subset large_string | id int64 | query large_string | answer large_string | GT int64 | pred int64 | correct int64 | original_answer large_string | gen_answer large_string | correct_parsed int64 |
|---|---|---|---|---|---|---|---|---|---|
AIME2025-I | 0 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
Problem:
Find the sum of all integer bases $b>9$ for which $17_{b}$ is a divisor of $97_{b}$.
Final Answer: | <think>
Okay, so I need to solve this AIME problem where I have to find the sum of all integer bases b > 9 for which 17 base b divides 97 base b. Hmm, let me think about how to approach this.
First, I remember that numbers in different bases can be converted to base 10 to make them easier to work with. So, let me star... | 70 | 70 | 1 | <think>
Okay, so I need to solve this AIME problem where I have to find the sum of all integer bases b > 9 for which 17 base b divides 97 base b. Hmm, let me think about how to approach this.
First, I remember that numbers in different bases can be converted to base 10 to make them easier to work with. So, let me star... | 70 | 1 |
AIME2025-I | 1 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
Problem:
On $\triangle ABC$ points $A,D,E$, and $B$ lie that order on side $\overline{AB}$ with $AD=4, DE=16$, and $EB=8$. Points $A,F,G$, and $C$ lie in that order on side $\overline{AC}$ with $AF=13, FG=52$, and $GC=26$. Let $M... | <think>
Okay, so I need to solve this AIME problem. Let me try to visualize the problem first. There's triangle ABC. On side AB, there are points A, D, E, B in that order, with AD = 4, DE = 16, and EB = 8. On side AC, there are points A, F, G, C in order, with AF = 13, FG = 52, and GC = 26. Then, M is the reflection of... | 588 | 30 | 0 | <think>
Okay, so I need to solve this AIME problem. Let me try to visualize the problem first. There's triangle ABC. On side AB, there are points A, D, E, B in that order, with AD = 4, DE = 16, and EB = 8. On side AC, there are points A, F, G, C in order, with AF = 13, FG = 52, and GC = 26. Then, M is the reflection of... | 588 | 1 |
AIME2025-I | 2 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
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 p... | <think>
Okay, so I need to solve this AIME problem. Let me read it again to make sure I understand.
There are 9 members of a baseball team who each had a single-scoop cone 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 ... | 16 | 16 | 1 | <think>
Okay, so I need to solve this AIME problem. Let me read it again to make sure I understand.
There are 9 members of a baseball team who each had a single-scoop cone 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 ... | 16 | 1 |
AIME2025-I | 3 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
Problem:
Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $100$, inclusive, such that $12x^{2}-xy-6y^{2}=0$.
Final Answer: | <think>
Okay, so I need to solve this AIME problem where I have to find the number of ordered pairs (x, y) with both x and y integers between -100 and 100 inclusive, such that 12x² - xy - 6y² = 0. Alright, let me start by understanding the equation and figuring out how to approach it.
First, the equation is 12x² - xy ... | 117 | 117 | 1 | <think>
Okay, so I need to solve this AIME problem where I have to find the number of ordered pairs (x, y) with both x and y integers between -100 and 100 inclusive, such that 12x² - xy - 6y² = 0. Alright, let me start by understanding the equation and figuring out how to approach it.
First, the equation is 12x² - xy ... | 117 | 1 |
AIME2025-I | 4 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
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.
... | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
The problem says: 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.
Alr... | 279 | 12 | 0 | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
The problem says: 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.
Alr... | 576 | 0 |
AIME2025-I | 5 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
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... | <think>
Okay, so I need to solve this AIME problem about a trapezoid with an inscribed circle. Let me try to recall what I know about trapezoids and incircles.
First, the problem says it's a trapezoid with an inscribed circle, which I think is called a tangential trapezoid. For a trapezoid to have an incircle, it mus... | 504 | 72 | 0 | <think>
Okay, so I need to solve this AIME problem about a trapezoid with an inscribed circle. Let me try to recall what I know about trapezoids and incircles.
First, the problem says it's a trapezoid with an inscribed circle, which I think is called a tangential trapezoid. For a trapezoid to have an incircle, it mus... | 504 | 1 |
AIME2025-I | 6 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
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 those... | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
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 placed next to each other in alphabetical order to form six two-letter words. Then these six words are listed alphabetic... | 821 | 1 | 0 | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
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 placed next to each other in alphabetical order to form six two-letter words. Then these six words are listed alphabetic... | 7 | 0 |
AIME2025-I | 7 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
Problem:
Let $k$ be real numbers such that the system $|25+20i-z|=5$ and $|z-4-k|=|z-3i-k|$ has exactly one complex solution $z$. The sum of all possible values of $k$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relati... | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
The problem says: Let k be real numbers such that the system |25 + 20i - z| = 5 and |z - 4 - k| = |z - 3i - k| has exactly one complex solution z. The sum of all possible values of k can be written as m/n, where m and n are relatively ... | 77 | 77 | 1 | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
The problem says: Let k be real numbers such that the system |25 + 20i - z| = 5 and |z - 4 - k| = |z - 3i - k| has exactly one complex solution z. The sum of all possible values of k can be written as m/n, where m and n are relatively ... | 77 | 1 |
AIME2025-I | 8 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
Problem:
The parabola with equation $y=x^{2}-4$ is rotated $60^{\circ}$ counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has $y$-coordinate $\frac{a-\... | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
The problem says: The parabola with equation y = x² - 4 is rotated 60 degrees counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has a y-coordinate of (a - √b... | 62 | 25 | 0 | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
The problem says: The parabola with equation y = x² - 4 is rotated 60 degrees counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has a y-coordinate of (a - √b... | 140 | 0 |
AIME2025-I | 9 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
Problem:
The 27 cells of a $3\times9$ grid are filled in using the numbers 1 through 9 so that each row contains 9 different numbers, and each of the three $3\times3$ blocks heavily outlined in the example below contains 9 differ... | <think>
Okay, so I need to solve this AIME problem about a 3x9 grid filled with numbers 1 through 9. Each row has to have all different numbers, and each of the three 3x3 blocks (like in Sudoku) must also have all different numbers. The question is asking for the number of different ways to fill such a grid, which can ... | 81 | 9 | 0 | <think>
Okay, so I need to solve this AIME problem about a 3x9 grid filled with numbers 1 through 9. Each row has to have all different numbers, and each of the three 3x3 blocks (like in Sudoku) must also have all different numbers. The question is asking for the number of different ways to fill such a grid, which can ... | 216 | 0 |
AIME2025-I | 10 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
Problem:
A piecewise linear periodic function is defined by $f(x)=\begin{cases}x&\text{if }x\in[-1,1)\\2-x&\text{if }x\in[1,3)\end{cases}$ and $f(x+4)=f(x)$ for all real numbers $x$. The graph of $f(x)$ has the sawtooth pattern. ... | <think>
Okay, so I need to solve this AIME problem. Let me try to understand what's being asked here.
First, there's a piecewise linear periodic function f(x) defined on intervals [-1, 1) and [1, 3), with f(x) = x on [-1, 1) and f(x) = 2 - x on [1, 3). Then it's periodic with period 4, meaning f(x + 4) = f(x) for all... | 259 | 3 | 0 | <think>
Okay, so I need to solve this AIME problem. Let me try to understand what's being asked here.
First, there's a piecewise linear periodic function f(x) defined on intervals [-1, 1) and [1, 3), with f(x) = x on [-1, 1) and f(x) = 2 - x on [1, 3). Then it's periodic with period 4, meaning f(x + 4) = f(x) for all... | 5 | 0 |
AIME2025-I | 11 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
Problem:
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 finit... | <think>
Okay, so I need to solve this AIME problem. Let me try to understand what it's asking. The problem is about points in 3D space that lie in the plane x + y + z = 75. These points also satisfy the inequalities x - yz < y - zx < z - xy. The set of such points forms three disjoint convex regions, and exactly one of... | 510 | 3 | 0 | <think>
Okay, so I need to solve this AIME problem. Let me try to understand what it's asking. The problem is about points in 3D space that lie in the plane x + y + z = 75. These points also satisfy the inequalities x - yz < y - zx < z - xy. The set of such points forms three disjoint convex regions, and exactly one of... | 48 | 0 |
AIME2025-I | 12 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
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 ... | <think>
Okay, so I need to solve this AIME problem about dividing a disk with line segments and finding the expected number of regions. Let me try to break it down step by step.
First, the problem says that Alex divides a disk into four quadrants with two perpendicular diameters. Then he draws 25 more line segments, ... | 204 | 2 | 0 | <think>
Okay, so I need to solve this AIME problem about dividing a disk with line segments and finding the expected number of regions. Let me try to break it down step by step.
First, the problem says that Alex divides a disk into four quadrants with two perpendicular diameters. Then he draws 25 more line segments, ... | 191 | 0 |
AIME2025-I | 13 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
Problem:
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 express... | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
We have a convex pentagon ABCDE with sides AB=14, BC=7, CD=24, DE=13, EA=26, and angles at B and E are both 60 degrees. For any point X in the plane, f(X) is defined as the sum of distances from X to each vertex: AX + BX + CX + DX + EX... | 60 | 13 | 0 | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
We have a convex pentagon ABCDE with sides AB=14, BC=7, CD=24, DE=13, EA=26, and angles at B and E are both 60 degrees. For any point X in the plane, f(X) is defined as the sum of distances from X to each vertex: AX + BX + CX + DX + EX... | 123 | 0 |
AIME2025-I | 14 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
Problem:
Let $N$ denote the number of ordered triples of positive integers $(a,b,c)$ such that $a,b,c\leq3^6$ and $a^3+b^3+c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$.
Final Answer: | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
The problem says: Let N denote the number of ordered triples of positive integers (a, b, c) such that a, b, c ≤ 3^6 and a³ + b³ + c³ is a multiple of 3^7. Find the remainder when N is divided by 1000.
First, let me parse the problem. ... | 735 | 1 | 0 | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
The problem says: Let N denote the number of ordered triples of positive integers (a, b, c) such that a, b, c ≤ 3^6 and a³ + b³ + c³ is a multiple of 3^7. Find the remainder when N is divided by 1000.
First, let me parse the problem. ... | 501 | 0 |
AIME2025-II | 0 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
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 ... | <think>
Okay, so I need to solve this AIME problem. Let me try to visualize the problem first. There are six points A, B, C, D, E, F lying on a straight line in that order. Point G is not on the line. The given lengths are AC = 26, BD = 22, CE = 31, DF = 33, AF = 73, CG = 40, and DG = 30. The goal is to find the area o... | 468 | 468 | 1 | <think>
Okay, so I need to solve this AIME problem. Let me try to visualize the problem first. There are six points A, B, C, D, E, F lying on a straight line in that order. Point G is not on the line. The given lengths are AC = 26, BD = 22, CE = 31, DF = 33, AF = 73, CG = 40, and DG = 30. The goal is to find the area o... | 468 | 1 |
AIME2025-II | 1 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
Problem:
Find the sum of all positive integers $ n $ such that $ n + 2 $ divides the product $ 3(n + 3)(n^2 + 9) $.
Final Answer: | <think>
Okay, so I need to solve this AIME problem: Find the sum of all positive integers n such that n + 2 divides the product 3(n + 3)(n² + 9). And then give the final answer as an integer between 0 and 999. Alright, let me try to figure this out step by step.
First, let me restate the problem in my own words. We ne... | 49 | 49 | 1 | <think>
Okay, so I need to solve this AIME problem: Find the sum of all positive integers n such that n + 2 divides the product 3(n + 3)(n² + 9). And then give the final answer as an integer between 0 and 999. Alright, let me try to figure this out step by step.
First, let me restate the problem in my own words. We ne... | 49 | 1 |
AIME2025-II | 2 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
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 a way that each unit square has 2 red sides and 2 blue sides. Find the ... | <think>
Okay, so I need to solve this AIME problem about coloring the sides of a 2x2 grid of unit squares. Each square has to have exactly 2 red sides and 2 blue sides. The question is asking for the number of such colorings. Let me try to break this down step by step.
First, let me visualize the grid. A 2x2 grid of s... | 82 | 1 | 0 | <think>
Okay, so I need to solve this AIME problem about coloring the sides of a 2x2 grid of unit squares. Each square has to have exactly 2 red sides and 2 blue sides. The question is asking for the number of such colorings. Let me try to break this down step by step.
First, let me visualize the grid. A 2x2 grid of s... | 24 | 0 |
AIME2025-II | 3 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
Problem:
The product $ \prod_{k=4}^{63} \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{... | <think>
Okay, so I need to solve this AIME problem where I have to compute this big product from k=4 to 63 of some logarithmic expressions. The problem gives me a specific product and tells me that it's equal to m/n where m and n are coprime, and I need to find m + n. Alright, let me try to break this down step by step... | 106 | 93 | 0 | <think>
Okay, so I need to solve this AIME problem where I have to compute this big product from k=4 to 63 of some logarithmic expressions. The problem gives me a specific product and tells me that it's equal to m/n where m and n are coprime, and I need to find m + n. Alright, let me try to break this down step by step... | 106 | 1 |
AIME2025-II | 4 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
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 $ \ove... | <think>
Okay, so I need to solve this AIME problem. Let me try to understand it step by step. The problem is about triangle ABC with angles given: angle at A is 84 degrees, angle at B is 60 degrees, and angle at C is 36 degrees. Then there are points D, E, F on sides BC, AC, and AB respectively. The circle of triangle ... | 336 | 0 | 0 | <think>
Okay, so I need to solve this AIME problem. Let me try to understand it step by step. The problem is about triangle ABC with angles given: angle at A is 84 degrees, angle at B is 60 degrees, and angle at C is 36 degrees. Then there are points D, E, F on sides BC, AC, and AB respectively. The circle of triangle ... | 252 | 0 |
AIME2025-II | 5 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
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... | <think>
Okay, let me try to tackle this AIME problem. First, I need to understand the problem step by step. Let me start by visualizing the given information.
We have two circles, ω₁ and ω₂. ω₁ has radius 6 and is centered at point A, and it's internally tangent to ω₂, which has radius 15. The point of tangency is B. ... | 293 | 0 | 0 | <think>
Okay, let me try to tackle this AIME problem. First, I need to understand the problem step by step. Let me start by visualizing the given information.
We have two circles, ω₁ and ω₂. ω₁ has radius 6 and is centered at point A, and it's internally tangent to ω₂, which has radius 15. The point of tangency is B. ... | 100 | 0 |
AIME2025-II | 6 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
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 elements i... | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
We have set A, which is the set of positive integer divisors of 2025. Then B is a randomly selected subset of A. We need to find the probability that B is a nonempty set where the least common multiple (LCM) of its elements is 2025. Th... | 237 | 237 | 1 | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
We have set A, which is the set of positive integer divisors of 2025. Then B is a randomly selected subset of A. We need to find the probability that B is a nonempty set where the least common multiple (LCM) of its elements is 2025. Th... | 237 | 1 |
AIME2025-II | 7 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
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 **gree... | <think>
Okay, so I need to solve this AIME problem about the greedy algorithm for making change with 1, 10, and 25 cent coins. The question is asking for the number of values of N between 1 and 1000 inclusive where the greedy algorithm actually gives the minimal number of coins.
First, let me make sure I understand t... | 610 | 1 | 0 | <think>
Okay, so I need to solve this AIME problem about the greedy algorithm for making change with 1, 10, and 25 cent coins. The question is asking for the number of values of N between 1 and 1000 inclusive where the greedy algorithm actually gives the minimal number of coins.
First, let me make sure I understand t... | 618 | 0 |
AIME2025-II | 8 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
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 ... | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
The problem says: There are n values of x in the interval 0 < x < 2π where f(x) = sin(7π·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.
Alright, so first, I need to find n, the ... | 149 | 5 | 0 | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
The problem says: There are n values of x in the interval 0 < x < 2π where f(x) = sin(7π·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.
Alright, so first, I need to find n, the ... | 139 | 0 |
AIME2025-II | 9 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
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 rem... | <think>
Okay, so I need to solve this AIME problem where 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. Wait, actually, the problem says "no person sits next to two other people." Hmm, maybe I need to parse that again. Wait, maybe it's that... | 907 | 3 | 0 | <think>
Okay, so I need to solve this AIME problem where 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. Wait, actually, the problem says "no person sits next to two other people." Hmm, maybe I need to parse that again. Wait, maybe it's that... | 907 | 1 |
AIME2025-II | 10 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
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.
Final Answer: | <think>
Okay, so I need to solve this AIME problem about a regular 24-gon. The question is asking for the number of ways to draw 12 segments of equal length such that each vertex is an endpoint of exactly one segment. Hmm, let me try to break this down.
First, let me visualize a regular 24-gon. It has 24 vertices, all... | 113 | 2 | 0 | <think>
Okay, so I need to solve this AIME problem about a regular 24-gon. The question is asking for the number of ways to draw 12 segments of equal length such that each vertex is an endpoint of exactly one segment. Hmm, let me try to break this down.
First, let me visualize a regular 24-gon. It has 24 vertices, all... | INVALID | 0 |
AIME2025-II | 11 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
Problem:
Let $ A_1A_2 \ldots A_{11} $ be an 11-sided non-convex simple polygon with the following properties:
* The area of $ A_iA_1A_{i+1} $ is 1 for each $ 2 \leq i \leq 10 $,
* $ \cos(\angle A_iA_1A_{i+1}) = \frac{12}{13} $ fo... | <think>
Okay, so I need to solve this AIME problem about an 11-sided non-convex simple polygon. Let me try to parse all the given information and figure out how to approach it.
First, the problem says that we have a polygon A₁A₂...A₁₁. It's non-convex and simple, which means it doesn't intersect itself and has at leas... | 19 | 1 | 0 | <think>
Okay, so I need to solve this AIME problem about an 11-sided non-convex simple polygon. Let me try to parse all the given information and figure out how to approach it.
First, the problem says that we have a polygon A₁A₂...A₁₁. It's non-convex and simple, which means it doesn't intersect itself and has at leas... | 118 | 0 |
AIME2025-II | 12 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
Problem:
Let the sequence of rationals $ x_1, x_2, \ldots $ 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 r... | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
We have a sequence of rational numbers x₁, x₂, ... where x₁ = 25/11, and each subsequent term is defined by the recurrence relation x_{k+1} = (1/3)(x_k + 1/x_k - 1). We need to find x_{2025} expressed as a reduced fraction m/n, then co... | 248 | 2 | 0 | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
We have a sequence of rational numbers x₁, x₂, ... where x₁ = 25/11, and each subsequent term is defined by the recurrence relation x_{k+1} = (1/3)(x_k + 1/x_k - 1). We need to find x_{2025} expressed as a reduced fraction m/n, then co... | 937 | 0 |
AIME2025-II | 13 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
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 that $ AK = AL = BK = CL = KL = 14. $ The area of the quadrilateral $ BK... | <think>
Okay, so I need to solve this AIME problem. Let me read it again to make sure I understand.
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. The area of quadrilateral BKLC is n√3... | 104 | 124 | 0 | <think>
Okay, so I need to solve this AIME problem. Let me read it again to make sure I understand.
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. The area of quadrilateral BKLC is n√3... | 150 | 0 |
AIME2025-II | 14 | Solve the following AIME problem.
Give ONLY the final answer as an integer from 0 to 999.
Problem:
There are exactly three positive real numbers $ k $ such that the function
$ f(x) = \frac{(x - 18)(x - 72)(x - 98)(x - k)}{x} $
defined over the positive real numbers achieves its minimum value at exactly two positive re... | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
The problem says: There are exactly three positive real numbers $ k $ such that the function
$$
f(x) = \frac{(x - 18)(x - 72)(x - 98)(x - k)}{x}
$$
defined over the positive real numbers achieves its minimum value at exactly two positi... | 240 | 18 | 0 | <think>
Okay, so I need to solve this AIME problem. Let me read it again carefully.
The problem says: There are exactly three positive real numbers $ k $ such that the function
$$
f(x) = \frac{(x - 18)(x - 72)(x - 98)(x - k)}{x}
$$
defined over the positive real numbers achieves its minimum value at exactly two positi... | 330 | 0 |
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