context listlengths 1 1 | gen_kwargs dict | repeat_idx int64 0 4 | request_idx int64 0 499 | task_name stringclasses 2
values | metadata dict | model_outputs stringlengths 1.7k 98k |
|---|---|---|---|---|---|---|
[
{
"content": "Problem: Find the sum of all integer bases $b > 9$ for which $17_b$ is a divisor of $97_b$.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 0 | AIME25 | {
"expected_answer": "70",
"problem_id": "I-1",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to solve this problem. The question is asking for the sum of all integer bases \( b > 9 \) such that \( 17_b \) is a divisor of \( 97_b \). Hmm, bases can be a bit tricky, so I need to recall how numbers are represented in different bases.
First, let me convert the numbers \( 17_... |
[
{
"content": "Problem: On $\\triangle ABC$, points $A$, $D$, $E$, and $B$ lie in 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... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 1 | AIME25 | {
"expected_answer": "588",
"problem_id": "I-2",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to tackle this geometry problem step by step. First, I need to understand the problem statement clearly.
We have triangle ABC. Points A, D, E, B are on side AB in that order. The lengths are given: AD = 4, DE = 16, EB = 8. Similarly, points A, F, G, C are on side AC in that order... |
[
{
"content": "Problem: 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... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 2 | AIME25 | {
"expected_answer": "16",
"problem_id": "I-3",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to tackle this problem step by step. So, we have 9 baseball players, each choosing a single scoop of chocolate, vanilla, or strawberry ice cream. The conditions are:
1. At least one player chose each flavor.
2. The number of chocolate scoops (let's call this C) is greater than the... |
[
{
"content": "Problem: Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between -100 and 100, inclusive, such that $12x^2 - xy - 6y^2 = 0$.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 3 | AIME25 | {
"expected_answer": "117",
"problem_id": "I-4",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the number of ordered pairs (x, y) where both x and y are integers between -100 and 100 inclusive, and they satisfy the equation 12x² - xy - 6y² = 0. Hmm, let's start by understanding the equation and figuring out how to approach this.
First, the equation is a quadratic in... |
[
{
"content": "Problem: 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.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 4 | AIME25 | {
"expected_answer": "279",
"problem_id": "I-5",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to figure out this problem. So, we need to find how many 8-digit numbers using each digit from 1 to 8 exactly once are divisible by 22. Then, subtract 2025 from that number and put the answer in a box. Hmm, okay. Let's start by understanding the problem step by step.
First, 8-digi... |
[
{
"content": "Problem: 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 $r^2 + s^2$.\nMark your solution with \\boxed\nAns... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 5 | AIME25 | {
"expected_answer": "504",
"problem_id": "I-6",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to solve this problem step by step. So, we have an isosceles trapezoid with an inscribed circle. The radius of the circle is 3, and the area is 72. We need to find \( r^2 + s^2 \) where \( r \) and \( s \) are the lengths of the parallel sides.
First, let me recall some propertie... |
[
{
"content": "Problem: 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 then those six words are listed alphabetical... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 6 | AIME25 | {
"expected_answer": "821",
"problem_id": "I-7",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to tackle this probability problem. So, we have twelve letters: A, B, C, D, E, F, G, H, I, J, K, L. They are randomly grouped into six pairs. Each pair is then arranged in alphabetical order to form two-letter words. Then these six words are listed alphabetically. We need to find t... |
[
{
"content": "Problem: Let $k$ be a real number such that the system\n$|25 + 20i - z| = 5$\n$|z - 4 - k| = |z - 3i - k|$\nhas 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 ... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 7 | AIME25 | {
"expected_answer": "77",
"problem_id": "I-8",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to tackle this problem step by step. The problem says that we need to find all real numbers k such that the system of equations:
1. |25 + 20i - z| = 5
2. |z - 4 - k| = |z - 3i - k|
has exactly one complex solution z. Then, we need to find the sum of all possible values of k and e... |
[
{
"content": "Problem: The parabola with equation $y = x^2 - 4$ is rotated $60°$ 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... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 8 | AIME25 | {
"expected_answer": "62",
"problem_id": "I-9",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the y-coordinate of the unique intersection point in the fourth quadrant between the original parabola y = x² - 4 and its rotated image after a 60° counterclockwise rotation around the origin. Then express this y-coordinate in the form (a - sqrt(b))/c and find a + b + c. Hm... |
[
{
"content": "Problem: The 27 cells of a $3 \\times 9$ grid are filled in using the numbers 1 through 9 so that each row contains 9 different numbers, and each of the three $3 \\times 3$ blocks heavily outlined in the example contains 9 different numbers, as in the first three rows of a Sudoku puzzle. The numbe... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 9 | AIME25 | {
"expected_answer": "81",
"problem_id": "I-10",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to figure out this problem. So, we have a 3x9 grid where each row is filled with numbers 1 through 9, and each of the three 3x3 blocks (the heavily outlined ones) also contain numbers 1 through 9 without repetition. The question is asking for the number of different ways to fill su... |
[
{
"content": "Problem: A piecewise linear periodic function is defined by\n$f(x)=\\begin{cases} x & \\text{if } x \\in [-1,1) \\\\ 2-x & \\text{if } x \\in [1,3) \\end{cases}$\nand $f(x+4)=f(x)$ for all real numbers $x$. The parabola $x=34y^2$ intersects the graph of $f(x)$ at finitely many points. The sum of t... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 10 | AIME25 | {
"expected_answer": "259",
"problem_id": "I-11",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to tackle this problem step by step. The problem involves a piecewise linear periodic function f(x) and a parabola x = 34y². We need to find the sum of the y-coordinates of their intersection points, expressed in a specific form, and then compute a + b + c + d where the sum is (a +... |
[
{
"content": "Problem: The set of points in 3-dimensional coordinate space that lie in the plane $x + y + z = 75$ whose coordinates satisfy the inequalities\n$x - yz < y - zx < z - xy$\nforms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expre... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 11 | AIME25 | {
"expected_answer": "510",
"problem_id": "I-12",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to tackle this problem step by step. The problem is about finding the area of a finite region formed by the intersection of the plane x + y + z = 75 with the inequalities x - yz < y - zx < z - xy. The answer needs to be expressed in the form a√b where a and b are integers with b sq... |
[
{
"content": "Problem: 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... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 12 | AIME25 | {
"expected_answer": "204",
"problem_id": "I-13",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the expected number of regions created by 27 line segments on a disk. The disk is divided into four quadrants by two perpendicular diameters. Then Alex adds 25 more line segments. Each of these 25 segments is drawn by selecting two points at random on the perimeter, but the... |
[
{
"content": "Problem: Let $ABCDE$ be a convex pentagon with $AB = 14$, $BC = 7$, $CD = 24$, $DE = 13$, $EA = 26$, and $\\angle ZBA = 60°$. 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 posi... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 13 | AIME25 | {
"expected_answer": "60",
"problem_id": "I-14",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the minimum value of f(X) = AX + BX + CX + DX + EX for a convex pentagon ABCDE with given side lengths: AB=14, BC=7, CD=24, DE=13, EA=26. Also, there's an angle given: angle ZBA = 60°, but I'm not sure what point Z is. Maybe a typo? The problem mentions "for each point X in... |
[
{
"content": "Problem: 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.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 14 | AIME25 | {
"expected_answer": "735",
"problem_id": "I-15",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the number of ordered triples (a, b, c) where each of a, b, c is a positive integer less than or equal to 3^6, and the sum of their cubes is a multiple of 3^7. Then, I have to find this number modulo 1000. Hmm, let's break this down step by step.
First, let me understand t... |
[
{
"content": "Problem: 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 $\\triangle BGE.$\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 15 | AIME25 | {
"expected_answer": "468",
"problem_id": "II-1",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to solve this geometry problem. So, we have six points A, B, C, D, E, F lying in a straight line in that order. There's a point G not on the line. We're given various distances: AC=26, BD=22, CE=31, DF=33, AF=73, CG=40, and DG=30. We need to find the area of triangle BGE.
First, l... |
[
{
"content": "Problem: Find the sum of all positive integers $n$ such that $n+2$ divides the product $3(n+3)(n^2+9)$.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 16 | AIME25 | {
"expected_answer": "49",
"problem_id": "II-2",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to solve this problem: Find the sum of all positive integers \( n \) such that \( n + 2 \) divides the product \( 3(n + 3)(n^2 + 9) \). Hmm, divisibility problem. Let me start by understanding what exactly is being asked here.
First, we need to find all positive integers \( n \) w... |
[
{
"content": "Problem: 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 way that each square has $2$ red sides and blue sides. One example is shown below (red is solid, blue is dashed). Find the number of such... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 17 | AIME25 | {
"expected_answer": "82",
"problem_id": "II-3",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to figure out this problem. So, we have a 2x2 grid of squares, which means there are 4 squares arranged in a 2x2 block. Each of the 12 unit line segments (the sides of the squares) is colored either red or blue. The condition is that each square must have exactly 2 red sides and 2 ... |
[
{
"content": "Problem: The product \\[\\prod^{63}_{k=4} \\frac{\\log_k (5^{k^2 - 1})}{\\log_{k + 1} (5^{k^2 - 4})} = \\frac{\\log_4 (5^{15})}{\\log_5 (5^{12})} \\cdot \\frac{\\log_5 (5^{24})}{\\log_6 (5^{21})}\\cdot \\frac{\\log_6 (5^{35})}{\\log_7 (5^{32})} \\cdots \\frac{\\log_{63} (5^{3968})}{\\log_{64} (5^{... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 18 | AIME25 | {
"expected_answer": "106",
"problem_id": "II-4",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to tackle this problem step by step. The problem is about a product of fractions, each involving logarithms with different bases and exponents. The goal is to simplify this product and express it as a fraction m/n where m and n are coprime, then find m + n.
First, let me parse the... |
[
{
"content": "Problem: 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 $\\triangle DEF$ intersects $\\overl... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 19 | AIME25 | {
"expected_answer": "336",
"problem_id": "II-5",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to tackle this geometry problem step by step. First, I need to understand the given information and what's being asked. So, we have triangle ABC with angles 84°, 60°, and 36°. The midpoints D, E, F of sides BC, AC, and AB respectively form triangle DEF. The circumcircle of DEF inte... |
[
{
"content": "Problem: 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... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 20 | AIME25 | {
"expected_answer": "293",
"problem_id": "II-6",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to tackle this geometry problem step by step. It seems a bit complex at first glance, but if I break it down, maybe I can figure it out. Let me start by understanding the problem statement thoroughly.
First, there are two circles, ω₁ and ω₂. ω₁ has a radius of 6 and is centered at... |
[
{
"content": "Problem: 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 ... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 21 | AIME25 | {
"expected_answer": "237",
"problem_id": "II-7",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to tackle this problem step by step. The problem says:
Let A be the set of positive integer divisors of 2025. Let B be a randomly selected subset of A. We need to find the probability that B is a nonempty set with the property that the least common multiple (LCM) of its elements i... |
[
{
"content": "Problem: 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 no... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 22 | AIME25 | {
"expected_answer": "610",
"problem_id": "II-8",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to figure out this problem. So, Silas is using the greedy algorithm to make change with 1, 10, and 25 cent coins. The question is asking how many values of N between 1 and 1000 inclusive will have the greedy algorithm succeed. That means, for those N, there's no way to make the tot... |
[
{
"content": "Problem: 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 $n+t$.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 23 | AIME25 | {
"expected_answer": "149",
"problem_id": "II-9",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to tackle this problem step by step. The question is about finding the number of solutions where a function f(x) = sin(7π sin(5x)) is zero, and among those solutions, how many are points where the graph is tangent to the x-axis. Then we need to find n + t where n is the total numbe... |
[
{
"content": "Problem: 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 $1000$.\nMark your solution with \\boxe... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 24 | AIME25 | {
"expected_answer": "907",
"problem_id": "II-10",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to tackle this problem step by step. The problem says there are 16 chairs arranged in a row, and eight people each select a chair such that no person sits next to two other people. We need to find the number of subsets of 16 chairs that could be selected, which is denoted as N, and... |
[
{
"content": "Problem: 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.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 25 | AIME25 | {
"expected_answer": "113",
"problem_id": "II-11",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the number of ways to draw 12 segments in a regular 24-gon such that each vertex is an endpoint of exactly one segment. Hmm, let me start by understanding the problem.
First, a regular 24-gon has 24 vertices. Each vertex needs to be paired with another vertex such that eac... |
[
{
"content": "Problem: Let $A_1A_2\\dots A_{11}$ be a non-convex $11$-gon such that \n- The area of $A_iA_1A_{i+1}$ is $1$ for each $2 \\le i \\le 10$, \n- $\\cos(\\angle A_iA_1A_{i+1})=\\frac{12}{13}$ for each $2 \\le i \\le 10$, \n- The perimeter of $A_1A_2\\dots A_{11}$ is $20$. \nIf $A_1A_2+A_1A_{11}$ can b... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 26 | AIME25 | {
"expected_answer": "19",
"problem_id": "II-12",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to tackle this geometry problem. It's about a non-convex 11-gon with some specific area and perimeter conditions. Hmm, let's break down the problem step by step.
First, the polygon is non-convex, which probably means that at least one of its internal angles is greater than 180 deg... |
[
{
"content": "Problem: 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 $\\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find the remainder when $m+n$ is div... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 27 | AIME25 | {
"expected_answer": "248",
"problem_id": "II-13",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to tackle this problem step by step. The problem is about a sequence of rational numbers defined recursively. The initial term is x₁ = 25/11, and each subsequent term is given by the formula x_{k+1} = (1/3)(x_k + 1/x_k - 1). We need to find x_{2025} expressed as a reduced fraction ... |
[
{
"content": "Problem: 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 $n.$\nMa... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 28 | AIME25 | {
"expected_answer": "104",
"problem_id": "II-14",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to tackle this geometry problem. So, we have a right triangle ABC with a right angle at A, and the hypotenuse BC is 38 units long. There are two points K and L inside the triangle such that AK = AL = BK = CL = KL = 14. We need to find the area of quadrilateral BKLC, which is expres... |
[
{
"content": "Problem: 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 $k$.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 29 | AIME25 | {
"expected_answer": "240",
"problem_id": "II-15",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to tackle this problem step by step. The function given is \( f(x) = \frac{(x - 18)(x - 72)(x - 98)(x - k)}{x} \). We need to find the three positive real values of \( k \) such that \( f \) has a minimum at exactly two real values of \( x \). Then, we have to find the sum of these... |
[
{
"content": "Problem: Find the sum of all integer bases $b > 9$ for which $17_b$ is a divisor of $97_b$.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 0 | AIME25 | {
"expected_answer": "70",
"problem_id": "I-1",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find all integer bases b greater than 9 where the number 17 in base b divides the number 97 in the same base. Then sum all those bases. Hmm, let me think step by step.
First, let me recall how numbers are represented in different bases. For example, 17_b means 1*b + 7 in decima... |
[
{
"content": "Problem: On $\\triangle ABC$, points $A$, $D$, $E$, and $B$ lie in 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... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 1 | AIME25 | {
"expected_answer": "588",
"problem_id": "I-2",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's tackle this geometry problem step by step. It's a bit complex, but I'll try to break it down. First, let me make sure I understand the problem correctly.
We have triangle ABC. Points A, D, E, B are on side AB in that order. The lengths are AD = 4, DE = 16, EB = 8. Similarly, points A,... |
[
{
"content": "Problem: 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... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 2 | AIME25 | {
"expected_answer": "16",
"problem_id": "I-3",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem where there are 9 baseball team members, each choosing a single scoop of chocolate, vanilla, or strawberry ice cream. The conditions are that at least one player chose each flavor, and the number of players who chose chocolate is greater than vanilla, which is... |
[
{
"content": "Problem: Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between -100 and 100, inclusive, such that $12x^2 - xy - 6y^2 = 0$.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 3 | AIME25 | {
"expected_answer": "117",
"problem_id": "I-4",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the number of ordered pairs (x, y) where both x and y are integers between -100 and 100 inclusive, and they satisfy the equation 12x² - xy - 6y² = 0. Hmm, let's start by understanding what this equation represents and how I can approach solving it.
First, the equation is a... |
[
{
"content": "Problem: 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.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 4 | AIME25 | {
"expected_answer": "279",
"problem_id": "I-5",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the number of eight-digit numbers using each digit from 1 to 8 exactly once that are divisible by 22. Then subtract 2025 from that number to get the answer. Let me start by understanding the problem step by step.
First, I know that 22 is 2 times 11. So, for a number to be ... |
[
{
"content": "Problem: 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 $r^2 + s^2$.\nMark your solution with \\boxed\nAns... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 5 | AIME25 | {
"expected_answer": "504",
"problem_id": "I-6",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem about an isosceles trapezoid with an inscribed circle. The radius of the circle is 3, and the area is 72. I have to find r squared plus s squared, where r and s are the lengths of the two parallel sides. Hmm, let's start by recalling what I know about trapezoi... |
[
{
"content": "Problem: 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 then those six words are listed alphabetical... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 6 | AIME25 | {
"expected_answer": "821",
"problem_id": "I-7",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the probability that the last word listed contains the letter G. The setup is that we have twelve letters grouped into six pairs. Each pair is arranged in alphabetical order, and then the six two-letter words are listed alphabetically. The question is asking for the probabi... |
[
{
"content": "Problem: Let $k$ be a real number such that the system\n$|25 + 20i - z| = 5$\n$|z - 4 - k| = |z - 3i - k|$\nhas 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 ... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 7 | AIME25 | {
"expected_answer": "77",
"problem_id": "I-8",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem where there's a complex number z that satisfies two equations, and the system has exactly one solution. Then, I have to find the sum of all possible k values, which is m/n, and then compute m + n. Alright, let's start by understanding each equation and how the... |
[
{
"content": "Problem: The parabola with equation $y = x^2 - 4$ is rotated $60°$ 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... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 8 | AIME25 | {
"expected_answer": "62",
"problem_id": "I-9",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem where a parabola y = x² - 4 is rotated 60 degrees counterclockwise around the origin, and then find the unique point in the fourth quadrant where the original parabola and the rotated one intersect. The y-coordinate of this intersection point is given in the f... |
[
{
"content": "Problem: The 27 cells of a $3 \\times 9$ grid are filled in using the numbers 1 through 9 so that each row contains 9 different numbers, and each of the three $3 \\times 3$ blocks heavily outlined in the example contains 9 different numbers, as in the first three rows of a Sudoku puzzle. The numbe... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 9 | AIME25 | {
"expected_answer": "81",
"problem_id": "I-10",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the number of ways to fill a 3x9 grid with numbers 1 through 9 such that each row has distinct numbers and each 3x3 block (there are three of them in a 3x9 grid) also has distinct numbers. Then, express this number in terms of prime factors and compute the weighted sum of t... |
[
{
"content": "Problem: A piecewise linear periodic function is defined by\n$f(x)=\\begin{cases} x & \\text{if } x \\in [-1,1) \\\\ 2-x & \\text{if } x \\in [1,3) \\end{cases}$\nand $f(x+4)=f(x)$ for all real numbers $x$. The parabola $x=34y^2$ intersects the graph of $f(x)$ at finitely many points. The sum of t... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 10 | AIME25 | {
"expected_answer": "259",
"problem_id": "I-11",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the sum of the y-coordinates where the parabola x = 34y² intersects the graph of the piecewise function f(x). The function f(x) is periodic with period 4, defined as x = y for x in [-1,1), and x = 2 - y for x in [1,3). Then it repeats every 4 units. The parabola x = 34y² is... |
[
{
"content": "Problem: The set of points in 3-dimensional coordinate space that lie in the plane $x + y + z = 75$ whose coordinates satisfy the inequalities\n$x - yz < y - zx < z - xy$\nforms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expre... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 11 | AIME25 | {
"expected_answer": "510",
"problem_id": "I-12",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem where we have points in 3D space lying in the plane x + y + z = 75. The coordinates x, y, z must satisfy the inequalities x - yz < y - zx < z - xy. These inequalities form three disjoint convex regions, and exactly one of them has finite area. The task is to f... |
[
{
"content": "Problem: 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... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 12 | AIME25 | {
"expected_answer": "204",
"problem_id": "I-13",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so Alex has a disk divided into four quadrants by two perpendicular diameters. Then he adds 25 more line segments by randomly selecting two points on the perimeter in different quadrants and connecting them. We need to find the expected number of regions these 27 line segments divide the dis... |
[
{
"content": "Problem: Let $ABCDE$ be a convex pentagon with $AB = 14$, $BC = 7$, $CD = 24$, $DE = 13$, $EA = 26$, and $\\angle ZBA = 60°$. 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 posi... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 13 | AIME25 | {
"expected_answer": "60",
"problem_id": "I-14",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the minimum value of f(X) = AX + BX + CX + DX + EX, where X is any point in the plane, and the pentagon ABCDE has given side lengths: AB=14, BC=7, CD=24, DE=13, EA=26. Also, there's an angle mentioned: angle ZBA = 60°. Wait, angle ZBA... Hmm, but the problem statement doesn... |
[
{
"content": "Problem: 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.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 14 | AIME25 | {
"expected_answer": "735",
"problem_id": "I-15",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the number of ordered triples (a, b, c) where each of a, b, c is a positive integer less than or equal to 3^6, and the sum of their cubes is a multiple of 3^7. Then, I have to find the remainder when this number is divided by 1000. Hmm, this seems like a problem involving m... |
[
{
"content": "Problem: 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 $\\triangle BGE.$\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 15 | AIME25 | {
"expected_answer": "468",
"problem_id": "II-1",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's see. We have six points A, B, C, D, E, F arranged in a straight line in that order. There's a point G not on the line, and we're given some distances: AC=26, BD=22, CE=31, DF=33, AF=73, CG=40, and DG=30. We need to find the area of triangle BGE. Hmm, okay. Let me try to visualize this.... |
[
{
"content": "Problem: Find the sum of all positive integers $n$ such that $n+2$ divides the product $3(n+3)(n^2+9)$.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 16 | AIME25 | {
"expected_answer": "49",
"problem_id": "II-2",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find all positive integers n such that n + 2 divides the product 3(n + 3)(n² + 9). Then, sum all those n's. Hmm, let me start by understanding the problem step by step.
First, the condition is that n + 2 divides 3(n + 3)(n² + 9). So, mathematically, this means that 3(n + 3)(n² ... |
[
{
"content": "Problem: 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 way that each square has $2$ red sides and blue sides. One example is shown below (red is solid, blue is dashed). Find the number of such... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 17 | AIME25 | {
"expected_answer": "82",
"problem_id": "II-3",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the number of colorings for a 2x2 grid of unit squares where each square has exactly 2 red sides and 2 blue sides. Each of the 12 unit line segments (the edges of the grid) is colored either red or blue. The example given has red solid and blue dashed, but the problem is as... |
[
{
"content": "Problem: The product \\[\\prod^{63}_{k=4} \\frac{\\log_k (5^{k^2 - 1})}{\\log_{k + 1} (5^{k^2 - 4})} = \\frac{\\log_4 (5^{15})}{\\log_5 (5^{12})} \\cdot \\frac{\\log_5 (5^{24})}{\\log_6 (5^{21})}\\cdot \\frac{\\log_6 (5^{35})}{\\log_7 (5^{32})} \\cdots \\frac{\\log_{63} (5^{3968})}{\\log_{64} (5^{... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 18 | AIME25 | {
"expected_answer": "106",
"problem_id": "II-4",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem where there's a product from k=4 to k=63 of some fractions involving logarithms, and then express the result as a reduced fraction m/n and find m + n. Let me start by understanding the structure of the product.
First, the product is given as:
\[
\prod^{63}_{... |
[
{
"content": "Problem: 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 $\\triangle DEF$ intersects $\\overl... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 19 | AIME25 | {
"expected_answer": "336",
"problem_id": "II-5",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this geometry problem. Let me read it again and make sure I understand all the parts.
We have triangle ABC with angles at A being 84 degrees, at B being 60 degrees, and at C being 36 degrees. Then, D, E, and F are midpoints of sides BC, AC, and AB respectively. The circum... |
[
{
"content": "Problem: 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... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 20 | AIME25 | {
"expected_answer": "293",
"problem_id": "II-6",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to tackle this problem step by step. It's a geometry problem involving circles, rectangles, and triangles with areas. Hmm, the problem seems a bit complex, but let's break it down.
First, let's parse the given information:
1. There's a circle ω₁ with radius 6 centered at point A.... |
[
{
"content": "Problem: 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 ... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 21 | AIME25 | {
"expected_answer": "237",
"problem_id": "II-7",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem where we have the set A of all positive integer divisors of 2025. Then, a subset B of A is randomly selected. We need to find the probability that B is a non-empty set where the least common multiple (LCM) of its elements is exactly 2025. The answer should be ... |
[
{
"content": "Problem: 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 no... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 22 | AIME25 | {
"expected_answer": "610",
"problem_id": "II-8",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to figure out how many values of N between 1 and 1000 inclusive will make the greedy algorithm succeed. The greedy algorithm here is choosing the highest denomination first, then the next, etc., to minimize the number of coins. But sometimes there's a smarter way with fewer coins. ... |
[
{
"content": "Problem: 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 $n+t$.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 23 | AIME25 | {
"expected_answer": "149",
"problem_id": "II-9",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's tackle this problem step by step. The question is about finding the sum of two numbers, n and t, where n is the number of solutions in the interval 0 < x < 2π for the equation f(x) = sin(7π sin(5x)) = 0, and t is the number of those solutions where the graph of y = f(x) is tangent to t... |
[
{
"content": "Problem: 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 $1000$.\nMark your solution with \\boxe... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 24 | AIME25 | {
"expected_answer": "907",
"problem_id": "II-10",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem where there are 16 chairs arranged in a row, and eight people each choose a chair such that no person sits next to another. Then, we need to find the number of subsets of these 16 chairs that could be selected, and the remainder when that number is divided by ... |
[
{
"content": "Problem: 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.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 25 | AIME25 | {
"expected_answer": "113",
"problem_id": "II-11",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the number of ways to draw 12 segments of equal lengths on a regular 24-gon such that each vertex is an endpoint of exactly one segment. Hmm, let me think.
First, a regular 24-gon has 24 vertices equally spaced around a circle. Each vertex is connected to its neighbors at... |
[
{
"content": "Problem: Let $A_1A_2\\dots A_{11}$ be a non-convex $11$-gon such that \n- The area of $A_iA_1A_{i+1}$ is $1$ for each $2 \\le i \\le 10$, \n- $\\cos(\\angle A_iA_1A_{i+1})=\\frac{12}{13}$ for each $2 \\le i \\le 10$, \n- The perimeter of $A_1A_2\\dots A_{11}$ is $20$. \nIf $A_1A_2+A_1A_{11}$ can b... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 26 | AIME25 | {
"expected_answer": "19",
"problem_id": "II-12",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem about a non-convex 11-gon with certain area and perimeter conditions. Let me try to break it down step by step.
First, let me restate the problem to make sure I understand it correctly. We have an 11-sided polygon, labeled A₁ through A₁₁, which is non-convex... |
[
{
"content": "Problem: 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 $\\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find the remainder when $m+n$ is div... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 27 | AIME25 | {
"expected_answer": "248",
"problem_id": "II-13",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the value of \( x_{2025} \) in the sequence defined by \( x_1 = \frac{25}{11} \) and the recurrence relation \( x_{k+1} = \frac{1}{3}\left(x_k + \frac{1}{x_k} - 1\right) \). Then, express \( x_{2025} \) as a reduced fraction \( \frac{m}{n} \) and compute \( m + n \mod 1000 ... |
[
{
"content": "Problem: 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 $n.$\nMa... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 28 | AIME25 | {
"expected_answer": "104",
"problem_id": "II-14",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this geometry problem about a right triangle ABC with a right angle at A, and BC = 38. There are points K and L inside the triangle such that AK = AL = BK = CL = KL = 14. The area of quadrilateral BKLC is supposed to be n√3, and I need to find n. Hmm, this seems a bit comp... |
[
{
"content": "Problem: 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 $k$.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 1,
"temperature": 0.7
} | 1 | 29 | AIME25 | {
"expected_answer": "240",
"problem_id": "II-15",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem where we have the function f(x) = [(x - 18)(x - 72)(x - 98)(x - k)] / x. The problem states that there exist exactly three positive real values of k such that the function f has a minimum at exactly two real values of x. Then, I need to find the sum of these t... |
[
{
"content": "Problem: Find the sum of all integer bases $b > 9$ for which $17_b$ is a divisor of $97_b$.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 0 | AIME25 | {
"expected_answer": "70",
"problem_id": "I-1",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find all integer bases b greater than 9 where the number 17 in base b divides the number 97 in base b. Then sum all those bases. Hmm, let's break this down step by step.
First, I remember that numbers in different bases can be converted to decimal by expanding each digit multi... |
[
{
"content": "Problem: On $\\triangle ABC$, points $A$, $D$, $E$, and $B$ lie in 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... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 1 | AIME25 | {
"expected_answer": "588",
"problem_id": "I-2",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the area of this heptagon AFNBCEM on triangle ABC. The problem gives me some points on the sides AB and AC with specific lengths, and mentions that the area of quadrilateral DEGF is 288. Let me start by visualizing the triangle and all the points mentioned.
First, let's sk... |
[
{
"content": "Problem: 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... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 2 | AIME25 | {
"expected_answer": "16",
"problem_id": "I-3",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem where there are 9 baseball team members, each choosing a single scoop of either chocolate, vanilla, or strawberry ice cream. The conditions are that each flavor is chosen by at least one player, and the number of players choosing chocolate is greater than thos... |
[
{
"content": "Problem: Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between -100 and 100, inclusive, such that $12x^2 - xy - 6y^2 = 0$.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 3 | AIME25 | {
"expected_answer": "117",
"problem_id": "I-4",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the number of ordered pairs (x, y) where both x and y are integers between -100 and 100 inclusive, such that 12x² - xy -6y² = 0. Hmm, let's start by understanding the equation. Maybe I can factor it or manipulate it somehow.
First, let me write down the equation again: 12x... |
[
{
"content": "Problem: 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.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 4 | AIME25 | {
"expected_answer": "279",
"problem_id": "I-5",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the number of eight-digit pandigital numbers (using each digit from 1 to 8 exactly once) that are divisible by 22. Then, subtract 2025 from that number to get the answer. Let me break this down step by step.
First, I know that a number is divisible by 22 if and only if it'... |
[
{
"content": "Problem: 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 $r^2 + s^2$.\nMark your solution with \\boxed\nAns... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 5 | AIME25 | {
"expected_answer": "504",
"problem_id": "I-6",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem about an isosceles trapezoid with an inscribed circle. The radius is 3, the area is 72, and I need to find r² + s² where r and s are the lengths of the parallel sides and r ≠ s. Hmm, let me start by recalling some properties of trapezoids with inscribed circle... |
[
{
"content": "Problem: 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 then those six words are listed alphabetical... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 6 | AIME25 | {
"expected_answer": "821",
"problem_id": "I-7",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this probability problem. Let me read it again carefully to make sure I understand all the parts.
We have twelve letters: A, B, C, D, E, F, G, H, I, J, K, L. They are randomly grouped into six pairs. Then, in each pair, the two letters are placed next to each other in alp... |
[
{
"content": "Problem: Let $k$ be a real number such that the system\n$|25 + 20i - z| = 5$\n$|z - 4 - k| = |z - 3i - k|$\nhas 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 ... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 7 | AIME25 | {
"expected_answer": "77",
"problem_id": "I-8",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem where we have a system of two equations involving complex numbers z, and we need to find all real numbers k such that the system has exactly one complex solution. Then, the sum of all possible k's should be expressed as a reduced fraction m/n, and we need to f... |
[
{
"content": "Problem: The parabola with equation $y = x^2 - 4$ is rotated $60°$ 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... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 8 | AIME25 | {
"expected_answer": "62",
"problem_id": "I-9",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem where a parabola y = x² - 4 is rotated 60 degrees counterclockwise around the origin, and then find the unique point in the fourth quadrant where the original parabola and the rotated image intersect. The answer should be in the form (a - sqrt(b))/c, and then ... |
[
{
"content": "Problem: The 27 cells of a $3 \\times 9$ grid are filled in using the numbers 1 through 9 so that each row contains 9 different numbers, and each of the three $3 \\times 3$ blocks heavily outlined in the example contains 9 different numbers, as in the first three rows of a Sudoku puzzle. The numbe... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 9 | AIME25 | {
"expected_answer": "81",
"problem_id": "I-10",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to figure out the number of ways to fill a 3x9 grid where each row has 9 distinct numbers from 1 to 9, and each of the three 3x3 blocks (which I assume are the three vertical 3x3 sections) also contains all distinct numbers. Then express that number in terms of primes raised to exp... |
[
{
"content": "Problem: A piecewise linear periodic function is defined by\n$f(x)=\\begin{cases} x & \\text{if } x \\in [-1,1) \\\\ 2-x & \\text{if } x \\in [1,3) \\end{cases}$\nand $f(x+4)=f(x)$ for all real numbers $x$. The parabola $x=34y^2$ intersects the graph of $f(x)$ at finitely many points. The sum of t... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 10 | AIME25 | {
"expected_answer": "259",
"problem_id": "I-11",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem where we have a piecewise linear periodic function f(x), and a parabola x = 34y². The question is asking for the sum of the y-coordinates of the intersection points between these two graphs, expressed in a specific form, and then find the sum of the coefficien... |
[
{
"content": "Problem: The set of points in 3-dimensional coordinate space that lie in the plane $x + y + z = 75$ whose coordinates satisfy the inequalities\n$x - yz < y - zx < z - xy$\nforms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expre... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 11 | AIME25 | {
"expected_answer": "510",
"problem_id": "I-12",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem where we have points in 3D space lying on the plane x + y + z = 75, and their coordinates satisfy the inequalities x - yz < y - zx < z - xy. The regions formed by these inequalities are three disjoint convex regions, and exactly one of them has finite area. Th... |
[
{
"content": "Problem: 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... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 12 | AIME25 | {
"expected_answer": "204",
"problem_id": "I-13",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the expected number of regions created by 27 line segments drawn on a disk. The disk is divided into four quadrants by two perpendicular diameters. Then, Alex adds 25 more line segments. Each of these segments is drawn by picking two points at random on the perimeter, in di... |
[
{
"content": "Problem: Let $ABCDE$ be a convex pentagon with $AB = 14$, $BC = 7$, $CD = 24$, $DE = 13$, $EA = 26$, and $\\angle ZBA = 60°$. 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 posi... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 13 | AIME25 | {
"expected_answer": "60",
"problem_id": "I-14",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the least possible value of the function f(X) = AX + BX + CX + DX + EX for a convex pentagon ABCDE with given side lengths: AB = 14, BC = 7, CD = 24, DE = 13, EA = 26, and angle ZBA = 60°. Hmm, first, let me make sure I understand the problem correctly. The pentagon is conv... |
[
{
"content": "Problem: 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.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 14 | AIME25 | {
"expected_answer": "735",
"problem_id": "I-15",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the number of ordered triples (a, b, c) where each of a, b, c is a positive integer less than or equal to 3^6, and the sum of their cubes is divisible by 3^7. Then, I have to find this number modulo 1000. Hmm, let me break this down step by step.
First, let me recall that ... |
[
{
"content": "Problem: 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 $\\triangle BGE.$\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 15 | AIME25 | {
"expected_answer": "468",
"problem_id": "II-1",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I have this geometry problem here. Let me read it again to make sure I understand all the details. There are six points A, B, C, D, E, and F lying in a straight line in that order. There's a point G not on the line. The distances given are: AC=26, BD=22, CE=31, DF=33, AF=73, CG=40, and DG... |
[
{
"content": "Problem: Find the sum of all positive integers $n$ such that $n+2$ divides the product $3(n+3)(n^2+9)$.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 16 | AIME25 | {
"expected_answer": "49",
"problem_id": "II-2",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find all positive integers n such that n + 2 divides the product 3(n + 3)(n² + 9). Then sum all those n's. Hmm, let's start by understanding the problem step by step.
First, let me restate the condition: n + 2 divides 3(n + 3)(n² + 9). That means when we divide 3(n + 3)(n² + 9)... |
[
{
"content": "Problem: 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 way that each square has $2$ red sides and blue sides. One example is shown below (red is solid, blue is dashed). Find the number of such... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 17 | AIME25 | {
"expected_answer": "82",
"problem_id": "II-3",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to figure out how many different ways there are to color the sides of a 2x2 grid of squares with red or blue such that each square has exactly two red sides and two blue sides. Hmm, let's start by understanding the problem better.
First, let me visualize the grid. A 2x2 grid of sq... |
[
{
"content": "Problem: The product \\[\\prod^{63}_{k=4} \\frac{\\log_k (5^{k^2 - 1})}{\\log_{k + 1} (5^{k^2 - 4})} = \\frac{\\log_4 (5^{15})}{\\log_5 (5^{12})} \\cdot \\frac{\\log_5 (5^{24})}{\\log_6 (5^{21})}\\cdot \\frac{\\log_6 (5^{35})}{\\log_7 (5^{32})} \\cdots \\frac{\\log_{63} (5^{3968})}{\\log_{64} (5^{... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 18 | AIME25 | {
"expected_answer": "106",
"problem_id": "II-4",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this product problem. Let me start by understanding what's given. The product is from k=4 to k=63 of [log_k (5^{k² -1}) divided by log_{k+1} (5^{k² -4})]. And then it's broken down into a series of fractions, each involving log base 4, 5, 6, ..., 63 of 5 raised to some exp... |
[
{
"content": "Problem: 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 $\\triangle DEF$ intersects $\\overl... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 19 | AIME25 | {
"expected_answer": "336",
"problem_id": "II-5",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this geometry problem about triangle ABC with specific angles and midpoints. Let me start by understanding the problem step by step.
First, triangle ABC has angles at A, B, and C as 84°, 60°, and 36° respectively. Then, D, E, and F are the midpoints of BC, AC, and AB. The... |
[
{
"content": "Problem: 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... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 20 | AIME25 | {
"expected_answer": "293",
"problem_id": "II-6",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to tackle this geometry problem step by step. It's a bit complex, so I need to break it down carefully. Let me start by understanding the given information and visualizing the figure.
First, there are two circles, ω₁ and ω₂. ω₁ has a radius of 6 and is centered at point A. It's in... |
[
{
"content": "Problem: 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 ... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 21 | AIME25 | {
"expected_answer": "237",
"problem_id": "II-7",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem where we have the set A of all positive integer divisors of 2025. Then, a subset B is randomly selected from A. We need to find the probability that B is a nonempty set where the least common multiple (LCM) of its elements is 2025. The answer should be in the ... |
[
{
"content": "Problem: 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 no... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 22 | AIME25 | {
"expected_answer": "610",
"problem_id": "II-8",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to figure out how many values of N between 1 and 1000 inclusive have the property that Silas's greedy algorithm (choosing coins starting with the highest denomination, 25 cents, then 10, then 1) uses the minimal number of coins. The problem states that the greedy algorithm succeeds... |
[
{
"content": "Problem: 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 $n+t$.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 23 | AIME25 | {
"expected_answer": "149",
"problem_id": "II-9",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's try to solve this problem step by step. The question is about finding the sum of n and t, where n is the number of x-values in the interval 0 < x < 2π where f(x) = sin(7π sin5x) = 0, and t is the number of those x-values where the graph of y = f(x) is tangent to the x-axis. So, n + t i... |
[
{
"content": "Problem: 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 $1000$.\nMark your solution with \\boxe... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 24 | AIME25 | {
"expected_answer": "907",
"problem_id": "II-10",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I have this problem where there are 16 chairs arranged in a row. Eight people each pick a chair, and the condition is that no person sits next to two other people. Then, we need to find the number of subsets of 16 chairs that could be selected, and then find N modulo 1000. Hmm, let me try... |
[
{
"content": "Problem: 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.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 25 | AIME25 | {
"expected_answer": "113",
"problem_id": "II-11",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I have this problem where I need to find the number of ways to draw 12 segments of equal lengths on a regular 24-gon such that each vertex is an endpoint of exactly one segment. Hmm, let's try to break this down step by step.
First, let me visualize a regular 24-gon. It has 24 vertices e... |
[
{
"content": "Problem: Let $A_1A_2\\dots A_{11}$ be a non-convex $11$-gon such that \n- The area of $A_iA_1A_{i+1}$ is $1$ for each $2 \\le i \\le 10$, \n- $\\cos(\\angle A_iA_1A_{i+1})=\\frac{12}{13}$ for each $2 \\le i \\le 10$, \n- The perimeter of $A_1A_2\\dots A_{11}$ is $20$. \nIf $A_1A_2+A_1A_{11}$ can b... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 26 | AIME25 | {
"expected_answer": "19",
"problem_id": "II-12",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem about a non-convex 11-gon. Let me try to parse all the given information and figure out how to approach this step by step.
First, the problem states that there's an 11-gon labeled A₁A₂…A₁₁. The key conditions are:
1. For each i from 2 to 10, the area of tri... |
[
{
"content": "Problem: 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 $\\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find the remainder when $m+n$ is div... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 27 | AIME25 | {
"expected_answer": "248",
"problem_id": "II-13",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the value of x_{2025} in the sequence defined by x_1 = 25/11 and the recursive formula x_{k+1} = (1/3)(x_k + 1/x_k - 1). Then, express x_{2025} as a reduced fraction m/n and find the remainder when m + n is divided by 1000. Hmm, let's start by understanding the problem step... |
[
{
"content": "Problem: 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 $n.$\nMa... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 28 | AIME25 | {
"expected_answer": "104",
"problem_id": "II-14",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I have this geometry problem here. Let me read it again to make sure I understand all the details. It says:
"Let triangle ABC be a right triangle with angle A = 90 degrees and BC = 38. There exist points K and L inside the triangle such that AK = AL = BK = CL = KL = 14. The area of the q... |
[
{
"content": "Problem: 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 $k$.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 29 | AIME25 | {
"expected_answer": "240",
"problem_id": "II-15",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem where we have the function f(x) = [(x - 18)(x - 72)(x - 98)(x - k)] / x. The problem states that there exist exactly three positive real values of k such that f has a minimum at exactly two real values of x. I need to find the sum of these three values of k.
... |
[
{
"content": "Problem: Find the sum of all integer bases $b > 9$ for which $17_b$ is a divisor of $97_b$.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 3,
"temperature": 0.7
} | 3 | 0 | AIME25 | {
"expected_answer": "70",
"problem_id": "I-1",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find all integer bases b greater than 9 such that 17_b divides 97_b. Hmm, let's start by understanding what these numbers mean in base b.
First, in any base b, the number 17_b represents 1*b + 7, right? Similarly, 97_b would be 9*b + 7. So, converting both numbers to base 10, ... |
[
{
"content": "Problem: On $\\triangle ABC$, points $A$, $D$, $E$, and $B$ lie in 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... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 3,
"temperature": 0.7
} | 3 | 1 | AIME25 | {
"expected_answer": "588",
"problem_id": "I-2",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's see. I need to find the area of the heptagon AFNBCEM on triangle ABC. The given information includes points along sides AB and AC with specific lengths, reflections of points D and G through F and E respectively, and the area of quadrilateral DEGF is 288. Hmm, this seems a bit complex,... |
[
{
"content": "Problem: 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... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 3,
"temperature": 0.7
} | 3 | 2 | AIME25 | {
"expected_answer": "16",
"problem_id": "I-3",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's see. I need to find the number of different assignments of ice cream flavors to the 9 players, given the conditions, and then find N mod 1000. The conditions are that each flavor is chosen by at least one player, the number of players choosing chocolate is greater than vanilla, which i... |
[
{
"content": "Problem: Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between -100 and 100, inclusive, such that $12x^2 - xy - 6y^2 = 0$.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 3,
"temperature": 0.7
} | 3 | 3 | AIME25 | {
"expected_answer": "117",
"problem_id": "I-4",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the number of ordered pairs (x, y) where both x and y are integers between -100 and 100 inclusive, such that the equation 12x² - xy - 6y² = 0 holds. Hmm, let's start by understanding the equation and how to approach this.
First, the equation is quadratic in both x and y. M... |
[
{
"content": "Problem: 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.\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 3,
"temperature": 0.7
} | 3 | 4 | AIME25 | {
"expected_answer": "279",
"problem_id": "I-5",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to find the number of eight-digit pandigital numbers (using each digit from 1 to 8 exactly once) that are divisible by 22, and then subtract 2025 from that number. Hmm, let's start by understanding the problem step by step.
First, divisibility by 22 means the number has to be divi... |
[
{
"content": "Problem: 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 $r^2 + s^2$.\nMark your solution with \\boxed\nAns... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 3,
"temperature": 0.7
} | 3 | 5 | AIME25 | {
"expected_answer": "504",
"problem_id": "I-6",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's see. I need to find the sum of the squares of the two parallel sides of an isosceles trapezoid that has an inscribed circle. The radius of the circle is 3, and the area of the trapezoid is 72. Hmm, let me start by recalling some properties of trapezoids with inscribed circles.
First, ... |
[
{
"content": "Problem: 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 then those six words are listed alphabetical... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 3,
"temperature": 0.7
} | 3 | 6 | AIME25 | {
"expected_answer": "821",
"problem_id": "I-7",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's see. I need to find the probability that the last word listed alphabetically contains the letter G. The setup is that we have 12 letters grouped into six pairs, each pair is ordered alphabetically, and then the six two-letter words are listed alphabetically. The example given is AB, CJ... |
[
{
"content": "Problem: Let $k$ be a real number such that the system\n$|25 + 20i - z| = 5$\n$|z - 4 - k| = |z - 3i - k|$\nhas 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 ... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 3,
"temperature": 0.7
} | 3 | 7 | AIME25 | {
"expected_answer": "77",
"problem_id": "I-8",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, let's see. I need to find the sum of all possible values of k such that the given system of equations has exactly one complex solution z. Hmm. Let me start by understanding what each equation represents geometrically.
First, the equation |25 + 20i - z| = 5. That should represent a circle i... |
[
{
"content": "Problem: The parabola with equation $y = x^2 - 4$ is rotated $60°$ 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... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 3,
"temperature": 0.7
} | 3 | 8 | AIME25 | {
"expected_answer": "62",
"problem_id": "I-9",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to solve this problem where a parabola y = x² - 4 is rotated 60 degrees counterclockwise around the origin. Then, I have to find the unique point in the fourth quadrant where the original parabola and its image intersect. The answer should be in the form (a - sqrt(b))/c, where a, b... |
[
{
"content": "Problem: The 27 cells of a $3 \\times 9$ grid are filled in using the numbers 1 through 9 so that each row contains 9 different numbers, and each of the three $3 \\times 3$ blocks heavily outlined in the example contains 9 different numbers, as in the first three rows of a Sudoku puzzle. The numbe... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 3,
"temperature": 0.7
} | 3 | 9 | AIME25 | {
"expected_answer": "81",
"problem_id": "I-10",
"reference_solution": ""
} | <|begin_of_thought|>
Okay, so I need to figure out the number of different ways to fill a 3x9 grid with numbers 1 through 9, following Sudoku-like rules. Each row has 9 different numbers, and each of the three 3x3 blocks (heavily outlined) must also contain 9 different numbers. Then, express the result as p^a * q^b * ... |
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