problem stringlengths 20 3.95k | answer stringclasses 532
values | task stringclasses 6
values | problem_number int64 1 1.31k | set_number int64 1 5 |
|---|---|---|---|---|
A piecewise linear function is defined by\[f(x) = \begin{cases} x & \operatorname{if} ~ -1 \leq x < 1 \ 2 - x & \operatorname{if} ~ 1 \leq x < 3\end{cases}\]and $f(x + 4) = f(x)$ for all real numbers $x$. The graph of $f(x)$ has the sawtooth pattern depicted below. The parabola $x = 34y^{2}$ intersects the graph of $f(... | 259 | AIME25 | 41 | 1 |
A piecewise linear function is defined by\[f(x) = \begin{cases} x & \operatorname{if} ~ -1 \leq x < 1 \ 2 - x & \operatorname{if} ~ 1 \leq x < 3\end{cases}\]and $f(x + 4) = f(x)$ for all real numbers $x$. The graph of $f(x)$ has the sawtooth pattern depicted below. The parabola $x = 34y^{2}$ intersects the graph of $f(... | 259 | AIME25 | 41 | 2 |
A piecewise linear function is defined by\[f(x) = \begin{cases} x & \operatorname{if} ~ -1 \leq x < 1 \ 2 - x & \operatorname{if} ~ 1 \leq x < 3\end{cases}\]and $f(x + 4) = f(x)$ for all real numbers $x$. The graph of $f(x)$ has the sawtooth pattern depicted below. The parabola $x = 34y^{2}$ intersects the graph of $f(... | 259 | AIME25 | 41 | 3 |
A piecewise linear function is defined by\[f(x) = \begin{cases} x & \operatorname{if} ~ -1 \leq x < 1 \ 2 - x & \operatorname{if} ~ 1 \leq x < 3\end{cases}\]and $f(x + 4) = f(x)$ for all real numbers $x$. The graph of $f(x)$ has the sawtooth pattern depicted below. The parabola $x = 34y^{2}$ intersects the graph of $f(... | 259 | AIME25 | 41 | 4 |
A piecewise linear function is defined by\[f(x) = \begin{cases} x & \operatorname{if} ~ -1 \leq x < 1 \ 2 - x & \operatorname{if} ~ 1 \leq x < 3\end{cases}\]and $f(x + 4) = f(x)$ for all real numbers $x$. The graph of $f(x)$ has the sawtooth pattern depicted below. The parabola $x = 34y^{2}$ intersects the graph of $f(... | 259 | AIME25 | 41 | 5 |
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 $b$... | 510 | AIME25 | 42 | 1 |
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 $b$... | 510 | AIME25 | 42 | 2 |
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 $b$... | 510 | AIME25 | 42 | 3 |
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 $b$... | 510 | AIME25 | 42 | 4 |
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 $b$... | 510 | AIME25 | 42 | 5 |
Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws $25$ more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expect... | 204 | AIME25 | 43 | 1 |
Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws $25$ more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expect... | 204 | AIME25 | 43 | 2 |
Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws $25$ more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expect... | 204 | AIME25 | 43 | 3 |
Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws $25$ more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expect... | 204 | AIME25 | 43 | 4 |
Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws $25$ more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expect... | 204 | AIME25 | 43 | 5 |
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 t... | 60 | AIME25 | 44 | 1 |
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 t... | 60 | AIME25 | 44 | 2 |
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 t... | 60 | AIME25 | 44 | 3 |
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 t... | 60 | AIME25 | 44 | 4 |
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 t... | 60 | AIME25 | 44 | 5 |
Let $N$ denote the number of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \leq 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$. | 735 | AIME25 | 45 | 1 |
Let $N$ denote the number of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \leq 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$. | 735 | AIME25 | 45 | 2 |
Let $N$ denote the number of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \leq 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$. | 735 | AIME25 | 45 | 3 |
Let $N$ denote the number of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \leq 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$. | 735 | AIME25 | 45 | 4 |
Let $N$ denote the number of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \leq 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$. | 735 | AIME25 | 45 | 5 |
Six points $A, B, C, D, E,$ and $F$ lie in a straight line in that order. Suppose that $G$ is a point not on the line and that $AC=26, BD=22, CE=31, DF=33, AF=73, CG=40,$ and $DG=30.$ Find the area of $\triangle BGE.$ | 468 | AIME25 | 46 | 1 |
Six points $A, B, C, D, E,$ and $F$ lie in a straight line in that order. Suppose that $G$ is a point not on the line and that $AC=26, BD=22, CE=31, DF=33, AF=73, CG=40,$ and $DG=30.$ Find the area of $\triangle BGE.$ | 468 | AIME25 | 46 | 2 |
Six points $A, B, C, D, E,$ and $F$ lie in a straight line in that order. Suppose that $G$ is a point not on the line and that $AC=26, BD=22, CE=31, DF=33, AF=73, CG=40,$ and $DG=30.$ Find the area of $\triangle BGE.$ | 468 | AIME25 | 46 | 3 |
Six points $A, B, C, D, E,$ and $F$ lie in a straight line in that order. Suppose that $G$ is a point not on the line and that $AC=26, BD=22, CE=31, DF=33, AF=73, CG=40,$ and $DG=30.$ Find the area of $\triangle BGE.$ | 468 | AIME25 | 46 | 4 |
Six points $A, B, C, D, E,$ and $F$ lie in a straight line in that order. Suppose that $G$ is a point not on the line and that $AC=26, BD=22, CE=31, DF=33, AF=73, CG=40,$ and $DG=30.$ Find the area of $\triangle BGE.$ | 468 | AIME25 | 46 | 5 |
Find the sum of all positive integers $n$ such that $n + 2$ divides the product $3(n + 3)(n^2 + 9)$. | 49 | AIME25 | 47 | 1 |
Find the sum of all positive integers $n$ such that $n + 2$ divides the product $3(n + 3)(n^2 + 9)$. | 49 | AIME25 | 47 | 2 |
Find the sum of all positive integers $n$ such that $n + 2$ divides the product $3(n + 3)(n^2 + 9)$. | 49 | AIME25 | 47 | 3 |
Find the sum of all positive integers $n$ such that $n + 2$ divides the product $3(n + 3)(n^2 + 9)$. | 49 | AIME25 | 47 | 4 |
Find the sum of all positive integers $n$ such that $n + 2$ divides the product $3(n + 3)(n^2 + 9)$. | 49 | AIME25 | 47 | 5 |
Four unit squares form a $2 \times 2$ grid. Each of the $12$ unit line segments forming the sides of the squares is colored either red or blue in such a say that each unit square has $2$ red sides and $2$ blue sides. One example is shown below (red is solid, blue is dashed). Find the number of such colorings. [asy] ... | 82 | AIME25 | 48 | 1 |
Four unit squares form a $2 \times 2$ grid. Each of the $12$ unit line segments forming the sides of the squares is colored either red or blue in such a say that each unit square has $2$ red sides and $2$ blue sides. One example is shown below (red is solid, blue is dashed). Find the number of such colorings. [asy] ... | 82 | AIME25 | 48 | 2 |
Four unit squares form a $2 \times 2$ grid. Each of the $12$ unit line segments forming the sides of the squares is colored either red or blue in such a say that each unit square has $2$ red sides and $2$ blue sides. One example is shown below (red is solid, blue is dashed). Find the number of such colorings. [asy] ... | 82 | AIME25 | 48 | 3 |
Four unit squares form a $2 \times 2$ grid. Each of the $12$ unit line segments forming the sides of the squares is colored either red or blue in such a say that each unit square has $2$ red sides and $2$ blue sides. One example is shown below (red is solid, blue is dashed). Find the number of such colorings. [asy] ... | 82 | AIME25 | 48 | 4 |
Four unit squares form a $2 \times 2$ grid. Each of the $12$ unit line segments forming the sides of the squares is colored either red or blue in such a say that each unit square has $2$ red sides and $2$ blue sides. One example is shown below (red is solid, blue is dashed). Find the number of such colorings. [asy] ... | 82 | AIME25 | 48 | 5 |
The product\[\prod^{63}_{k=4} rac{\log_k (5^{k^2 - 1})}{\log_{k + 1} (5^{k^2 - 4})} = rac{\log_4 (5^{15})}{\log_5 (5^{12})} \cdot rac{\log_5 (5^{24})}{\log_6 (5^{21})}\cdot rac{\log_6 (5^{35})}{\log_7 (5^{32})} \cdots rac{\log_{63} (5^{3968})}{\log_{64} (5^{3965})}\]is equal to $\tfrac mn,$ where $m$ and $n$ are r... | 106 | AIME25 | 49 | 1 |
The product\[\prod^{63}_{k=4} rac{\log_k (5^{k^2 - 1})}{\log_{k + 1} (5^{k^2 - 4})} = rac{\log_4 (5^{15})}{\log_5 (5^{12})} \cdot rac{\log_5 (5^{24})}{\log_6 (5^{21})}\cdot rac{\log_6 (5^{35})}{\log_7 (5^{32})} \cdots rac{\log_{63} (5^{3968})}{\log_{64} (5^{3965})}\]is equal to $\tfrac mn,$ where $m$ and $n$ are r... | 106 | AIME25 | 49 | 2 |
The product\[\prod^{63}_{k=4} rac{\log_k (5^{k^2 - 1})}{\log_{k + 1} (5^{k^2 - 4})} = rac{\log_4 (5^{15})}{\log_5 (5^{12})} \cdot rac{\log_5 (5^{24})}{\log_6 (5^{21})}\cdot rac{\log_6 (5^{35})}{\log_7 (5^{32})} \cdots rac{\log_{63} (5^{3968})}{\log_{64} (5^{3965})}\]is equal to $\tfrac mn,$ where $m$ and $n$ are r... | 106 | AIME25 | 49 | 3 |
The product\[\prod^{63}_{k=4} rac{\log_k (5^{k^2 - 1})}{\log_{k + 1} (5^{k^2 - 4})} = rac{\log_4 (5^{15})}{\log_5 (5^{12})} \cdot rac{\log_5 (5^{24})}{\log_6 (5^{21})}\cdot rac{\log_6 (5^{35})}{\log_7 (5^{32})} \cdots rac{\log_{63} (5^{3968})}{\log_{64} (5^{3965})}\]is equal to $\tfrac mn,$ where $m$ and $n$ are r... | 106 | AIME25 | 49 | 4 |
The product\[\prod^{63}_{k=4} rac{\log_k (5^{k^2 - 1})}{\log_{k + 1} (5^{k^2 - 4})} = rac{\log_4 (5^{15})}{\log_5 (5^{12})} \cdot rac{\log_5 (5^{24})}{\log_6 (5^{21})}\cdot rac{\log_6 (5^{35})}{\log_7 (5^{32})} \cdots rac{\log_{63} (5^{3968})}{\log_{64} (5^{3965})}\]is equal to $\tfrac mn,$ where $m$ and $n$ are r... | 106 | AIME25 | 49 | 5 |
Suppose $\triangle ABC$ has angles $\angle BAC = 84^\circ, \angle ABC=60^\circ,$ and $\angle ACB = 36^\circ.$ Let $D, E,$ and $F$ be the midpoints of sides $\overline{BC}, \overline{AC},$ and $\overline{AB},$ respectively. The circumcircle of $ riangle DEF$ intersects $\overline{BD}, \overline{AE},$ and $\overline{AF}$... | 336 | AIME25 | 50 | 1 |
Suppose $\triangle ABC$ has angles $\angle BAC = 84^\circ, \angle ABC=60^\circ,$ and $\angle ACB = 36^\circ.$ Let $D, E,$ and $F$ be the midpoints of sides $\overline{BC}, \overline{AC},$ and $\overline{AB},$ respectively. The circumcircle of $ riangle DEF$ intersects $\overline{BD}, \overline{AE},$ and $\overline{AF}$... | 336 | AIME25 | 50 | 2 |
Suppose $\triangle ABC$ has angles $\angle BAC = 84^\circ, \angle ABC=60^\circ,$ and $\angle ACB = 36^\circ.$ Let $D, E,$ and $F$ be the midpoints of sides $\overline{BC}, \overline{AC},$ and $\overline{AB},$ respectively. The circumcircle of $ riangle DEF$ intersects $\overline{BD}, \overline{AE},$ and $\overline{AF}$... | 336 | AIME25 | 50 | 3 |
Suppose $\triangle ABC$ has angles $\angle BAC = 84^\circ, \angle ABC=60^\circ,$ and $\angle ACB = 36^\circ.$ Let $D, E,$ and $F$ be the midpoints of sides $\overline{BC}, \overline{AC},$ and $\overline{AB},$ respectively. The circumcircle of $ riangle DEF$ intersects $\overline{BD}, \overline{AE},$ and $\overline{AF}$... | 336 | AIME25 | 50 | 4 |
Suppose $\triangle ABC$ has angles $\angle BAC = 84^\circ, \angle ABC=60^\circ,$ and $\angle ACB = 36^\circ.$ Let $D, E,$ and $F$ be the midpoints of sides $\overline{BC}, \overline{AC},$ and $\overline{AB},$ respectively. The circumcircle of $ riangle DEF$ intersects $\overline{BD}, \overline{AE},$ and $\overline{AF}$... | 336 | AIME25 | 50 | 5 |
Circle $\omega_1$ with radius $6$ centered at point $A$ is internally tangent at point $B$ to circle $\omega_2$ with radius $15$. Points $C$ and $D$ lie on $\omega_2$ such that $\overline{BC}$ is a diameter of $\omega_2$ and ${\overline{BC} \perp \overline{AD}}$. The rectangle $EFGH$ is inscribed in $\omega_1$ such tha... | 293 | AIME25 | 51 | 1 |
Circle $\omega_1$ with radius $6$ centered at point $A$ is internally tangent at point $B$ to circle $\omega_2$ with radius $15$. Points $C$ and $D$ lie on $\omega_2$ such that $\overline{BC}$ is a diameter of $\omega_2$ and ${\overline{BC} \perp \overline{AD}}$. The rectangle $EFGH$ is inscribed in $\omega_1$ such tha... | 293 | AIME25 | 51 | 2 |
Circle $\omega_1$ with radius $6$ centered at point $A$ is internally tangent at point $B$ to circle $\omega_2$ with radius $15$. Points $C$ and $D$ lie on $\omega_2$ such that $\overline{BC}$ is a diameter of $\omega_2$ and ${\overline{BC} \perp \overline{AD}}$. The rectangle $EFGH$ is inscribed in $\omega_1$ such tha... | 293 | AIME25 | 51 | 3 |
Circle $\omega_1$ with radius $6$ centered at point $A$ is internally tangent at point $B$ to circle $\omega_2$ with radius $15$. Points $C$ and $D$ lie on $\omega_2$ such that $\overline{BC}$ is a diameter of $\omega_2$ and ${\overline{BC} \perp \overline{AD}}$. The rectangle $EFGH$ is inscribed in $\omega_1$ such tha... | 293 | AIME25 | 51 | 4 |
Circle $\omega_1$ with radius $6$ centered at point $A$ is internally tangent at point $B$ to circle $\omega_2$ with radius $15$. Points $C$ and $D$ lie on $\omega_2$ such that $\overline{BC}$ is a diameter of $\omega_2$ and ${\overline{BC} \perp \overline{AD}}$. The rectangle $EFGH$ is inscribed in $\omega_1$ such tha... | 293 | AIME25 | 51 | 5 |
Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 237 | AIME25 | 52 | 1 |
Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 237 | AIME25 | 52 | 2 |
Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 237 | AIME25 | 52 | 3 |
Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 237 | AIME25 | 52 | 4 |
Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 237 | AIME25 | 52 | 5 |
From an unlimited supply of 1-cent coins, 10-cent coins, and 25-cent coins, Silas wants to find a collection of coins that has a total value of $N$ cents, where $N$ is a positive integer. He uses the so-called greedy algorithm, successively choosing the coin of greatest value that does not cause the value of his collec... | 610 | AIME25 | 53 | 1 |
From an unlimited supply of 1-cent coins, 10-cent coins, and 25-cent coins, Silas wants to find a collection of coins that has a total value of $N$ cents, where $N$ is a positive integer. He uses the so-called greedy algorithm, successively choosing the coin of greatest value that does not cause the value of his collec... | 610 | AIME25 | 53 | 2 |
From an unlimited supply of 1-cent coins, 10-cent coins, and 25-cent coins, Silas wants to find a collection of coins that has a total value of $N$ cents, where $N$ is a positive integer. He uses the so-called greedy algorithm, successively choosing the coin of greatest value that does not cause the value of his collec... | 610 | AIME25 | 53 | 3 |
From an unlimited supply of 1-cent coins, 10-cent coins, and 25-cent coins, Silas wants to find a collection of coins that has a total value of $N$ cents, where $N$ is a positive integer. He uses the so-called greedy algorithm, successively choosing the coin of greatest value that does not cause the value of his collec... | 610 | AIME25 | 53 | 4 |
From an unlimited supply of 1-cent coins, 10-cent coins, and 25-cent coins, Silas wants to find a collection of coins that has a total value of $N$ cents, where $N$ is a positive integer. He uses the so-called greedy algorithm, successively choosing the coin of greatest value that does not cause the value of his collec... | 610 | AIME25 | 53 | 5 |
There are $n$ values of $x$ in the interval $0<x<2\pi$ where $f(x)=\sin(7\pi\cdot\sin(5x))=0$. For $t$ of these $n$ values of $x$, the graph of $y=f(x)$ is tangent to the $x$-axis. Find $n+t$. | 149 | AIME25 | 54 | 1 |
There are $n$ values of $x$ in the interval $0<x<2\pi$ where $f(x)=\sin(7\pi\cdot\sin(5x))=0$. For $t$ of these $n$ values of $x$, the graph of $y=f(x)$ is tangent to the $x$-axis. Find $n+t$. | 149 | AIME25 | 54 | 2 |
There are $n$ values of $x$ in the interval $0<x<2\pi$ where $f(x)=\sin(7\pi\cdot\sin(5x))=0$. For $t$ of these $n$ values of $x$, the graph of $y=f(x)$ is tangent to the $x$-axis. Find $n+t$. | 149 | AIME25 | 54 | 3 |
There are $n$ values of $x$ in the interval $0<x<2\pi$ where $f(x)=\sin(7\pi\cdot\sin(5x))=0$. For $t$ of these $n$ values of $x$, the graph of $y=f(x)$ is tangent to the $x$-axis. Find $n+t$. | 149 | AIME25 | 54 | 4 |
There are $n$ values of $x$ in the interval $0<x<2\pi$ where $f(x)=\sin(7\pi\cdot\sin(5x))=0$. For $t$ of these $n$ values of $x$, the graph of $y=f(x)$ is tangent to the $x$-axis. Find $n+t$. | 149 | AIME25 | 54 | 5 |
Sixteen chairs are arranged in a row. Eight people each select a chair in which to sit so that no person sits next to two other people. Let $N$ be the number of subsets of $16$ chairs that could be selected. Find the remainder when $N$ is divided by $1000$. | 907 | AIME25 | 55 | 1 |
Sixteen chairs are arranged in a row. Eight people each select a chair in which to sit so that no person sits next to two other people. Let $N$ be the number of subsets of $16$ chairs that could be selected. Find the remainder when $N$ is divided by $1000$. | 907 | AIME25 | 55 | 2 |
Sixteen chairs are arranged in a row. Eight people each select a chair in which to sit so that no person sits next to two other people. Let $N$ be the number of subsets of $16$ chairs that could be selected. Find the remainder when $N$ is divided by $1000$. | 907 | AIME25 | 55 | 3 |
Sixteen chairs are arranged in a row. Eight people each select a chair in which to sit so that no person sits next to two other people. Let $N$ be the number of subsets of $16$ chairs that could be selected. Find the remainder when $N$ is divided by $1000$. | 907 | AIME25 | 55 | 4 |
Sixteen chairs are arranged in a row. Eight people each select a chair in which to sit so that no person sits next to two other people. Let $N$ be the number of subsets of $16$ chairs that could be selected. Find the remainder when $N$ is divided by $1000$. | 907 | AIME25 | 55 | 5 |
Let $S$ be the set of vertices of a regular $24$-gon. Find the number of ways to draw $12$ segments of equal lengths so that each vertex in $S$ is an endpoint of exactly one of the $12$ segments. | 113 | AIME25 | 56 | 1 |
Let $S$ be the set of vertices of a regular $24$-gon. Find the number of ways to draw $12$ segments of equal lengths so that each vertex in $S$ is an endpoint of exactly one of the $12$ segments. | 113 | AIME25 | 56 | 2 |
Let $S$ be the set of vertices of a regular $24$-gon. Find the number of ways to draw $12$ segments of equal lengths so that each vertex in $S$ is an endpoint of exactly one of the $12$ segments. | 113 | AIME25 | 56 | 3 |
Let $S$ be the set of vertices of a regular $24$-gon. Find the number of ways to draw $12$ segments of equal lengths so that each vertex in $S$ is an endpoint of exactly one of the $12$ segments. | 113 | AIME25 | 56 | 4 |
Let $S$ be the set of vertices of a regular $24$-gon. Find the number of ways to draw $12$ segments of equal lengths so that each vertex in $S$ is an endpoint of exactly one of the $12$ segments. | 113 | AIME25 | 56 | 5 |
Let $A_1A_2\dots A_{11}$ be a non-convex $11$-gon such that The area of $A_iA_1A_{i+1}$ is $1$ for each $2 \le i \le 10$, $\cos(\angle A_iA_1A_{i+1})=\frac{12}{13}$ for each $2 \le i \le 10$, The perimeter of $A_1A_2\dots A_{11}$ is $20$. If $A_1A_2+A_1A_{11}$ can be expressed as $\frac{m\sqrt{n}-p}{q}$ for positive in... | 19 | AIME25 | 57 | 1 |
Let $A_1A_2\dots A_{11}$ be a non-convex $11$-gon such that The area of $A_iA_1A_{i+1}$ is $1$ for each $2 \le i \le 10$, $\cos(\angle A_iA_1A_{i+1})=\frac{12}{13}$ for each $2 \le i \le 10$, The perimeter of $A_1A_2\dots A_{11}$ is $20$. If $A_1A_2+A_1A_{11}$ can be expressed as $\frac{m\sqrt{n}-p}{q}$ for positive in... | 19 | AIME25 | 57 | 2 |
Let $A_1A_2\dots A_{11}$ be a non-convex $11$-gon such that The area of $A_iA_1A_{i+1}$ is $1$ for each $2 \le i \le 10$, $\cos(\angle A_iA_1A_{i+1})=\frac{12}{13}$ for each $2 \le i \le 10$, The perimeter of $A_1A_2\dots A_{11}$ is $20$. If $A_1A_2+A_1A_{11}$ can be expressed as $\frac{m\sqrt{n}-p}{q}$ for positive in... | 19 | AIME25 | 57 | 3 |
Let $A_1A_2\dots A_{11}$ be a non-convex $11$-gon such that The area of $A_iA_1A_{i+1}$ is $1$ for each $2 \le i \le 10$, $\cos(\angle A_iA_1A_{i+1})=\frac{12}{13}$ for each $2 \le i \le 10$, The perimeter of $A_1A_2\dots A_{11}$ is $20$. If $A_1A_2+A_1A_{11}$ can be expressed as $\frac{m\sqrt{n}-p}{q}$ for positive in... | 19 | AIME25 | 57 | 4 |
Let $A_1A_2\dots A_{11}$ be a non-convex $11$-gon such that The area of $A_iA_1A_{i+1}$ is $1$ for each $2 \le i \le 10$, $\cos(\angle A_iA_1A_{i+1})=\frac{12}{13}$ for each $2 \le i \le 10$, The perimeter of $A_1A_2\dots A_{11}$ is $20$. If $A_1A_2+A_1A_{11}$ can be expressed as $\frac{m\sqrt{n}-p}{q}$ for positive in... | 19 | AIME25 | 57 | 5 |
Let the sequence of rationals $x_1,x_2,\dots$ be defined such that $x_1=\frac{25}{11}$ and\[x_{k+1}=\frac{1}{3}\left(x_k+\frac{1}{x_k}-1\right).\]$x_{2025}$ can be expressed as $ rac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find the remainder when $m+n$ is divided by $1000$. | 248 | AIME25 | 58 | 1 |
Let the sequence of rationals $x_1,x_2,\dots$ be defined such that $x_1=\frac{25}{11}$ and\[x_{k+1}=\frac{1}{3}\left(x_k+\frac{1}{x_k}-1\right).\]$x_{2025}$ can be expressed as $ rac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find the remainder when $m+n$ is divided by $1000$. | 248 | AIME25 | 58 | 2 |
Let the sequence of rationals $x_1,x_2,\dots$ be defined such that $x_1=\frac{25}{11}$ and\[x_{k+1}=\frac{1}{3}\left(x_k+\frac{1}{x_k}-1\right).\]$x_{2025}$ can be expressed as $ rac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find the remainder when $m+n$ is divided by $1000$. | 248 | AIME25 | 58 | 3 |
Let the sequence of rationals $x_1,x_2,\dots$ be defined such that $x_1=\frac{25}{11}$ and\[x_{k+1}=\frac{1}{3}\left(x_k+\frac{1}{x_k}-1\right).\]$x_{2025}$ can be expressed as $ rac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find the remainder when $m+n$ is divided by $1000$. | 248 | AIME25 | 58 | 4 |
Let the sequence of rationals $x_1,x_2,\dots$ be defined such that $x_1=\frac{25}{11}$ and\[x_{k+1}=\frac{1}{3}\left(x_k+\frac{1}{x_k}-1\right).\]$x_{2025}$ can be expressed as $ rac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find the remainder when $m+n$ is divided by $1000$. | 248 | AIME25 | 58 | 5 |
Let ${\triangle ABC}$ be a right triangle with $\angle A = 90^\circ$ and $BC = 38.$ There exist points $K$ and $L$ inside the triangle such\[AK = AL = BK = CL = KL = 14.\]The area of the quadrilateral $BKLC$ can be expressed as $n\sqrt3$ for some positive integer $n.$ Find $n.$ | 104 | AIME25 | 59 | 1 |
Let ${\triangle ABC}$ be a right triangle with $\angle A = 90^\circ$ and $BC = 38.$ There exist points $K$ and $L$ inside the triangle such\[AK = AL = BK = CL = KL = 14.\]The area of the quadrilateral $BKLC$ can be expressed as $n\sqrt3$ for some positive integer $n.$ Find $n.$ | 104 | AIME25 | 59 | 2 |
Let ${\triangle ABC}$ be a right triangle with $\angle A = 90^\circ$ and $BC = 38.$ There exist points $K$ and $L$ inside the triangle such\[AK = AL = BK = CL = KL = 14.\]The area of the quadrilateral $BKLC$ can be expressed as $n\sqrt3$ for some positive integer $n.$ Find $n.$ | 104 | AIME25 | 59 | 3 |
Let ${\triangle ABC}$ be a right triangle with $\angle A = 90^\circ$ and $BC = 38.$ There exist points $K$ and $L$ inside the triangle such\[AK = AL = BK = CL = KL = 14.\]The area of the quadrilateral $BKLC$ can be expressed as $n\sqrt3$ for some positive integer $n.$ Find $n.$ | 104 | AIME25 | 59 | 4 |
Let ${\triangle ABC}$ be a right triangle with $\angle A = 90^\circ$ and $BC = 38.$ There exist points $K$ and $L$ inside the triangle such\[AK = AL = BK = CL = KL = 14.\]The area of the quadrilateral $BKLC$ can be expressed as $n\sqrt3$ for some positive integer $n.$ Find $n.$ | 104 | AIME25 | 59 | 5 |
Let\[f(x)=\frac{(x-18)(x-72)(x-98)(x-k)}{x}.\]There exist exactly three positive real values of $k$ such that $f$ has a minimum at exactly two real values of $x$. Find the sum of these three values of $k$. | 240 | AIME25 | 60 | 1 |
Let\[f(x)=\frac{(x-18)(x-72)(x-98)(x-k)}{x}.\]There exist exactly three positive real values of $k$ such that $f$ has a minimum at exactly two real values of $x$. Find the sum of these three values of $k$. | 240 | AIME25 | 60 | 2 |
Let\[f(x)=\frac{(x-18)(x-72)(x-98)(x-k)}{x}.\]There exist exactly three positive real values of $k$ such that $f$ has a minimum at exactly two real values of $x$. Find the sum of these three values of $k$. | 240 | AIME25 | 60 | 3 |
Let\[f(x)=\frac{(x-18)(x-72)(x-98)(x-k)}{x}.\]There exist exactly three positive real values of $k$ such that $f$ has a minimum at exactly two real values of $x$. Find the sum of these three values of $k$. | 240 | AIME25 | 60 | 4 |
Let\[f(x)=\frac{(x-18)(x-72)(x-98)(x-k)}{x}.\]There exist exactly three positive real values of $k$ such that $f$ has a minimum at exactly two real values of $x$. Find the sum of these three values of $k$. | 240 | AIME25 | 60 | 5 |
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