problems/Problem 8|Solution]]
 
[[2015 AMC 12A Problems/Problem 8|Solution]]
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A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?
 
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?
  
−
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}}\ \frac13 \qquad\textbf{(E)}\ \frac12</math>
+
<math> \textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12</math>
  
 
[[2015 AMC 12A Problems/Problem 9|Solution]]
 
[[2015 AMC 12A Problems/Problem 9|Solution]]
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Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?
 
Integers <math>x</math> and <math>y</math> with <math>x>y>0</math> satisfy <math>x+y+xy=80</math>. What is <math>x</math>?
  
−
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}}\ 18 \qquad\textbf{(E)}\ 26</math>
+
<math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math>
  
 
[[2015 AMC 12A Problems/Problem 10|Solution]]
 
[[2015 AMC 12A Problems/Problem 10|Solution]]
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==Problem 11==
 
==Problem 11==
  
−
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k >= 0</math> lines. How many different values of <math>k</math> are possible?
+
On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible?
  
−
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6</math>
+
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math>
  
 
[[2015 AMC 12A Problems/Problem 11|Solution]]
 
[[2015 AMC 12A Problems/Problem 11|Solution]]
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The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?
 
The parabolas <math>y=ax^2 - 2</math> and <math>y=4 - bx^2</math> intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area <math>12</math>. What is <math>a+b</math>?
  
−
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 2.5\qquad\textbf{(E)}\ 3</math>
+
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3</math>
  
 
[[2015 AMC 12A Problems/Problem 12|Solution]]
 
[[2015 AMC 12A Problems/Problem 12|Solution]]
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\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\
 
\qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\
 
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\
 
\qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\
−
\qquad\textbf{(D)}}\ \text{The sum of the scores must be at least }100\text{.}\\
+
\qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\
 
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math>
 
\qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}</math>
  
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==Problem 14==
 
==Problem 14==
  
−
What is the value of <math>a</math> for which <math>\frac{1}{\text{log}_2a} + \frac{1}{\text{log}_3a} + \frac{1}{\text{log}_4a} = 1</math>?
+
What is the value of <math>a</math> for which <math>\frac{1}{\log_2 a} + \frac{1}{\log_3 a} + \frac{1}{\log_4 a} = 1</math>?
  
−
<math> \textbf{(A)}\ 9\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}}\ 24\qquad\textbf{(E)}\ 36</math>
+
<math>\textbf{(A)}\ 9\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 36</math>
  
 
[[2015 AMC 12A Problems/Problem 14|Solution]]
 
[[2015 AMC 12A Problems/Problem 14|Solution]]
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What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?
 
What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal?
  
−
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}}\ 30\qquad\textbf{(E)}\ 104</math>
+
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math>
  
 
[[2015 AMC 12A Problems/Problem 15|Solution]]
 
[[2015 AMC 12A Problems/Problem 15|Solution]]
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Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?
 
Tetrahedron <math>ABCD</math> has <math>AB=5,AC=3,BC=4,BD=4,AD=3,</math> and <math>CD=\frac{12}{5}\sqrt{2}</math>. What is the volume of the tetrahedron?
  
−
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math>
+
<math> \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}</math>
  
 
[[2015 AMC 12A Problems/Problem 16|Solution]]
 
[[2015 AMC 12A Problems/Problem 16|Solution]]
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Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
 
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
  
−
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math>
+
<math> \textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}</math>
  
 
[[2015 AMC 12A Problems/Problem 17|Solution]]
 
[[2015 AMC 12A Problems/Problem 17|Solution]]
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The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?
 
The zeros of the function <math>f(x) = x^2-ax+2a</math> are integers. What is the sum of the possible values of <math>a</math>?
  
−
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}}\ 17 \qquad\textbf{(E)}\ 18</math>
+
<math> \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 17 \qquad\textbf{(E)}\ 18</math>
  
 
[[2015 AMC 12A Problems/Problem 18|Solution]]
 
[[2015 AMC 12A Problems/Problem 18|Solution]]
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For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?
 
For some positive integers <math>p</math>, there is a quadrilateral <math>ABCD</math> with positive integer side lengths, perimeter <math>p</math>, right angles at <math>B</math> and <math>C</math>, <math>AB=2</math>, and <math>CD=AD</math>. How many different values of <math>p<2015</math> are possible?
  
−
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math>
+
<math> \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math>
  
 
[[2015 AMC 12A Problems/Problem 19|Solution]]
 
[[2015 AMC 12A Problems/Problem 19|Solution]]
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Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?
 
Isosceles triangles <math>T</math> and <math>T'</math> are not congruent but have the same area and the same perimeter. The sides of <math>T</math> have lengths of <math>5,5,</math> and <math>8</math>, while those of <math>T'</math> have lengths of <math>a,a,</math> and <math>b</math>. Which of the following numbers is closest to <math>b</math>?
  
−
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}}\ 6 \qquad\textbf{(E)}\ 8</math>
+
<math> \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math>
  
 
[[2015 AMC 12A Problems/Problem 20|Solution]]
 
[[2015 AMC 12A Problems/Problem 20|Solution]]
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A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?
 
A circle of radius <math>r</math> passes through both foci of, and exactly four points on, the ellipse with equation <math>x^2+16y^2=16</math>. The set of all possible values of <math>r</math> is an interval <math>[a,b)</math>. What is <math>a+b</math>?
  
−
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math>
+
<math> \textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12</math>
  
 
[[2015 AMC 12A Problems/Problem 21|Solution]]
 
[[2015 AMC 12A Problems/Problem 21|Solution]]
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For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?
 
For each positive integer <math>n</math>, let <math>S(n)</math> be the number of sequences of length <math>n</math> consisting solely of the letters <math>A</math> and <math>B</math>, with no more than three <math>A</math>s in a row and no more than three <math>B</math>s in a row. What is the remainder when <math>S(2015)</math> is divided by 12?
  
−
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}}\ 8 \qquad\textbf{(E)}\ 10</math>
+
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 10</math>
  
 
[[2015 AMC 12A Problems/Problem 22|Solution]]
 
[[2015 AMC 12A Problems/Problem 22|Solution]]
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Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?
 
Let <math>S</math> be a square of side length 1. Two points are chosen independently at random on the sides of <math>S</math>. The probability that the straight-line distance between the points is at least <math>\frac12</math> is <math>\frac{a-b\pi}{c}</math>, where <math>a,b,</math> and <math>c</math> are positive integers and <math>\text{gcd}(a,b,c) = 1</math>. What is <math>a+b+c</math>?
  
−
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}}\ 62 \qquad\textbf{(E)}\ 63</math>
+
<math> \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63</math>
  
 
[[2015 AMC 12A Problems/Problem 23|Solution]]
 
[[2015 AMC 12A Problems/Problem 23|Solution]]
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==Problem 24==
 
==Problem 24==
  
−
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 <= d <= 5</math>. What is the probability that
+
Rational numbers <math>a</math> and <math>b</math> are chosen at random among all rational numbers in the interval <math>[0,2)</math> that can be written as fractions <math>\frac{n}{d}</math> where <math>n</math> and <math>d</math> are integers with <math>1 \le d \le 5</math>. What is the probability that
 
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath>
 
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath>
 
is a real number?
 
is a real number?
  
−
<math> \textbf{(A)}\ \frac{3}{5} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math>
+
<math> \textbf{(A)}\ \frac{3}{50} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}</math>
  
 
[[2015 AMC 12A Problems/Problem 24|Solution]]
 
[[2015 AMC 12A Problems/Problem 24|Solution]]
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==Problem 25==
 
==Problem 25==
  
−
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k>=1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is
+
A collection of circles in the upper half-plane, all tangent to the <math>x</math>-axis, is constructed in layers as follows. Layer <math>L_0</math> consists of two circles of radii <math>70^2</math> and <math>73^2</math> that are externally tangent. For <math>k\ge1</math>, the circles in <math>\bigcup_{j=0}^{k-1}L_j</math> are ordered according to their points of tangency with the <math>x</math>-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j</math>, and for every circle <math>C</math> denote by <math>r(C)</math> its radius. What is
 
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath>
 
<cmath>\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?</cmath>
  
−
DIAGRAM NEEDED
+
<asy>
 +
import olympiad;
 +
size(350);
 +
defaultpen(linewidth(0.7));
 +
// define a bunch of arrays and starting points
 +
pair[] coord = new pair[65];
 +
int[] trav = {32,16,8,4,2,1};
 +
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);
 +
// draw the big circles and the bottom line
 +
path arc1 = arc(coord[0],coord[0].y,260,360);
 +
path arc2 = arc(coord[64],coord[64].y,175,280);
 +
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));
 +
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));
 +
draw(arc1^^arc2);
 +
draw((-930,0)--(70^2+73^2+850,0));
 +
// We now apply the findCenter function 63 times to get
 +
// the location of the centers of all 63 constructed circles.
 +
// The complicated array setup ensures that all the circles
 +
// will be taken in the right order
 +
for(int i = 0;i<=5;i=i+1)
 +
{
 +
int skip = trav[i];
 +
for(int k=skip;k<=64 - skip; k = k + 2*skip)
 +
{
 +
pair cent1 = coord[k-skip], cent2 = coord[k+skip];
 +
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);
 +
real shiftx = cent1.x + sqrt(4*r1*rn);
 +
coord[k] = (shiftx,rn);
 +
}
 +
// Draw the remaining 63 circles
 +
}
 +
for(int i=1;i<=63;i=i+1)
 +
{
 +
filldraw(circle(coord[i],coord[i].y),gray(0.75));
 +
}
 +
</asy>
 +
 
 +
 
 +
<math> \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73}\qquad\textbf{(D)}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}</math>
  
−
<math> \textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73} \qquad\textbf{(D)}}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}</math>
 
  
 
[[2015 AMC 12A Problems/Problem 25|Solution]]
 
[[2015 AMC 12A Problems/Problem 25|Solution]]
 +
 
== See also ==
 
== See also ==
 +
{{AMC12 box|year=2015|ab=A|before=[[2014 AMC 12B Problems]]|after=[[2015 AMC 12B Problems]]}}
 +
 
* [[AMC Problems and Solutions]]
 
* [[AMC Problems and Solutions]]
 +
 +
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 11:33, 24 August 2023

2015 AMC 12A (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the test if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 minutes working time to complete the test.
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

Problem 1

What is the value of $(2^0-1+5^2-0)^{-1}\times5?$

$\textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}\ \frac{5}{24}\qquad\textbf{(E)}\ 25$

Solution

Problem 2

Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?

$\textbf{(A)}\ 52\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 67\qquad\textbf{(E)}\ 72$

Solution

Problem 3

Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. After he graded Payton's test, the class average became 81. What was Payton's score on the test?

$\textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 94\qquad\textbf{(E)}\ 95$

Solution

Problem 4

The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?

$\textbf{(A)}\ \frac54 \qquad\textbf{(B)}\ \frac32 \qquad\textbf{(C)}\ \frac95 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \frac52$

Solution

Problem 5

Amelia needs to estimate the quantity $\frac{a}{b} - c$, where $a, b,$ and $c$ are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of $\frac{a}{b} - c$?

$\textbf{(A)}\ \text{She rounds all three numbers up.}\\ \qquad\textbf{(B)}\ \text{She rounds } a \text{ and } b \text{ up, and she rounds } c \text{ down.}\\ \qquad\textbf{(C)}\ \text{She rounds } a \text{ and } c \text{ up, and she rounds } b \text{ down.} \\ \qquad\textbf{(D)}\ \text{She rounds } a \text{ up, and she rounds } b \text{ and } c \text{ down.}\\ \qquad\textbf{(E)}\ \text{She rounds } c \text{ up, and she rounds } a \text{ and } b \text{ down.}$

Solution

Problem 6

Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2 : 1$?

$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8$

Solution

Problem 7

Two right circular cylinders have the same volume. The radius of the second cylinder is $10\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders?

$\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \qquad\textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.} \\ \qquad\textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \qquad\textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.} \\ \qquad\textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}$

Solution

Problem 8

The ratio of the length to the width of a rectangle is $4$ : $3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$?

$\textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac37 \qquad\textbf{(C)}\ \frac{12}{25} \qquad\textbf{(D)}\ \frac{16}{25} \qquad\textbf{(E)}\ \frac34$

Solution

Problem 9

A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?

$\textbf{(A)}\ \frac{1}{10} \qquad\textbf{(B)}\ \frac16 \qquad\textbf{(C)}\ \frac15 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac12$

Solution

Problem 10

Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=80$. What is $x$?

$\textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26$

Solution

Problem 11

On a sheet of paper, Isabella draws a circle of radius $2$, a circle of radius $3$, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly $k \ge 0$ lines. How many different values of $k$ are possible?

$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

Solution

Problem 12

The parabolas $y=ax^2 - 2$ and $y=4 - bx^2$ intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area $12$. What is $a+b$?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 1.5\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 2.5\qquad\textbf{(E)}\ 3$

Solution

Problem 13

A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?

$\textbf{(A)}\ \text{There must be an even number of odd scores.}\\ \qquad\textbf{(B)}\ \text{There must be an even number of even scores.}\\ \qquad\textbf{(C)}\ \text{There cannot be two scores of }0\text{.}\\ \qquad\textbf{(D)}\ \text{The sum of the scores must be at least }100\text{.}\\ \qquad\textbf{(E)}\ \text{The highest score must be at least }12\text{.}$

Solution

Problem 14

What is the value of $a$ for which $\frac{1}{\log_2 a} + \frac{1}{\log_3 a} + \frac{1}{\log_4 a} = 1$?

$\textbf{(A)}\ 9\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 36$

Solution

Problem 15

What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{123456789}{2^{26}\cdot 5^4}$ as a decimal?

$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104$

Solution

Problem 16

Tetrahedron $ABCD$ has $AB=5,AC=3,BC=4,BD=4,AD=3,$ and $CD=\frac{12}{5}\sqrt{2}$. What is the volume of the tetrahedron?

$\textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ \frac{24}{5}\qquad\textbf{(D)}\ 3\sqrt{3}\qquad\textbf{(E)}\ \frac{24}{5}\sqrt{2}$

Solution

Problem 17

Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?

$\textbf{(A)}\ \frac{47}{256} \qquad\textbf{(B)}\ \frac{3}{16} \qquad\textbf{(C)}\ \frac{49}{256} \qquad\textbf{(D)}\ \frac{25}{128} \qquad\textbf{(E)}\ \frac{51}{256}$

Solution

Problem 18

The zeros of the function $f(x) = x^2-ax+2a$ are integers. What is the sum of the possible values of $a$?

$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 17 \qquad\textbf{(E)}\ 18$

Solution

Problem 19

For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible?

$\textbf{(A)}\ 30 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63$

Solution

Problem 20

Isosceles triangles $T$ and $T'$ are not congruent but have the same area and the same perimeter. The sides of $T$ have lengths of $5,5,$ and $8$, while those of $T'$ have lengths of $a,a,$ and $b$. Which of the following numbers is closest to $b$?

$\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8$

Solution

Problem 21

A circle of radius $r$ passes through both foci of, and exactly four points on, the ellipse with equation $x^2+16y^2=16$. The set of all possible values of $r$ is an interval $[a,b)$. What is $a+b$?

$\textbf{(A)}\ 5\sqrt{2}+4 \qquad\textbf{(B)}\ \sqrt{17}+7 \qquad\textbf{(C)}\ 6\sqrt{2}+3 \qquad\textbf{(D)}\ \sqrt{15}+8 \qquad\textbf{(E)}\ 12$

Solution

Problem 22

For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in a row. What is the remainder when $S(2015)$ is divided by 12?

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 10$

Solution

Problem 23

Let $S$ be a square of side length 1. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\frac12$ is $\frac{a-b\pi}{c}$, where $a,b,$ and $c$ are positive integers and $\text{gcd}(a,b,c) = 1$. What is $a+b+c$?

$\textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ 63$

Solution

Problem 24

Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\frac{n}{d}$ where $n$ and $d$ are integers with $1 \le d \le 5$. What is the probability that \[(\text{cos}(a\pi)+i\text{sin}(b\pi))^4\] is a real number?

$\textbf{(A)}\ \frac{3}{50} \qquad\textbf{(B)}\ \frac{4}{25} \qquad\textbf{(C)}\ \frac{41}{200} \qquad\textbf{(D)}\ \frac{6}{25} \qquad\textbf{(E)}\ \frac{13}{50}$

Solution

Problem 25

A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k\ge1$, the circles in $\bigcup_{j=0}^{k-1}L_j$ are ordered according to their points of tangency with the $x$-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer $L_k$ consists of the $2^{k-1}$ circles constructed in this way. Let $S=\bigcup_{j=0}^{6}L_j$, and for every circle $C$ denote by $r(C)$ its radius. What is \[\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?\]

[asy] import olympiad; size(350); defaultpen(linewidth(0.7)); // define a bunch of arrays and starting points pair[] coord = new pair[65]; int[] trav = {32,16,8,4,2,1}; coord[0] = (0,73^2); coord[64] = (2*73*70,70^2); // draw the big circles and the bottom line path arc1 = arc(coord[0],coord[0].y,260,360); path arc2 = arc(coord[64],coord[64].y,175,280); fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75)); fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75)); draw(arc1^^arc2); draw((-930,0)--(70^2+73^2+850,0)); // We now apply the findCenter function 63 times to get // the location of the centers of all 63 constructed circles. // The complicated array setup ensures that all the circles // will be taken in the right order for(int i = 0;i<=5;i=i+1) { int skip = trav[i]; for(int k=skip;k<=64 - skip; k = k + 2*skip) { pair cent1 = coord[k-skip], cent2 = coord[k+skip]; real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2); real shiftx = cent1.x + sqrt(4*r1*rn); coord[k] = (shiftx,rn); } // Draw the remaining 63 circles } for(int i=1;i<=63;i=i+1) { filldraw(circle(coord[i],coord[i].y),gray(0.75)); } [/asy]


$\textbf{(A)}\ \frac{286}{35} \qquad\textbf{(B)}\ \frac{583}{70} \qquad\textbf{(C)}\ \frac{715}{73}\qquad\textbf{(D)}\ \frac{143}{14} \qquad\textbf{(E)}\ \frac{1573}{146}$


Solution

See also

2015 AMC 12A (Problems • Answer Key • Resources)
Preceded by
2014 AMC 12B Problems 		Followed by
2015 AMC 12B Problems
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
All AMC 12 Problems and Solutions


The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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