Found problems: 110
2024 AMC 12/AHSME, 10
Let $\alpha$ be the radian measure of the smallest angle in a $3{-}4{-}5$ right triangle. Let $\beta$ be the radian measure of the smallest angle in a $7{-}24{-}25$ right triangle. In terms of $\alpha$, what is $\beta$?
$
\textbf{(A) }\frac{\alpha}{3}\qquad
\textbf{(B) }\alpha - \frac{\pi}{8}\qquad
\textbf{(C) }\frac{\pi}{2} - 2\alpha \qquad
\textbf{(D) }\frac{\alpha}{2}\qquad
\textbf{(E) }\pi - 4\alpha\qquad
$
2021 AMC 10 Spring, 10
Which of the following is equivalent to $$(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?$$
$\textbf{(A) }3^{127}+2^{127} \qquad \textbf{(B) }3^{127}+2^{127}+2\cdot 3^{63}+3\cdot 2^{63} \qquad \textbf{(C) }3^{128}-2^{128} \qquad \textbf{(D) }3^{128}+2^{128} \qquad \textbf{(E) }5^{127}$
2021 AMC 12/AHSME Spring, 7
What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 2$
2020 AMC 12/AHSME, 10
There is a unique positive integer $n$ such that \[\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}.\]
What is the sum of the digits of $n?$
$\textbf{(A) } 4 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 13$
2013 AMC 12/AHSME, 8
Given that $x$ and $y$ are distinct nonzero real numbers such that $x+\tfrac{2}{x} = y + \tfrac{2}{y}$, what is $xy$?
$ \textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 4\qquad $
2017 AMC 12/AHSME, 2
The sum of two nonzero real numbers is $4$ times their product. What is the sum of the reciprocals of the two numbers?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12$
2017 AMC 10, 12
Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$?
$\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$
2012 AMC 10, 1
Cagney can frost a cupcake every $20$ seconds and Lacey can frost a cupcake every $30$ seconds. Working together, how many cupcakes can they frost in $5$ minutes?
$ \textbf{(A)}\ 10
\qquad\textbf{(B)}\ 15
\qquad\textbf{(C)}\ 20
\qquad\textbf{(D)}\ 25
\qquad\textbf{(E)}\ 30
$
2021 AMC 12/AHSME Spring, 14
What is the value of $$\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right)\cdot\left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)?$$
$\textbf{(A) }21 \qquad \textbf{(B) }100\log_5 3 \qquad \textbf{(C) }200\log_3 5 \qquad \textbf{(D) }2,200\qquad \textbf{(E) }21,000$
2020 AMC 12/AHSME, 15
In the complex plane, let $A$ be the set of solutions to $z^3 - 8 = 0$ and let $B$ be the set of solutions to $z^3 - 8z^2 - 8z + 64 = 0$. What is the greatest distance between a point of $A$ and a point of $B?$
$\textbf{(A) } 2\sqrt{3} \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 2\sqrt{21} \qquad \textbf{(E) } 9 + \sqrt{3}$
2020 AMC 12/AHSME, 25
The number $a = \tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying $$\lfloor x \rfloor \cdot \{x\} = a \cdot x^2$$ is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p + q?$
$\textbf{(A) } 245 \qquad \textbf{(B) } 593 \qquad \textbf{(C) } 929 \qquad \textbf{(D) } 1331 \qquad \textbf{(E) } 1332$
2012 AMC 12/AHSME, 2
Cagney can frost a cupcake every $20$ seconds and Lacey can frost a cupcake every $30$ seconds. Working together, how many cupcakes can they frost in $5$ minutes?
$ \textbf{(A)}\ 10
\qquad\textbf{(B)}\ 15
\qquad\textbf{(C)}\ 20
\qquad\textbf{(D)}\ 25
\qquad\textbf{(E)}\ 30
$
2020 AMC 10, 25
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7$. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
$\textbf{(A) } \frac{7}{36} \qquad\textbf{(B) } \frac{5}{24} \qquad\textbf{(C) } \frac{2}{9} \qquad\textbf{(D) } \frac{17}{72} \qquad\textbf{(E) } \frac{1}{4}$
2018 AMC 12/AHSME, 14
The solution to the equation $\log_{3x} 4 = \log_{2x} 8$, where $x$ is a positive real number other than $\tfrac{1}{3}$ or $\tfrac{1}{2}$, can be written as $\tfrac {p}{q}$ where $p$ and $q$ are relatively prime positive integers. What is $p + q$?
$\textbf{(A) } 5 \qquad
\textbf{(B) } 13 \qquad
\textbf{(C) } 17 \qquad
\textbf{(D) } 31 \qquad
\textbf{(E) } 35 $
2018 AMC 12/AHSME, 20
Triangle $ABC$ is an isosceles right triangle with $AB=AC=3$. Let $M$ be the midpoint of hypotenuse $\overline{BC}$. Points $I$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $AI>AE$ and $AIME$ is a cyclic quadrilateral. Given that triangle $EMI$ has area $2$, the length $CI$ can be written as $\frac{a-\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is the value of $a+b+c$?
$
\textbf{(A) }9 \qquad
\textbf{(B) }10 \qquad
\textbf{(C) }11 \qquad
\textbf{(D) }12 \qquad
\textbf{(E) }13 \qquad
$
2020 AMC 12/AHSME, 6
In the plane figure shown below, $3$ of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry$?$
[asy]
import olympiad;
unitsize(25);
filldraw((1,3)--(1,4)--(2,4)--(2,3)--cycle, gray(0.7));
filldraw((2,1)--(2,2)--(3,2)--(3,1)--cycle, gray(0.7));
filldraw((4,0)--(5,0)--(5,1)--(4,1)--cycle, gray(0.7));
for (int i = 0; i < 5; ++i) {
for (int j = 0; j < 6; ++j) {
pair A = (j,i);
}
}
for (int i = 0; i < 5; ++i) {
for (int j = 0; j < 6; ++j) {
if (j != 5) {
draw((j,i)--(j+1,i));
}
if (i != 4) {
draw((j,i)--(j,i+1));
}
}
}
[/asy]
$\textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$
2017 AMC 10, 21
A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed so that one side of the square lies on the hypotenuse of the triangle. What is $\frac{x}{y}$?
$\textbf{(A)}\ \frac{12}{13}\qquad\textbf{(B)}\ \frac{35}{37}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{37}{35}\qquad\textbf{(E)}\ \frac{13}{12}$
2019 AMC 12/AHSME, 12
Positive real numbers $x \neq 1$ and $y \neq 1$ satisfy $\log_2{x} = \log_y{16}$ and $xy = 64$. What is $(\log_2{\tfrac{x}{y}})^2$?
$\textbf{(A) } \frac{25}{2} \qquad\textbf{(B) } 20 \qquad\textbf{(C) } \frac{45}{2} \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 32$
2018 AMC 12/AHSME, 6
For positive integers $m$ and $n$ such that $m+10<n+1$, both the mean and the median of the set $\{m, m+4, m+10, n+1, n+2, 2n\}$ are equal to $n$. What is $m+n$?
$\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24$
2017 AMC 12/AHSME, 16
In the figure below, semicircles with centers at $A$ and $B$ and with radii $2$ and $1$, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter $\overline{JK}$. The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at $P$ is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle. What is the radius of the circle centered at $P$?
[asy]
size(8cm);
draw(arc((0,0),3,0,180));
draw(arc((2,0),1,0,180));
draw(arc((-1,0),2,0,180));
draw((-3,0)--(3,0));
pair P = (-1,0)+(2+6/7)*dir(36.86989);
draw(circle(P,6/7));
dot((-1,0)); dot((2,0)); dot((-3,0)); dot((3,0)); dot(P);
label("$J$",(-3,0),W);
label("$A$",(-1,0),NW);
label("$B$",(2,0),NE);
label("$K$",(3,0),E);
label("$P$",P,NW);
[/asy]
$ \textbf{(A)}\ \frac{3}{4}
\qquad \textbf{(B)}\ \frac{6}{7}
\qquad\textbf{(C)}\ \frac{1}{2}\sqrt{3}
\qquad\textbf{(D)}\ \frac{5}{8}\sqrt{2}
\qquad\textbf{(E)}\ \frac{11}{12} $
2018 AMC 12/AHSME, 8
All of the triangles in the diagram below are similar to iscoceles triangle $ABC$, in which $AB=AC$. Each of the 7 smallest triangles has area 1, and $\triangle ABC$ has area 40. What is the area of trapezoid $DBCE$?
[asy]
unitsize(5);
dot((0,0));
dot((60,0));
dot((50,10));
dot((10,10));
dot((30,30));
draw((0,0)--(60,0)--(50,10)--(30,30)--(10,10)--(0,0));
draw((10,10)--(50,10));
label("$B$",(0,0),SW);
label("$C$",(60,0),SE);
label("$E$",(50,10),E);
label("$D$",(10,10),W);
label("$A$",(30,30),N);
draw((10,10)--(15,15)--(20,10)--(25,15)--(30,10)--(35,15)--(40,10)--(45,15)--(50,10));
draw((15,15)--(45,15));
[/asy]
$\textbf{(A) } 16 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 22 \qquad \textbf{(E) } 24 $
2016 AMC 12/AHSME, 25
Let $k$ be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows. Bernardo starts by writing the smallest perfect square with $k+1$ digits. Every time Bernardo writes a number, Silvia erases the last $k$ digits of it. Bernardo then writes the next perfect square, Silvia erases the last $k$ digits of it, and this process continues until the last two numbers that remain on the board differ by at least $2$. Let $f(k)$ be the smallest positive integer not written on the board. For example, if $k = 1$, then the numbers that Bernardo writes are $16$, $25$, $36$, $49$, and $64$, and the numbers showing on the board after Silvia erases are $1$, $2$, $3$, $4$, and $6$, and thus $f(1) = 5$. What is the sum of the digits of $f(2) + f(4) + f(6) + \cdots + f(2016)$?
$\textbf{(A) } 7986 \qquad\textbf{(B) } 8002 \qquad\textbf{(C) } 8030 \qquad\textbf{(D) } 8048 \qquad\textbf{(E) } 8064$
2018 AMC 10, 7
For how many (not necessarily positive) integer values of $n$ is the value of $4000\cdot \left(\tfrac{2}{5}\right)^n$ an integer?
$
\textbf{(A) }3 \qquad
\textbf{(B) }4 \qquad
\textbf{(C) }6 \qquad
\textbf{(D) }8 \qquad
\textbf{(E) }9 \qquad
$
2021 AMC 12/AHSME Spring, 6
A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is $\frac13$. When $4$ black cards are added to the deck, the probability of choosing red becomes $\frac14$. How many cards were in the deck originally.
$\textbf{(A) }6 \qquad \textbf{(B) }9 \qquad \textbf{(C) }12 \qquad \textbf{(D) }15 \qquad \textbf{(E) }18$
2021 AMC 12/AHSME Spring, 11
A laser is placed at the point (3,5). The laser bean travels in a straight line. Larry wants the beam to hit and bounce off the $y$-axis, then hit and bounce off the $x$-axis, then hit the point $(7,5)$. What is the total distance the beam will travel along this path?
$\textbf{(A) }2\sqrt{10} \qquad \textbf{(B) }5\sqrt2 \qquad \textbf{(C) }10\sqrt2 \qquad \textbf{(D) }15\sqrt2 \qquad \textbf{(E) }10\sqrt5$