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Difference between revisions of "2024 AMC 12B Problems"

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==Problem 1==
 
==Problem 1==
−
In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in line?
+
In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?
  
 
<math>\textbf{(A) } 2021 \qquad\textbf{(B) } 2022 \qquad\textbf{(C) } 2023 \qquad\textbf{(D) } 2024 \qquad\textbf{(E) } 2025</math>
 
<math>\textbf{(A) } 2021 \qquad\textbf{(B) } 2022 \qquad\textbf{(C) } 2023 \qquad\textbf{(D) } 2024 \qquad\textbf{(E) } 2025</math>
Line 19: Line 19:
  
 
<math>\textbf{(A) }16 \qquad\textbf{(B) }17\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21</math>
 
<math>\textbf{(A) }16 \qquad\textbf{(B) }17\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21</math>
 +
 
[[2024 AMC 12B Problems/Problem 3|Solution]]
 
[[2024 AMC 12B Problems/Problem 3|Solution]]
  
Line 40: Line 41:
 
\textbf{(E) }18 \qquad
 
\textbf{(E) }18 \qquad
 
</math>
 
</math>
 +
 
[[2024 AMC 12B Problems/Problem 5|Solution]]
 
[[2024 AMC 12B Problems/Problem 5|Solution]]
  
Line 52: Line 54:
 
\textbf{(E) }26 \qquad
 
\textbf{(E) }26 \qquad
 
</math>
 
</math>
 +
 
[[2024 AMC 12B Problems/Problem 6|Solution]]
 
[[2024 AMC 12B Problems/Problem 6|Solution]]
  
Line 57: Line 60:
 
In the figure below <math>WXYZ</math> is a rectangle with <math>WX=4</math> and <math>WZ=8</math>. Point <math>M</math> lies <math>\overline{XY}</math>, point <math>A</math> lies on <math>\overline{YZ}</math>, and <math>\angle WMA</math> is a right angle. The areas of <math>\triangle WXM</math> and <math>\triangle WAZ</math> are equal. What is the area of <math>\triangle WMA</math>?
 
In the figure below <math>WXYZ</math> is a rectangle with <math>WX=4</math> and <math>WZ=8</math>. Point <math>M</math> lies <math>\overline{XY}</math>, point <math>A</math> lies on <math>\overline{YZ}</math>, and <math>\angle WMA</math> is a right angle. The areas of <math>\triangle WXM</math> and <math>\triangle WAZ</math> are equal. What is the area of <math>\triangle WMA</math>?
  
−
[asy]
+
<asy>
 
pair X = (0, 0);
 
pair X = (0, 0);
 
pair W = (0, 4);
 
pair W = (0, 4);
 
pair Y = (8, 0);
 
pair Y = (8, 0);
 
pair Z = (8, 4);
 
pair Z = (8, 4);
−
label("<math>X</math>", X, dir(180));
+
label("$X$", X, dir(180));
−
label("<math>W</math>", W, dir(180));
+
label("$W$", W, dir(180));
−
label("<math>Y</math>", Y, dir(0));
+
label("$Y$", Y, dir(0));
−
label("<math>Z</math>", Z, dir(0));
+
label("$Z$", Z, dir(0));
  
 
draw(W--X--Y--Z--cycle);
 
draw(W--X--Y--Z--cycle);
Line 74: Line 77:
 
pair M = (2, 0);
 
pair M = (2, 0);
 
pair A = (8, 3);
 
pair A = (8, 3);
−
label("<math>A</math>", A, dir(0));
+
label("$A$", A, dir(0));
 
dot(M);
 
dot(M);
 
dot(A);
 
dot(A);
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markscalefactor = 0.05;
 
markscalefactor = 0.05;
 
draw(rightanglemark(W, M, A));
 
draw(rightanglemark(W, M, A));
−
label("<math>M</math>", M, dir(-90));
+
label("$M$", M, dir(-90));
−
[/asy]
+
</asy>
  
 
<math>
 
<math>
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\textbf{(E) }17 \qquad
 
\textbf{(E) }17 \qquad
 
</math>
 
</math>
 +
 
[[2024 AMC 12B Problems/Problem 7|Solution]]
 
[[2024 AMC 12B Problems/Problem 7|Solution]]
  
Line 101: Line 105:
 
\textbf{(E) }48\qquad
 
\textbf{(E) }48\qquad
 
</math>
 
</math>
 +
 
[[2024 AMC 12B Problems/Problem 8|Solution]]
 
[[2024 AMC 12B Problems/Problem 8|Solution]]
  
 
==Problem 9==
 
==Problem 9==
−
A dartboard is the region B in the coordinate plane consisting of points <math>(x, y)</math> such that <math>|x| + |y| \le 8</math>. A target T is the region where <math>(x^2 + y^2 - 25)^2 \le 49</math>. A dart is thrown at a random point in B. The probability that the dart lands in T can be expressed as <math>\frac{m}{n} \pi</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m + n</math>?
+
A dartboard is the region <math>B</math> in the coordinate plane consisting of points <math>(x,y)</math> such that <math>|x| + |y| \le 8</math> . A target <math>T</math> is the region where <math>(x^2 + y^2 - 25)^2 \le 49.</math> A dart is thrown and lands at a random point in <math>B</math>. The probability that the dart lands in <math>T</math> can be expressed as <math>\frac{m}{n} \cdot \pi,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m + n?</math>
  
 
<math>
 
<math>
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==Problem 10==
 
==Problem 10==
  
−
A list of 9 real numbers consists of <math>1</math>, <math>2.2 </math>, <math>3.2 </math>, <math>5.2 </math>, <math>6.2 </math>, <math>7</math>, as well as <math>x, y,z</math> with <math>x\leq y\leq z</math>. The range of the list is <math>7</math>, and the mean and median are both positive integers. How many ordered triples <math>(x,y,z)</math> are possible?
+
A list of 9 real numbers consists of <math>1</math>, <math>2.2 </math>, <math>3.2 </math>, <math>5.2 </math>, <math>6.2 </math>, and <math>7</math>, as well as <math>x, y,z</math> with <math>x\leq y\leq z</math>. The range of the list is <math>7</math>, and the mean and median are both positive integers. How many ordered triples <math>(x,y,z)</math> are possible?
  
−
<math>\textbf{(A) }1 \qquad\textbf{(B) }2 \qquad\textbf{(C) }3 \qquad\textbf{(D) }4 \qquad\textbf{(E) \text{infinitely many}}\qquad</math>
+
<math>\textbf{(A) }1 \qquad\textbf{(B) }2 \qquad\textbf{(C) }3 \qquad\textbf{(D) }4 \qquad\textbf{(E) }\text{infinitely many}\qquad</math>
  
 
[[2024 AMC 12B Problems/Problem 10|Solution]]
 
[[2024 AMC 12B Problems/Problem 10|Solution]]
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==Problem 11==
 
==Problem 11==
 
Let <math>x_{n} = \sin^2(n^\circ)</math>. What is the mean of <math>x_{1}, x_{2}, x_{3}, \cdots, x_{90}</math>?
 
Let <math>x_{n} = \sin^2(n^\circ)</math>. What is the mean of <math>x_{1}, x_{2}, x_{3}, \cdots, x_{90}</math>?
−
 
  
 
<math>
 
<math>
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==Problem 12==
 
==Problem 12==
 +
Suppose <math>z</math> is a complex number with positive imaginary part, with real part greater than <math>1</math>, and with <math>|z| = 2</math>. In the complex plane, the four points <math>0</math>, <math>z</math>, <math>z^{2}</math>, and <math>z^{3}</math> are the vertices of a quadrilateral with area <math>15</math>. What is the imaginary part of <math>z</math>?
 +
 +
<math>\textbf{(A)}~\frac{3}{4}\qquad\textbf{(B)}~1\qquad\textbf{(C)}~\frac{4}{3}\qquad\textbf{(D)}~\frac{3}{2}\qquad\textbf{(E)}~\frac{5}{3}</math>
 +
 
[[2024 AMC 12B Problems/Problem 12|Solution]]
 
[[2024 AMC 12B Problems/Problem 12|Solution]]
  
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A group of <math>16</math> people will be partitioned into <math>4</math> indistinguishable <math>4</math>-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as <math>3^{r}M</math>, where <math>r</math> and <math>M</math> are positive integers and <math>M</math> is not divisible by <math>3</math>. What is <math>r</math>?
 
A group of <math>16</math> people will be partitioned into <math>4</math> indistinguishable <math>4</math>-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as <math>3^{r}M</math>, where <math>r</math> and <math>M</math> are positive integers and <math>M</math> is not divisible by <math>3</math>. What is <math>r</math>?
−
 
  
 
<math>
 
<math>
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[[2024 AMC 12B Problems/Problem 16|Solution]]
 
[[2024 AMC 12B Problems/Problem 16|Solution]]
−
 
 
==Problem 17==
 
==Problem 17==
 
Integers <math>a</math> and <math>b</math> are randomly chosen without replacement from the set of integers with absolute value not exceeding <math>10</math>. What is the probability that the polynomial <math>x^3 + ax^2 + bx + 6</math> has <math>3</math> distinct integer roots?
 
Integers <math>a</math> and <math>b</math> are randomly chosen without replacement from the set of integers with absolute value not exceeding <math>10</math>. What is the probability that the polynomial <math>x^3 + ax^2 + bx + 6</math> has <math>3</math> distinct integer roots?
  
−
<math>\textbf{(A)} \frac{1}{240} \qquad \textbf{(B)} \frac{1}{221} \qquad \textbf{(C)} \frac{1}{105} \qquad \textbf{(D)} \frac{1}{84} \qquad \textbf{(E)} \frac{1}{63}</math>.
+
<math>\textbf{(A) }\frac{1}{240} \qquad \textbf{(B) }\frac{1}{221} \qquad \textbf{(C) }\frac{1}{105} \qquad \textbf{(D) }\frac{1}{84} \qquad \textbf{(E) }\frac{1}{63}</math>
  
 
[[2024 AMC 12B Problems/Problem 17|Solution]]
 
[[2024 AMC 12B Problems/Problem 17|Solution]]
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==Problem 19==
 
==Problem 19==
 
Equilateral <math>\triangle ABC</math> with side length <math>14</math> is rotated about its center by angle <math>\theta</math>, where <math>0 < \theta < 60^{\circ}</math>, to form <math>\triangle DEF</math>. See the figure. The area of hexagon <math>ADBECF</math> is <math>91\sqrt{3}</math>. What is <math>\tan\theta</math>?
 
Equilateral <math>\triangle ABC</math> with side length <math>14</math> is rotated about its center by angle <math>\theta</math>, where <math>0 < \theta < 60^{\circ}</math>, to form <math>\triangle DEF</math>. See the figure. The area of hexagon <math>ADBECF</math> is <math>91\sqrt{3}</math>. What is <math>\tan\theta</math>?
 +
<asy>
 +
// Credit to shihan for this diagram.
 +
 +
defaultpen(fontsize(13)); size(200);
 +
pair O=(0,0),A=dir(225),B=dir(-15),C=dir(105),D=rotate(38.21,O)*A,E=rotate(38.21,O)*B,F=rotate(38.21,O)*C;
 +
draw(A--B--C--A,gray+0.4);draw(D--E--F--D,gray+0.4); draw(A--D--B--E--C--F--A,black+0.9); dot(O); dot("$A$",A,dir(A)); dot("$B$",B,dir(B)); dot("$C$",C,dir(C)); dot("$D$",D,dir(D)); dot("$E$",E,dir(E)); dot("$F$",F,dir(F));
 +
</asy>
 +
 +
<math>\textbf{(A)}~\frac{3}{4}\qquad\textbf{(B)}~\frac{5\sqrt{3}}{11}\qquad\textbf{(C)}~\frac{4}{5}\qquad\textbf{(D)}~\frac{11}{13}\qquad\textbf{(E)}~\frac{7\sqrt{3}}{13}</math>
  
 
[[2024 AMC 12B Problems/Problem 19|Solution]]
 
[[2024 AMC 12B Problems/Problem 19|Solution]]
Line 216: Line 231:
 
\textbf{(E) }913\qquad
 
\textbf{(E) }913\qquad
 
</math>
 
</math>
 +
 
[[2024 AMC 12B Problems/Problem 20|Solution]]
 
[[2024 AMC 12B Problems/Problem 20|Solution]]
  
Line 228: Line 244:
 
\textbf{(E) }208 \qquad
 
\textbf{(E) }208 \qquad
 
</math>
 
</math>
 +
 
[[2024 AMC 12B Problems/Problem 21|Solution]]
 
[[2024 AMC 12B Problems/Problem 21|Solution]]
  
Line 240: Line 257:
 
\textbf{(E) }17 \qquad
 
\textbf{(E) }17 \qquad
 
</math>
 
</math>
 +
 
[[2024 AMC 12B Problems/Problem 22|Solution]]
 
[[2024 AMC 12B Problems/Problem 22|Solution]]
  
Line 252: Line 270:
 
\textbf{(E) }\frac{2+\sqrt2}{3} \qquad
 
\textbf{(E) }\frac{2+\sqrt2}{3} \qquad
 
</math>
 
</math>
 +
 
[[2024 AMC 12B Problems/Problem 23|Solution]]
 
[[2024 AMC 12B Problems/Problem 23|Solution]]
  
Line 264: Line 283:
 
\textbf{(E) }6\qquad
 
\textbf{(E) }6\qquad
 
</math>
 
</math>
 +
 
[[2024 AMC 12B Problems/Problem 24|Solution]]
 
[[2024 AMC 12B Problems/Problem 24|Solution]]
  

Latest revision as of 11:30, 15 November 2024

2024 AMC 12B (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

In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?

$\textbf{(A) } 2021 \qquad\textbf{(B) } 2022 \qquad\textbf{(C) } 2023 \qquad\textbf{(D) } 2024 \qquad\textbf{(E) } 2025$

Solution

Problem 2

What is $10! - 7! \cdot 6!$?

$\textbf{(A) }-120 \qquad\textbf{(B) }0 \qquad\textbf{(C) }120 \qquad\textbf{(D) }600 \qquad\textbf{(E) }720 \qquad$

Solution

Problem 3

For how many integer values of $x$ is $|2x|\leq 7\pi?$

$\textbf{(A) }16 \qquad\textbf{(B) }17\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21$

Solution

Problem 4

Balls numbered $1,2,3,\ldots$ are deposited in $5$ bins, labeled $A,B,C,D,$ and $E$, using the following procedure. Ball $1$ is deposited in bin $A$, and balls $2$ and $3$ are deposited in $B$. The next three balls are deposited in bin $C$, the next $4$ in bin $D$, and so on, cycling back to bin $A$ after balls are deposited in bin $E$. (For example, $22,23,\ldots,28$ are deposited in bin $B$ at step 7 of this process.) In which bin is ball $2024$ deposited?

$\textbf{(A) }A\qquad\textbf{(B) }B\qquad\textbf{(C) }C\qquad\textbf{(D) }D\qquad\textbf{(E) }E$

Solution

Problem 5

In the following expression, Melanie changed some of the plus signs to minus signs:\[1 + 3+5+7+\cdots+97+99\]When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?

$\textbf{(A) }14 \qquad \textbf{(B) }15 \qquad \textbf{(C) }16 \qquad \textbf{(D) }17 \qquad \textbf{(E) }18 \qquad$

Solution

Problem 6

The national debt of the United States is on track to reach $5 \cdot 10^{13}$ dollars by $2033$. How many digits does this number of dollars have when written as a numeral in base $5$? (The approximation of $\log_{10} 5$ as $0.7$ is sufficient for this problem.)

$\textbf{(A) }18 \qquad \textbf{(B) }20 \qquad \textbf{(C) }22 \qquad \textbf{(D) }24 \qquad \textbf{(E) }26 \qquad$

Solution

Problem 7

In the figure below $WXYZ$ is a rectangle with $WX=4$ and $WZ=8$. Point $M$ lies $\overline{XY}$, point $A$ lies on $\overline{YZ}$, and $\angle WMA$ is a right angle. The areas of $\triangle WXM$ and $\triangle WAZ$ are equal. What is the area of $\triangle WMA$?

[asy] pair X = (0, 0); pair W = (0, 4); pair Y = (8, 0); pair Z = (8, 4); label("$X$", X, dir(180)); label("$W$", W, dir(180)); label("$Y$", Y, dir(0)); label("$Z$", Z, dir(0)); draw(W--X--Y--Z--cycle); dot(X); dot(Y); dot(W); dot(Z); pair M = (2, 0); pair A = (8, 3); label("$A$", A, dir(0)); dot(M); dot(A); draw(W--M--A--cycle); markscalefactor = 0.05; draw(rightanglemark(W, M, A)); label("$M$", M, dir(-90)); [/asy]

$\textbf{(A) }13 \qquad \textbf{(B) }14 \qquad \textbf{(C) }15 \qquad \textbf{(D) }16 \qquad \textbf{(E) }17 \qquad$

Solution

Problem 8

What value of $x$ satisfies\[\frac{\log_2x\cdot\log_3x}{\log_2x+\log_3x}=2?\] $\textbf{(A) }25\qquad \textbf{(B) }32\qquad \textbf{(C) }36\qquad \textbf{(D) }42\qquad \textbf{(E) }48\qquad$

Solution

Problem 9

A dartboard is the region $B$ in the coordinate plane consisting of points $(x,y)$ such that $|x| + |y| \le 8$ . A target $T$ is the region where $(x^2 + y^2 - 25)^2 \le 49.$ A dart is thrown and lands at a random point in $B$. The probability that the dart lands in $T$ can be expressed as $\frac{m}{n} \cdot \pi,$ where $m$ and $n$ are relatively prime positive integers. What is $m + n?$

$\textbf{(A) }39 \qquad \textbf{(B) }71 \qquad \textbf{(C) }73 \qquad \textbf{(D) }75 \qquad \textbf{(E) }135 \qquad$

Solution

Problem 10

A list of 9 real numbers consists of $1$, $2.2$, $3.2$, $5.2$, $6.2$, and $7$, as well as $x, y,z$ with $x\leq y\leq z$. The range of the list is $7$, and the mean and median are both positive integers. How many ordered triples $(x,y,z)$ are possible?

$\textbf{(A) }1 \qquad\textbf{(B) }2 \qquad\textbf{(C) }3 \qquad\textbf{(D) }4 \qquad\textbf{(E) }\text{infinitely many}\qquad$

Solution

Problem 11

Let $x_{n} = \sin^2(n^\circ)$. What is the mean of $x_{1}, x_{2}, x_{3}, \cdots, x_{90}$?

$\textbf{(A) }\frac{11}{45} \qquad \textbf{(B) }\frac{22}{45} \qquad \textbf{(C) }\frac{89}{180} \qquad \textbf{(D) }\frac{1}{2} \qquad \textbf{(E) }\frac{91}{180} \qquad$

Solution

Problem 12

Suppose $z$ is a complex number with positive imaginary part, with real part greater than $1$, and with $|z| = 2$. In the complex plane, the four points $0$, $z$, $z^{2}$, and $z^{3}$ are the vertices of a quadrilateral with area $15$. What is the imaginary part of $z$?

$\textbf{(A)}~\frac{3}{4}\qquad\textbf{(B)}~1\qquad\textbf{(C)}~\frac{4}{3}\qquad\textbf{(D)}~\frac{3}{2}\qquad\textbf{(E)}~\frac{5}{3}$

Solution

Problem 13

There are real numbers $x,y,h$ and $k$ that satisfy the system of equations\[x^2 + y^2 - 6x - 8y = h\]\[x^2 + y^2 - 10x + 4y = k\]What is the minimum possible value of $h+k$?

$\textbf{(A) }-54 \qquad \textbf{(B) }-46 \qquad \textbf{(C) }-34 \qquad \textbf{(D) }-16 \qquad \textbf{(E) }16 \qquad$

Solution

Problem 14

How many different remainders can result when the $100$th power of an integer is divided by $125$?

$\textbf{(A) }1 \qquad\textbf{(B) }2 \qquad\textbf{(C) }5 \qquad\textbf{(D) }25 \qquad\textbf{(E) }125 \qquad$

Solution

Problem 15

A triangle in the coordinate plane has vertices $A(\log_21,\log_22)$, $B(\log_23,\log_24)$, and $C(\log_27,\log_28)$. What is the area of $\triangle ABC$?

$\textbf{(A) }\log_2\frac{\sqrt3}7\qquad \textbf{(B) }\log_2\frac3{\sqrt7}\qquad \textbf{(C) }\log_2\frac7{\sqrt3}\qquad \textbf{(D) }\log_2\frac{11}{\sqrt7}\qquad \textbf{(E) }\log_2\frac{11}{\sqrt3}\qquad$

Solution

Problem 16

A group of $16$ people will be partitioned into $4$ indistinguishable $4$-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^{r}M$, where $r$ and $M$ are positive integers and $M$ is not divisible by $3$. What is $r$?

$\textbf{(A) }5 \qquad \textbf{(B) }6 \qquad \textbf{(C) }7 \qquad \textbf{(D) }8 \qquad \textbf{(E) }9 \qquad$

Solution

Problem 17

Integers $a$ and $b$ are randomly chosen without replacement from the set of integers with absolute value not exceeding $10$. What is the probability that the polynomial $x^3 + ax^2 + bx + 6$ has $3$ distinct integer roots?

$\textbf{(A) }\frac{1}{240} \qquad \textbf{(B) }\frac{1}{221} \qquad \textbf{(C) }\frac{1}{105} \qquad \textbf{(D) }\frac{1}{84} \qquad \textbf{(E) }\frac{1}{63}$

Solution

Problem 18

The Fibonacci numbers are defined by $F_1=1,$ $F_2=1,$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3.$ What is\[\dfrac{F_2}{F_1}+\dfrac{F_4}{F_2}+\dfrac{F_6}{F_3}+\cdots+\dfrac{F_{20}}{F_{10}}?\] $\textbf{(A) }318 \qquad\textbf{(B) }319\qquad\textbf{(C) }320\qquad\textbf{(D) }321\qquad\textbf{(E) }322$

Solution

Problem 19

Equilateral $\triangle ABC$ with side length $14$ is rotated about its center by angle $\theta$, where $0 < \theta < 60^{\circ}$, to form $\triangle DEF$. See the figure. The area of hexagon $ADBECF$ is $91\sqrt{3}$. What is $\tan\theta$? [asy] // Credit to shihan for this diagram. defaultpen(fontsize(13)); size(200); pair O=(0,0),A=dir(225),B=dir(-15),C=dir(105),D=rotate(38.21,O)*A,E=rotate(38.21,O)*B,F=rotate(38.21,O)*C; draw(A--B--C--A,gray+0.4);draw(D--E--F--D,gray+0.4); draw(A--D--B--E--C--F--A,black+0.9); dot(O); dot("$A$",A,dir(A)); dot("$B$",B,dir(B)); dot("$C$",C,dir(C)); dot("$D$",D,dir(D)); dot("$E$",E,dir(E)); dot("$F$",F,dir(F)); [/asy]

$\textbf{(A)}~\frac{3}{4}\qquad\textbf{(B)}~\frac{5\sqrt{3}}{11}\qquad\textbf{(C)}~\frac{4}{5}\qquad\textbf{(D)}~\frac{11}{13}\qquad\textbf{(E)}~\frac{7\sqrt{3}}{13}$

Solution

Problem 20

Suppose $A$, $B$, and $C$ are points in the plane with $AB=40$ and $AC=42$, and let $x$ be the length of the line segment from $A$ to the midpoint of $\overline{BC}$. Define a function $f$ by letting $f(x)$ be the area of $\triangle ABC$. Then the domain of $f$ is an open interval $(p,q)$, and the maximum value $r$ of $f(x)$ occurs at $x=s$. What is $p+q+r+s$?

$\textbf{(A) }909\qquad \textbf{(B) }910\qquad \textbf{(C) }911\qquad \textbf{(D) }912\qquad \textbf{(E) }913\qquad$

Solution

Problem 21

The measures of the smallest angles of three different right triangles sum to $90^\circ$. All three triangles have side lengths that are primitive Pythagorean triples. Two of them are $3-4-5$ and $5-12-13$. What is the perimeter of the third triangle?

$\textbf{(A) }40 \qquad \textbf{(B) }126 \qquad \textbf{(C) }154 \qquad \textbf{(D) }176 \qquad \textbf{(E) }208 \qquad$

Solution

Problem 22

Let $\triangle{ABC}$ be a triangle with integer side lengths and the property that $\angle{B} = 2\angle{A}$. What is the least possible perimeter of such a triangle?

$\textbf{(A) }13 \qquad \textbf{(B) }14 \qquad \textbf{(C) }15 \qquad \textbf{(D) }16 \qquad \textbf{(E) }17 \qquad$

Solution

Problem 23

A right pyramid has regular octagon $ABCDEFGH$ with side length $1$ as its base and apex $V.$ Segments $\overline{AV}$ and $\overline{DV}$ are perpendicular. What is the square of the height of the pyramid?

$\textbf{(A) }1 \qquad \textbf{(B) }\frac{1+\sqrt2}{2} \qquad \textbf{(C) }\sqrt2 \qquad \textbf{(D) }\frac32 \qquad \textbf{(E) }\frac{2+\sqrt2}{3} \qquad$

Solution

Problem 24

What is the number of ordered triples $(a,b,c)$ of positive integers, with $a\le b\le c\le 9$, such that there exists a (non-degenerate) triangle $\triangle ABC$ with an integer inradius for which $a$, $b$, and $c$ are the lengths of the altitudes from $A$ to $\overline{BC}$, $B$ to $\overline{AC}$, and $C$ to $\overline{AB}$, respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.)

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

Solution

Problem 25

Pablo will decorate each of $6$ identical white balls with either a striped or a dotted pattern, using either red or blue paint. He will decide on the color and pattern for each ball by flipping a fair coin for each of the $12$ decisions he must make. After the paint dries, he will place the $6$ balls in an urn. Frida will randomly select one ball from the urn and note its color and pattern. The events "the ball Frida selects is red" and "the ball Frida selects is striped" may or may not be independent, depending on the outcome of Pablo's coin flips. The probability that these two events are independent can be written as $\frac mn,$ where $m$ and $n$ are relatively prime positive integers. What is $m?$ (Recall that two events $A$ and $B$ are independent if $P(A \text{ and }B) = P(A) \cdot P(B).$)

$\textbf{(A) } 243 \qquad \textbf{(B) } 245 \qquad \textbf{(C) } 247 \qquad \textbf{(D) } 249\qquad \textbf{(E) } 251$

Solution

See also

2024 AMC 12B (Problems • Answer Key • Resources)
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2024 AMC 12A Problems 		Followed by
2025 AMC 12A Problems
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All AMC 12 Problems and Solutions

AMC 12

AMC 12 Problems and Solutions

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