context listlengths 2 2 | gen_kwargs dict | repeat_idx int64 0 9 | request_idx int64 0 29 | task_name stringclasses 1
value | metadata dict | model_outputs stringlengths 5.63k 17.5k |
|---|---|---|---|---|---|---|
[
{
"content": "<think>",
"role": "system"
},
{
"content": "Problem: Find the largest possible real part of \\[(75+117i)z+\\frac{96+144i}{z}\\] where $z$ is a complex number with $|z|=4$ .\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 8,
"temperature": 0.7
} | 8 | 21 | AIME24 | {
"expected_answer": "540",
"problem_id": "aime24-21",
"reference_solution": "Let $z=a+bi$ such that $a^2+b^2=4^2=16$ . The expression becomes: \\[(75+117i)(a+bi)+\\dfrac{96+144i}{a+bi}.\\] Call this complex number $w$ . We simplify this expression. \\begin{align*}\nw&=(75+117i)(a+bi)+\\dfrac{96+144i}{a+bi} \... | Okay, let's see. I need to find the largest possible real part of the expression (75 + 117i)z + (96 + 144i)/z, where z is a complex number with |z| = 4. Hmm, okay. Let me try to break this down step by step.
First, since z is a complex number with magnitude 4, I can represent z in polar form. Let's recall that any com... |
[
{
"content": "<think>",
"role": "system"
},
{
"content": "Problem: Let $A$ , $B$ , $C$ , and $D$ be points on the hyperbola $\\frac{x^2}{20}- \\frac{y^2}{24} = 1$ such that $ABCD$ is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is less than $BD^2$ ... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 9,
"temperature": 0.7
} | 9 | 23 | AIME24 | {
"expected_answer": "480",
"problem_id": "aime24-23",
"reference_solution": "A quadrilateral is a rhombus if and only if its two diagonals bisect each other and are perpendicular to each other. The first condition is automatically satisfied because of the hyperbola's symmetry about the origin. To satisfy the sec... | Okay, let's see. The problem is about a rhombus ABCD on the hyperbola x²/20 - y²/24 = 1, with the diagonals intersecting at the origin. We need to find the greatest real number that is less than BD² for all such rhombi. Hmm, so BD is one of the diagonals, and we need the infimum of BD², but the problem says "the greate... |
[
{
"content": "<think>",
"role": "system"
},
{
"content": "Problem: A list of positive integers has the following properties: $\\bullet$ The sum of the items in the list is $30$ . $\\bullet$ The unique mode of the list is $9$ . $\\bullet$ The median of the list is a positive integer that does no... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 2,
"temperature": 0.7
} | 2 | 1 | AIME24 | {
"expected_answer": "236",
"problem_id": "aime24-1",
"reference_solution": "The third condition implies that the list's size must be an even number, as if it were an odd number, the median of the list would surely appear in the list itself. Therefore, we can casework on what even numbers work. Say the size is 2.... | Okay, let's try to solve this problem step by step. The problem is about a list of positive integers with three properties: the sum is 30, the unique mode is 9, and the median is a positive integer that doesn't appear in the list. We need to find the sum of the squares of all the items in the list.
First, let me reca... |
[
{
"content": "<think>",
"role": "system"
},
{
"content": "Problem: Let \\(b\\ge 2\\) be an integer. Call a positive integer \\(n\\) \\(b\\text-\\textit{eautiful}\\) if it has exactly two digits when expressed in base \\(b\\) and these two digits sum to \\(\\sqrt n\\). For example, \\(81\\) is \\(13... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 8,
"temperature": 0.7
} | 8 | 13 | AIME24 | {
"expected_answer": "211",
"problem_id": "aime24-13",
"reference_solution": "We write the base- $b$ two-digit integer as $\\left( xy \\right)_b$ .\nThus, this number satisfies\n \\[ \\left( x + y \\right)^2 = b x + y \\] \nwith $x \\in \\left\\{ 1, 2, \\cdots , b-1 \\right\\}$ and $y \\in \\left\\{ 0, 1, \\... | Okay, let's tackle this problem step by step. The question is about finding the smallest base \( b \geq 2 \) such that there are more than ten \( b \)-eautiful numbers. A number \( n \) is \( b \)-eautiful if it has exactly two digits in base \( b \), and the sum of those two digits equals \( \sqrt{n} \).
First, let m... |
[
{
"content": "<think>",
"role": "system"
},
{
"content": "Problem: There exist real numbers $x$ and $y$ , both greater than 1, such that $\\log_x\\left(y^x\\right)=\\log_y\\left(x^{4y}\\right)=10$ . Find $xy$ .\nMark your solution with \\boxed\nAnswer:",
"role": "user"
}
] | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 5,
"temperature": 0.7
} | 5 | 16 | AIME24 | {
"expected_answer": "25",
"problem_id": "aime24-16",
"reference_solution": "By properties of logarithms, we can simplify the given equation to $x\\log_xy=4y\\log_yx=10$ . Let us break this into two separate equations: \\[x\\log_xy=10\\] \n \\[4y\\log_yx=10.\\] \nWe multiply the two equations to get: \n \\[4xy\\... | Okay, let's see. The problem says there are real numbers x and y, both greater than 1, such that log base x of y^x equals log base y of x^{4y} equals 10. We need to find xy. Hmm, so both log expressions are equal to 10. Let me write that down properly.
First, let's translate the logarithmic equations into something mo... |
[
{
"content": "<think>",
"role": "system"
},
{
"content": "Problem: Define $f(x)=|| x|-\\tfrac{1}{2}|$ and $g(x)=|| x|-\\tfrac{1}{4}|$ . Find the number of intersections of the graphs of \\[y=4 g(f(\\sin (2 \\pi x))) \\quad\\text{ and }\\quad x=4 g(f(\\cos (3 \\pi y))).\\]\nMark your solution wit... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 26 | AIME24 | {
"expected_answer": "385",
"problem_id": "aime24-26",
"reference_solution": "If we graph $4g(f(x))$ , we see it forms a sawtooth graph that oscillates between $0$ and $1$ (for values of $x$ between $-1$ and $1$ , which is true because the arguments are between $-1$ and $1$ ). Thus by precariously dr... | Alright, so I need to solve this problem where I have two functions defined in terms of nested absolute values and trigonometric functions, and I need to find the number of intersections between their graphs. The functions are:
\[ y = 4g(f(\sin(2\pi x))) \]
and
\[ x = 4g(f(\cos(3\pi y))) \]
where \( f(x) = ||x| - \fr... |
[
{
"content": "<think>",
"role": "system"
},
{
"content": "Problem: Torus $T$ is the surface produced by revolving a circle with radius $3$ around an axis in the plane of the circle that is a distance $6$ from the center of the circle (so like a donut). Let $S$ be a sphere with a radius $11$... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 4,
"temperature": 0.7
} | 4 | 7 | AIME24 | {
"expected_answer": "127",
"problem_id": "aime24-7",
"reference_solution": "First, let's consider a section $\\mathcal{P}$ of the solids, along the axis.\nBy some 3D-Geomerty thinking, we can simply know that the axis crosses the sphere center. So, that is saying, the $\\mathcal{P}$ we took crosses one of th... | Okay, so I need to solve this problem about a torus and a sphere. The torus T is created by revolving a circle of radius 3 around an axis in its plane that's 6 units away from the center. So, like a donut shape. The sphere S has a radius of 11. When the torus rests inside the sphere, it's tangent along a circle with ra... |
[
{
"content": "<think>",
"role": "system"
},
{
"content": "Problem: Find the number of rectangles that can be formed inside a fixed regular dodecagon ( $12$ -gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. The diagram below shows three of those rectangles. ... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 9,
"temperature": 0.7
} | 9 | 14 | AIME24 | {
"expected_answer": "315",
"problem_id": "aime24-14",
"reference_solution": "By Furaken There are two kinds of such rectangles: those whose sides are parallel to some edges of the regular 12-gon (Case 1), and those whose sides are not (Case 2). For Case 1, WLOG assume that the rectangle's sides are horizontal an... | Okay, so I need to find the number of rectangles that can be formed inside a regular dodecagon (12-gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. Hmm, let me think.
First, I know that in a regular polygon, rectangles can be formed by choosing pairs of parallel sides or dia... |
[
{
"content": "<think>",
"role": "system"
},
{
"content": "Problem: Rectangles $ABCD$ and $EFGH$ are drawn such that $D,E,C,F$ are collinear. Also, $A,D,H,G$ all lie on a circle. If $BC=16$ , $AB=107$ , $FG=17$ , and $EF=184$ , what is the length of $CE$ ? [asy] import graph; unitsize(0.1c... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 0,
"temperature": 0.7
} | 0 | 19 | AIME24 | {
"expected_answer": "104",
"problem_id": "aime24-19",
"reference_solution": "We use simple geometry to solve this problem. [asy] import graph; unitsize(0.1cm); pair A = (0,0);pair B = (107,0);pair C = (107,16);pair D = (0,16);pair E = (3,16);pair F = (187,16);pair G = (187,33);pair H = (3,33); label(\"$A$\", A,... | Alright, so I need to find the length of CE in this problem. Let me start by understanding the given information and the figure.
First, there are two rectangles, ABCD and EFGH. The points D, E, C, F are collinear. That means they lie on the same straight line. Also, the points A, D, H, G all lie on a circle, which sug... |
[
{
"content": "<think>",
"role": "system"
},
{
"content": "Problem: Find the number of rectangles that can be formed inside a fixed regular dodecagon ( $12$ -gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. The diagram below shows three of those rectangles. ... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 8,
"temperature": 0.7
} | 8 | 14 | AIME24 | {
"expected_answer": "315",
"problem_id": "aime24-14",
"reference_solution": "By Furaken There are two kinds of such rectangles: those whose sides are parallel to some edges of the regular 12-gon (Case 1), and those whose sides are not (Case 2). For Case 1, WLOG assume that the rectangle's sides are horizontal an... | Okay, let's see. I need to find the number of rectangles that can be formed inside a regular dodecagon (12-gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. Hmm, the Asymptote code shows a dodecagon with some diagonals and three example rectangles. So, rectangles are defined by... |
[
{
"content": "<think>",
"role": "system"
},
{
"content": "Problem: Let \\(O=(0,0)\\), \\(A=\\left(\\tfrac{1}{2},0\\right)\\), and \\(B=\\left(0,\\tfrac{\\sqrt{3}}{2}\\right)\\) be points in the coordinate plane. Let \\(\\mathcal{F}\\) be the family of segments \\(\\overline{PQ}\\) of unit length lyi... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 6,
"temperature": 0.7
} | 6 | 11 | AIME24 | {
"expected_answer": "23",
"problem_id": "aime24-11",
"reference_solution": "Begin by finding the equation of the line $\\overline{AB}$ : $y= -\\sqrt{3}x+\\frac{\\sqrt{3}}{2}$ Now, consider the general equation of all lines that belong to $\\mathcal{F}$ . Let $P$ be located at $(a,0)$ and $Q$ be located... | Okay, let's try to solve this problem step by step. First, I need to understand what's being asked here. We have points O, A, and B with coordinates given. The family of segments F consists of unit length segments PQ in the first quadrant where P is on the x-axis and Q is on the y-axis. We need to find a point C on AB,... |
[
{
"content": "<think>",
"role": "system"
},
{
"content": "Problem: Consider the paths of length $16$ that follow the lines from the lower left corner to the upper right corner on an $8\\times 8$ grid. Find the number of such paths that change direction exactly four times, as in the examples show... | {
"do_sample": false,
"max_new_tokens": 32768,
"seed": 9,
"temperature": 0.7
} | 9 | 20 | AIME24 | {
"expected_answer": "294",
"problem_id": "aime24-20",
"reference_solution": "We divide the path into eight “ $R$ ” movements and eight “ $U$ ” movements. Five sections of alternative $RURUR$ or $URURU$ are necessary in order to make four “turns.” We use the first case and multiply by $2$ . <br/>\nFor $U$ ,... | Okay, so I need to find the number of paths from the lower left corner to the upper right corner on an 8x8 grid that change direction exactly four times. The paths are of length 16, which makes sense because moving from (0,0) to (8,8) requires 8 moves right and 8 moves up, totaling 16 moves.
First, let me make sure I... |
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chengfu0118/DeepSeek-R1_1757071649_eval_22be
Precomputed model outputs for evaluation.
Evaluation Results
AIME24
- Average Accuracy: 24.33% ± 1.77%
- Number of Runs: 10
| Run | Accuracy | Questions Solved | Total Questions |
|---|---|---|---|
| 1 | 26.67% | 8 | 30 |
| 2 | 30.00% | 9 | 30 |
| 3 | 33.33% | 10 | 30 |
| 4 | 23.33% | 7 | 30 |
| 5 | 20.00% | 6 | 30 |
| 6 | 20.00% | 6 | 30 |
| 7 | 30.00% | 9 | 30 |
| 8 | 16.67% | 5 | 30 |
| 9 | 26.67% | 8 | 30 |
| 10 | 16.67% | 5 | 30 |
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