problem stringlengths 28 991 | solution stringlengths 11 36 | source stringclasses 5 values | year int64 2.02k 2.03k | exam_type stringclasses 2 values |
|---|---|---|---|---|
Cities $A$ and $B$ are $45$ miles apart. Alicia lives in $A$ and Beth lives in $B$. Alicia bikes towards $B$ at 18 miles per hour. Leaving at the same time, Beth bikes toward $A$ at 12 miles per hour. How many miles from City $A$ will they be when they meet? | $\boxed{27.0}$ | AMC 2023 | 2,023 | AMC |
Positive real numbers $x$ and $y$ satisfy $y^3=x^2$ and $(y-x)^2=4y^2$. What is $x+y$? | $\boxed{36.0}$ | AMC 2023 | 2,023 | AMC |
What is the degree measure of the acute angle formed by lines with slopes $2$ and $\frac{1}{3}$? | $\boxed{45.0}$ | AMC 2023 | 2,023 | AMC |
What is the value of
\[2^3 - 1^3 + 4^3 - 3^3 + 6^3 - 5^3 + \dots + 18^3 - 17^3?\] | $\boxed{3159.0}$ | AMC 2023 | 2,023 | AMC |
In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was $40\%$ more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played? | $\boxed{36.0}$ | AMC 2023 | 2,023 | AMC |
How many complex numbers satisfy the equation $z^5=\overline{z}$, where $\overline{z}$ is the conjugate of the complex number $z$? | $\boxed{7.0}$ | AMC 2023 | 2,023 | AMC |
Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\tfrac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | $\boxed{21.0}$ | AMC 2023 | 2,023 | AMC |
Flora the frog starts at 0 on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\frac{1}{2^m}$.
What is the probability that Flora will eventually land at 10? Write the answer as a simplified fraction $\frac{m}{n}$, find $m+n$ | $\boxed{3.0}$ | AMC 2023 | 2,023 | AMC |
What is the product of all solutions to the equation
\[\log_{7x}2023\cdot \log_{289x}2023=\log_{2023x}2023\] | $\boxed{1.0}$ | AMC 2023 | 2,023 | AMC |
The weight of $\frac{1}{3}$ of a large pizza together with $3 \frac{1}{2}$ cups of orange slices is the same as the weight of $\frac{3}{4}$ of a large pizza together with $\frac{1}{2}$ cup of orange slices. A cup of orange slices weighs $\frac{1}{4}$ of a pound. What is the weight, in pounds, of a large pizza? The answer can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m-n$? | $\boxed{4.0}$ | AMC 2023 | 2,023 | AMC |
Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below.
1
1 1
1 3 1
1 5 5 1
1 7 11 7 1
Each row after the first row is formed by placing a 1 at each end of the row, and each interior entry is 1 greater than the sum of the two numbers diagonally above it in the previous row. What is the units digits of the sum of the 2023 numbers in the 2023rd row? | $\boxed{5.0}$ | AMC 2023 | 2,023 | AMC |
If $A$ and $B$ are vertices of a polyhedron, define the distance $d(A,B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\overline{AC}$ and $\overline{CB}$ are edges and $\overline{AB}$ is not an edge, then $d(A, B) = 2$. Let $Q$, $R$, and $S$ be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). Find the probability that $d(Q, R) > d(R, S)$. The final answer can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | $\boxed{29.0}$ | AMC 2023 | 2,023 | AMC |
Let $f$ be the unique function defined on the positive integers such that \[\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1\] for all positive integers $n$. What is $f(2023)$? | $\boxed{96.0}$ | AMC 2023 | 2,023 | AMC |
How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation
\[(1+2a)(2+2b)(2a+b) = 32ab?\] | $\boxed{1.0}$ | AMC 2023 | 2,023 | AMC |
Let $K$ be the number of sequences $A_1$, $A_2$, $\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\{\}$, $\{5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 6, 7, 9\}$ is one such sequence, with $n = 5$.What is the remainder when $K$ is divided by $10$? | $\boxed{5.0}$ | AMC 2023 | 2,023 | AMC |
There is a unique sequence of integers $a_1, a_2, \cdots a_{2023}$ such that
\[\tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x}\]whenever $\tan 2023x$ is defined. What is $a_{2023}?$ | $\boxed{-1.0}$ | AMC 2023 | 2,023 | AMC |
How many positive perfect squares less than $2023$ are divisible by $5$? | $\boxed{8.0}$ | AMC 2023 | 2,023 | AMC |
How many digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$? | $\boxed{18.0}$ | AMC 2023 | 2,023 | AMC |
Janet rolls a standard $6$-sided die $4$ times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal $3$? The final answer can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | $\boxed{265.0}$ | AMC 2023 | 2,023 | AMC |
Points $A$ and $B$ lie on the graph of $y=\log_{2}x$. The midpoint of $\overline{AB}$ is $(6, 2)$. What is the positive difference between the $x$-coordinates of $A$ and $B$? The final answer can be written in the form $m \sqrt{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | $\boxed{9.0}$ | AMC 2023 | 2,023 | AMC |
A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in $2023$ will each digit appear an even number of times in the 8-digital display for that date? | $\boxed{9.0}$ | AMC 2023 | 2,023 | AMC |
Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an $11$ on the next quiz, her mean will increase by $1$. If she scores an $11$ on each of the next three quizzes, her mean will increase by $2$. What is the mean of her quiz scores currently? | $\boxed{7.0}$ | AMC 2023 | 2,023 | AMC |
Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only $\frac{1}{3}$ full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice? The final answer can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | $\boxed{7.0}$ | AMC 2023 | 2,023 | AMC |
In the $xy$-plane, a circle of radius $4$ with center on the positive $x$-axis is tangent to the $y$-axis at the origin, and a circle with radius $10$ with center on the positive $y$-axis is tangent to the $x$-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect? The final answer can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | $\boxed{7.0}$ | AMC 2023 | 2,023 | AMC |
Calculate the maximum area of an isosceles trapezoid that has legs of length $1$ and one base twice as long as the other. The final answer can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m^2+n^2$? | $\boxed{13.0}$ | AMC 2023 | 2,023 | AMC |
For complex number $u = a+bi$ and $v = c+di$ (where $i=\sqrt{-1}$), define the binary operation
$u \otimes v = ac + bdi$
Suppose $z$ is a complex number such that $z\otimes z = z^{2}+40$. What is $|z|^2$? | $\boxed{50.0}$ | AMC 2023 | 2,023 | AMC |
A rectangular box $P$ has distinct edge lengths $a$, $b$, and $c$. The sum of the lengths of all $12$ edges of $P$ is $13$, the areas of all $6$ faces of $P$ is $\frac{11}{2}$, and the volume of $P$ is $\frac{1}{2}$. Find the length of the longest interior diagonal connecting two vertices of $P$. The final answer can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | $\boxed{13.0}$ | AMC 2023 | 2,023 | AMC |
For how many ordered pairs $(a,b)$ of integers does the polynomial $x^3+ax^2+bx+6$ have $3$ distinct integer roots? | $\boxed{5.0}$ | AMC 2023 | 2,023 | AMC |
In the state of Coinland, coins have values $6,10,$ and $15$ cents. Suppose $x$ is the value in cents of the most expensive item in Coinland that cannot be purchased using these coins with exact change. What is the sum of the digits of $x?$ | $\boxed{11.0}$ | AMC 2023 | 2,023 | AMC |
Triangle $ABC$ has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has an angle of $120^\circ,$ Find the area of $ABC$. The final answer can be simplified in the form $m \sqrt{n}$, where $m$ and $n$ are positive integers and $n$ without square factore. What is $m+n$? | $\boxed{18.0}$ | AMC 2023 | 2,023 | AMC |
Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by $20\%$ on every pair of shoes. Carlos also knew that he had to pay a $7.5\%$ sales tax on the discounted price. He had $$43$ dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy? | $\boxed{50.0}$ | AMC 2023 | 2,023 | AMC |
When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n$? | $\boxed{11.0}$ | AMC 2023 | 2,023 | AMC |
Suppose that $a$, $b$, $c$ and $d$ are positive integers satisfying all of the following relations.
\[abcd=2^6\cdot 3^9\cdot 5^7\]
\[\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3\]
\[\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3\]
\[\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3\]
\[\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2\]
\[\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2\]
\[\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2\]
What is $\text{gcd}(a,b,c,d)$? | $\boxed{3.0}$ | AMC 2023 | 2,023 | AMC |
A $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. Find the ratio of the area of circle $A$ to the area of circle $B$. The final answer can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | $\boxed{194.0}$ | AMC 2023 | 2,023 | AMC |
Jackson's paintbrush makes a narrow strip with a width of $6.5$ millimeters. Jackson has enough paint to make a strip $25$ meters long. How many square centimeters of paper could Jackson cover with paint? | $\boxed{1625.0}$ | AMC 2023 | 2,023 | AMC |
You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle? | $\boxed{4.0}$ | AMC 2023 | 2,023 | AMC |
When the roots of the polynomial
\[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot (x-10)^{10}\]
are removed from the number line, what remains is the union of $11$ disjoint open intervals. On how many of these intervals is $P(x)$ positive? | $\boxed{6.0}$ | AMC 2023 | 2,023 | AMC |
For how many integers $n$ does the expression\[\sqrt{\frac{\log (n^2) - (\log n)^2}{\log n - 3}}\]represent a real number, where log denotes the base $10$ logarithm? | $\boxed{901.0}$ | AMC 2023 | 2,023 | AMC |
How many nonempty subsets $B$ of ${0, 1, 2, 3, \cdots, 12}$ have the property that the number of elements in $B$ is equal to the least element of $B$? For example, $B = {4, 6, 8, 11}$ satisfies the condition. | $\boxed{144.0}$ | AMC 2023 | 2,023 | AMC |
What is the area of the region in the coordinate plane defined by
$| | x | - 1 | + | | y | - 1 | \le 1$? | $\boxed{8.0}$ | AMC 2023 | 2,023 | AMC |
What is the value of $9901 \cdot 101 - 99 \cdot 10101$? | $\boxed{2}$ | 2024 AMC 12A | 2,024 | AMC |
A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form $T = aL + bG$, where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take $69$ minutes to hike to the top if a trail is $1.5$ miles long and ascends $800$ feet, as well as if a trail is $1.2$ miles long and ascends $1100$ feet. How many minutes does the model estimates it will take to hike to the top if the trail is $4.2$ miles long and ascends $4000$ feet? | $\boxed{246}$ | 2024 AMC 12A | 2,024 | AMC |
The number $2024$ is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum? | $\boxed{21}$ | 2024 AMC 12A | 2,024 | AMC |
What is the least value of $n$ such that $n!$ is a multiple of $2024$? | $\boxed{23}$ | 2024 AMC 12A | 2,024 | AMC |
A data set containing $20$ numbers, some of which are $6$, has mean $45$. When all the 6s are removed, the data set has mean $66$. How many 6s were in the original data set? | $\boxed{7}$ | 2024 AMC 12A | 2,024 | AMC |
The product of three integers is $60$. What is the least possible positive sum of the three integers? | $\boxed{3}$ | 2024 AMC 12A | 2,024 | AMC |
In $\triangle ABC$, $\angle ABC = 90^\circ$ and $BA = BC = \sqrt{2}$. Points $P_1, P_2, \ldots, P_{2024}$ lie on hypotenuse $\overline{AC}$ so that $AP_1 = P_1P_2 = P_2P_3 = \cdots = P_{2023}P_{2024} = P_{2024}C$. What is the length of the vector sum $\overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \cdots + \overrightarrow{BP_{2024}}$? | $\boxed{2024}$ | 2024 AMC 12A | 2,024 | AMC |
How many angles $\theta$ with $0 \leq \theta \leq 2\pi$ satisfy $\log(\sin(3\theta)) + \log(\cos(2\theta)) = 0$? | $\boxed{0}$ | 2024 AMC 12A | 2,024 | AMC |
Let $M$ be the greatest integer such that both $M + 1213$ and $M + 3773$ are perfect squares. What is the units digit of $M$? | $\boxed{8}$ | 2024 AMC 12A | 2,024 | AMC |
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 $\beta$? | $\boxed{\frac{\pi}{2} - 2\alpha}$ | 2024 AMC 12A | 2,024 | AMC |
There are exactly $K$ positive integers $b$ with $5 \leq b \leq 2024$ such that the base-$b$ integer $2024_b$ is divisible by $16$ (where $16$ is in base ten). What is the sum of the digits of $K$? | $\boxed{20}$ | 2024 AMC 12A | 2,024 | AMC |
The first three terms of a geometric sequence are the integers $a$, $720$, and $b$, where $a < 720 < b$. What is the sum of the digits of the least possible value of $b$? | $\boxed{21}$ | 2024 AMC 12A | 2,024 | AMC |
The graph of $y = e^{x+1} + e^{-x} - 2$ has an axis of symmetry. What is the reflection of the point $(-1, \frac{1}{2})$ over this axis? | $\boxed{(0, \frac{1}{2})}$ | 2024 AMC 12A | 2,024 | AMC |
The roots of $x^3 + 2x^2 - x + 3$ are $p$, $q$, and $r$. What is the value of $(p^2 + 4)(q^2 + 4)(r^2 + 4)$? | $\boxed{125}$ | 2024 AMC 12A | 2,024 | AMC |
A set of $12$ tokens — $3$ red, $2$ white, $1$ blue, and $6$ black — is to be distributed at random to $3$ game players, $4$ tokens per player. The probability that some player gets all the red tokens, another gets all the white tokens, and the remaining player gets the blue token can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? | $\boxed{389}$ | 2024 AMC 12A | 2,024 | AMC |
Integers $a$, $b$, and $c$ satisfy $ab + c = 100$, $bc + a = 87$, and $ca + b = 60$. What is $ab + bc + ca$? | $\boxed{276}$ | 2024 AMC 12A | 2,024 | AMC |
Cyclic quadrilateral $ABCD$ has lengths $BC = CD = 3$ and $DA = 5$ with $\angle CDA = 120^\circ$. What is the length of the shorter diagonal of $ABCD$? | $\boxed{\frac{39}{7}}$ | 2024 AMC 12A | 2,024 | AMC |
Points $P$ and $Q$ are chosen uniformly and independently at random on sides $\overline{AB}$ and $\overline{AC}$, respectively, of equilateral triangle $\triangle ABC$. Which of the following intervals contains the probability that the area of $\triangle APQ$ is less than half the area of $\triangle ABC$? | $\boxed{[\frac{3}{4}, \frac{7}{8}]}$ | 2024 AMC 12A | 2,024 | AMC |
Suppose that $a_1 = 2$ and the sequence $(a_n)$ satisfies the recurrence relation $\frac{a_n - 1}{n - 1} = \frac{a_{n-1} + 1}{(n - 1) + 1}$ for all $n \geq 2$. What is the greatest integer less than or equal to $\sum_{n=1}^{100} a_n^2$? | $\boxed{338551}$ | 2024 AMC 12A | 2,024 | AMC |
What is the value of $\tan^2 \frac{\pi}{16}\cdot\tan^2 \frac{3\pi}{16} + \tan^2 \frac{\pi}{16}\cdot\tan^2 \frac{5\pi}{16} + \tan^2 \frac{3\pi}{16}\cdot\tan^2 \frac{7\pi}{16} + \tan^2 \frac{5\pi}{16}\cdot\tan^2 \frac{7\pi}{16}$? | $\boxed{68}$ | 2024 AMC 12A | 2,024 | AMC |
A disphenoid is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths? | $\boxed{15\sqrt{7}}$ | 2024 AMC 12A | 2,024 | AMC |
A graph is symmetric about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers $(a, b, c, d)$, where $|a|, |b|, |c|, |d| \leq 5$ and $c$ and $d$ are not both $0$, is the graph of $y = \frac{ax + b}{cx + d}$ symmetric about the line $y = x$? | $\boxed{1292}$ | 2024 AMC 12A | 2,024 | AMC |
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? | $\boxed{2022}$ | 2024 AMC 12B | 2,024 | AMC |
What is $10! - 7! \cdot 6!$? | $\boxed{0}$ | 2024 AMC 12B | 2,024 | AMC |
For how many integer values of $x$ is $|2x| \leq 7\pi$? | $\boxed{21}$ | 2024 AMC 12B | 2,024 | AMC |
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? | $\boxed{D}$ | 2024 AMC 12B | 2,024 | AMC |
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? | $\boxed{15}$ | 2024 AMC 12B | 2,024 | AMC |
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.) | $\boxed{20}$ | 2024 AMC 12B | 2,024 | AMC |
What value of $x$ satisfies $\frac{\log_2 x \cdot \log_3 x}{\log_2 x + \log_3 x} = 2$? | $\boxed{36}$ | 2024 AMC 12B | 2,024 | AMC |
A dartboard is the region $B$ in the coordinate plane consisting of points $(x, y)$ such that $|x| + |y| \leq 8$. A target $T$ is the region where $(x^2 + y^2 - 25)^2 \leq 49$. 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 $m + n$? | $\boxed{71}$ | 2024 AMC 12B | 2,024 | AMC |
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? | $\boxed{3}$ | 2024 AMC 12B | 2,024 | AMC |
Let $x_n = \sin^2 (n^5)$. What is the mean of $x_1, x_2, x_3, \cdots, x_{90}$? | $\boxed{\frac{91}{180}}$ | 2024 AMC 12B | 2,024 | AMC |
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 $z$? | $\boxed{\frac{3}{2}}$ | 2024 AMC 12B | 2,024 | AMC |
There are real numbers $x, y, h$ and $k$ that satisfy the system of equations $x^2 + y^2 - 6x - 8y = h$ and $x^2 + y^2 - 10x + 4y = k$. What is the minimum possible value of $h + k$? | $\boxed{-34}$ | 2024 AMC 12B | 2,024 | AMC |
How many different remainders can result when the $100$th power of an integer is divided by $125$? | $\boxed{5}$ | 2024 AMC 12B | 2,024 | AMC |
A triangle in the coordinate plane has vertices $A(\log_2 1, \log_2 2), B(\log_2 3, \log_2 4)$, and $C(\log_2 7, \log_2 8)$. What is the area of $\triangle ABC$? | $\boxed{\log_2 \frac{7}{\sqrt{3}}}$ | 2024 AMC 12B | 2,024 | AMC |
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 $r$? | $\boxed{5}$ | 2024 AMC 12B | 2,024 | AMC |
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? | $\boxed{\frac{1}{105}}$ | 2024 AMC 12B | 2,024 | AMC |
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 $\frac{F_2}{F_1} + \frac{F_4}{F_2} + \frac{F_6}{F_3} + \cdots + \frac{F_{20}}{F_{10}}$? | $\boxed{320}$ | 2024 AMC 12B | 2,024 | AMC |
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 $x = 8$. What is $p + q + r + s$? | $\boxed{911}$ | 2024 AMC 12B | 2,024 | AMC |
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? | $\boxed{154}$ | 2024 AMC 12B | 2,024 | AMC |
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? | $\boxed{15}$ | 2024 AMC 12B | 2,024 | AMC |
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? | $\boxed{\frac{1 + \sqrt{2}}{2}}$ | 2024 AMC 12B | 2,024 | AMC |
What is the number of ordered triples $(a, b, c)$ of positive integers, with $a \leq b \leq c \leq 9$, 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.) | $\boxed{3}$ | 2024 AMC 12B | 2,024 | AMC |
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 $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m$? (Recall that two events $A$ and $B$ are independent if $P(A \text{ and } B) = P(A) \cdot P(B)$.) | $\boxed{247}$ | 2024 AMC 12B | 2,024 | AMC |
Find the sum of all integer bases $b>9$ for which $17_{b}$ is a divisor of $97_{b}$. | $\boxed{70}$ | AIME 2025-I | 2,025 | AIME |
On $\triangle ABC$ points $A,D,E$, and $B$ lie that order on side $\overline{AB}$ with $AD=4, DE=16$, and $EB=8$. Points $A,F,G$, and $C$ lie in that order on side $\overline{AC}$ with $AF=13, FG=52$, and $GC=26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ through $E$. Quadrilateral $DEGF$ has area 288. Find the area of heptagon $AFNBCEM$. | $\boxed{588}$ | AIME 2025-I | 2,025 | AIME |
The 9 members of a baseball team went to an ice cream parlor after their game. Each player had a singlescoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose vanilla, which was greater than the number of players who chose strawberry. Let $N$ be the number of different assignments of flavors to players that meet these conditions. Find the remainder when $N$ is divided by 1000. | $\boxed{16}$ | AIME 2025-I | 2,025 | AIME |
Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $100$, inclusive, such that $12x^{2}-xy-6y^{2}=0$. | $\boxed{117}$ | AIME 2025-I | 2,025 | AIME |
There are $8!=40320$ eight-digit positive integers that use each of the digits $1,2,3,4,5,6,7,8$ exactly once. Let $N$ be the number of these integers that are divisible by 22. Find the difference between $N$ and 2025. | $\boxed{279}$ | AIME 2025-I | 2,025 | AIME |
An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is 3, and the area of the trapezoid is 72. Let the parallel sides of the trapezoid have lengths $r$ and $s$, with $r \neq s$. Find $r^{2}+s^{2}$. | $\boxed{504}$ | AIME 2025-I | 2,025 | AIME |
The twelve letters $A,B,C,D,E,F,G,H,I,J,K$, and $L$ are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and those six words are listed alphabetically. For example, a possible result is $AB,CJ,DG,EK,FL,HI$. The probability that the last word listed contains $G$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | $\boxed{821}$ | AIME 2025-I | 2,025 | AIME |
Let $k$ be real numbers such that the system $|25+20i-z|=5$ and $|z-4-k|=|z-3i-k|$ has exactly one complex solution $z$. The sum of all possible values of $k$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. Here $i=\sqrt{-1}$. | $\boxed{77}$ | AIME 2025-I | 2,025 | AIME |
The parabola with equation $y=x^{2}-4$ is rotated $60^{\circ}$ counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has $y$-coordinate $\frac{a-\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, and $a$ and $c$ are relatively prime. Find $a+b+c$. | $\boxed{62}$ | AIME 2025-I | 2,025 | AIME |
The 27 cells of a $3\times9$ grid are filled in using the numbers 1 through 9 so that each row contains 9 different numbers, and each of the three $3\times3$ blocks heavily outlined in the example below contains 9 different numbers, as in the first three rows of a Sudoku puzzle.
| 4 | 2 | 8 | 9 | 6 | 3 | 1 | 7 | 5 |
| 3 | 7 | 9 | 5 | 2 | 1 | 6 | 8 | 4 |
| 5 | 6 | 1 | 8 | 4 | 7 | 9 | 2 | 3 |
The number of different ways to fill such a grid can be written as $p^a\cdot q^b\cdot r^c\cdot s^d$, where $p,q,r,$ and $s$ are distinct prime numbers and $a,b,c,$ and $d$ are positive integers. Find $p\cdot a+q\cdot b+r\cdot c+s\cdot d$. | $\boxed{81}$ | AIME 2025-I | 2,025 | AIME |
A piecewise linear periodic function is defined by $f(x)=\begin{cases}x&\text{if }x\in[-1,1)\\2-x&\text{if }x\in[1,3)\end{cases}$ and $f(x+4)=f(x)$ for all real numbers $x$. The graph of $f(x)$ has the sawtooth pattern. The parabola $x=34y^2$ intersects the graph of $f(x)$ at finitely many points. The sum of the $y$-coordinates of these intersection points can be expressed in the form $\frac{a+b\sqrt{c}}{d}$, where $a,b,c,$ and $d$ are positive integers, $a,b,$ and $d$ have greatest common divisor equal to 1, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$. | $\boxed{259}$ | AIME 2025-I | 2,025 | AIME |
The set of points in 3-dimensional coordinate space that lie in the plane $x+y+z=75$ whose coordinates satisfy the inequalities $x-yz<y-zx<z-xy$ forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form $a\sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b$. | $\boxed{510}$ | AIME 2025-I | 2,025 | AIME |
Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws 25 more line segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting those two points. Find the expected number of regions into which these 27 line segments divide the disk. | $\boxed{204}$ | AIME 2025-I | 2,025 | AIME |
Let $ABCDE$ be a convex pentagon with $AB=14, BC=7, CD=24, DE=13, EA=26,$ and $\angle B=\angle E=60^\circ$. For each point $X$ in the plane, define $f(X)=AX+BX+CX+DX+EX$. The least possible value of $f(X)$ can be expressed as $m+n\sqrt{p}$, where $m$ and $n$ are positive integers and $p$ is not divisible by the square of any prime. Find $m+n+p$. | $\boxed{60}$ | AIME 2025-I | 2,025 | AIME |
Let $N$ denote the number of ordered triples of positive integers $(a,b,c)$ such that $a,b,c\leq3^6$ and $a^3+b^3+c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$. | $\boxed{735}$ | AIME 2025-I | 2,025 | AIME |
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