data_source large_stringclasses 7
values | prompt listlengths 2 2 | target listlengths 1 1 | ability large_stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|---|
olympiads | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "7.207. $\\log _{\\sqrt{a}} \\frac{\\sqrt{2 a-x}}{a}-\\log _{1 / a} x=0$.\n\n7.207. $\\log _{\\sqrt{a}} \\frac{\\sqrt{2 a-x}}{a}-\\log _{1 / a} x=0$.",
"role": "user"
}
] | [
{
"content": "<think>\nOkay, let's try to solve this logarithmic equation step by step. The equation is:\n\n\\[\n\\log _{\\sqrt{a}} \\frac{\\sqrt{2 a - x}}{a} - \\log _{1/a} x = 0\n\\]\n\nFirst, I need to recall some logarithm properties to simplify this equation. Logarithms with different bases can sometimes b... | {
"ground_truth": "a",
"style": "rule"
} | {
"index": -1,
"split": "default"
} | |
olympiads | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "8. (10 points) In $\\triangle A B C$, $B D=D E=E C$, $C F: A C=1: 3$. If the area of $\\triangle A D H$ is 24 square centimeters more than the area of $\\triangle H E F$, find th... | [
{
"content": "<think>\nOkay, let's see. I need to find the area of triangle ABC. The problem says that BD = DE = EC, and CF : AC = 1:3. Also, the area of triangle ADH is 24 square centimeters more than the area of triangle HEF. Hmm, okay. Let me try to visualize the triangle first.\n\nSo, triangle ABC. Let me s... | {
"ground_truth": "108",
"style": "rule"
} | {
"index": -1,
"split": "default"
} | |
olympiads | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "2.1. (14 points) When going from the first to the third floor, Petya walks 36 steps. When going from the first floor to his own floor in the same entrance, Vasya walks 72 steps. ... | [
{
"content": "<think>\nOkay, so I need to figure out on which floor Vasya lives based on the number of steps he takes compared to Petya. Let me start by understanding the problem step by step.\n\nFirst, both Petya and Vasya are in the same entrance of a building. When Petya goes from the first floor to the thir... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": -1,
"split": "default"
} | |
olympiads | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "6. Person A and Person B play a \"Guess the Number\" game with a fair die (the six faces of the die are numbered $1,2, \\cdots, 6$). A and B each think of a number on the die, de... | [
{
"content": "<think>\nOkay, let's try to figure out this probability problem. So, we have two people, A and B, each thinking of a number from 1 to 6 on a die. They are considered \"in sync\" if the absolute difference between their numbers is less than or equal to 1. We need to find the probability of that hap... | {
"ground_truth": "\\frac{4}{9}",
"style": "rule"
} | {
"index": -1,
"split": "default"
} | |
olympiads | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "## Task B-3.1.\n\nCalculate: $\\frac{1}{\\left(337^{\\frac{1}{2}}+6^{\\frac{1}{2}}\\right)^{-2}}-\\left(\\frac{\\sqrt{337}-\\sqrt{6}}{337^{\\frac{3}{2}}-6^{\\frac{3}{2}}}\\right)... | [
{
"content": "<think>\nOkay, let's see. I have to calculate this expression: \n\n1 divided by (sqrt(337) + sqrt(6)) squared inverse, minus the inverse of ( (sqrt(337) - sqrt(6)) divided by (337^(3/2) - 6^(3/2)) ). Hmm, that's a mouthful. Let me break it down step by step.\n\nFirst, let's rewrite everything to m... | {
"ground_truth": "\\sqrt{2022}",
"style": "rule"
} | {
"index": -1,
"split": "default"
} | |
olympiads | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Problem 3. Martin is 3 times younger than his father, and together they are 44 years old. In how many years will Martin be 2 times younger than his father.",
"role": "user"
... | [
{
"content": "<think>\nOkay, let's see. The problem says that Martin is 3 times younger than his father, and together they are 44 years old. We need to find out in how many years Martin will be 2 times younger than his father. Hmm. Let's break this down step by step.\n\nFirst, let's parse the age relationships.... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": -1,
"split": "default"
} | |
aops_forum | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Six chairs sit in a row. Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right,... | [
{
"content": "<think>\nOkay, let's tackle this probability problem. So, we have six chairs in a row, six people seating themselves randomly. Then each person randomly chooses either to keep their feet on the floor, cross their legs to the right, or cross their legs to the left. The only problem is if two people... | {
"ground_truth": "1106",
"style": "rule"
} | {
"index": -1,
"split": "default"
} | |
olympiads | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "3. (7p) On a line, consider the distinct points $A_{0}, A_{1}, A_{2}, \\ldots, A_{n}$, in this order, such that $A_{0} A_{1}=10 \\text{~cm}, A_{1} A_{2}=2 \\cdot A_{0} A_{1}, A_{... | [
{
"content": "<think>\nOkay, so I need to find the natural number n such that the distance from A₀ to Aₙ is 2016 times 10085 centimeters. The points are arranged in order A₀, A₁, A₂, ..., Aₙ on a line, and each segment between consecutive points is a multiple of the first segment A₀A₁, which is 10 cm. Specifica... | {
"ground_truth": "2016",
"style": "rule"
} | {
"index": -1,
"split": "default"
} | |
olympiads | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "7. Let $A\\left(x_{1}, y_{1}\\right)$ and $B\\left(x_{2}, y_{2}\\right)$ be two points on the curve $C: x^{2}-y^{2}=2(x>0)$, then the minimum value of $f=\\overrightarrow{O A} \\... | [
{
"content": "<think>\nOkay, let's try to solve this problem. So, we have two points A and B on the curve C: x² - y² = 2 with x > 0. We need to find the minimum value of the dot product f = OA · OB, where OA and OB are vectors from the origin to points A and B, respectively.\n\nFirst, let's recall that the dot ... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": -1,
"split": "default"
} | |
cn_contest | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "7. The function $f$ is defined on the set of integers, satisfying\n$$\nf(n)=\\left\\{\\begin{array}{ll}\nn-3, & \\text { when } n \\geqslant 1000, \\\\\nf(f(n+5)), & \\text { whe... | [
{
"content": "<think>\nOkay, so I need to find f(84) for this function f that's defined on the set of integers. The function has two cases: if n is greater than or equal to 1000, then f(n) is just n minus 3. But if n is less than 1000, then f(n) is defined recursively as f(f(n+5)). Hmm, that sounds a bit tricky... | {
"ground_truth": "997",
"style": "rule"
} | {
"index": -1,
"split": "default"
} | |
olympiads | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Example 1 Given a curve $C: x^{2}+y^{2}-2 x-2 y+1=0$ and a tangent line $l$ that intersects the $x$-axis and $y$-axis at points $A$ and $B$ respectively. $O$ is the origin, $|O A... | [
{
"content": "<think>\nOkay, so I have this problem about a curve and a tangent line. Let me try to parse it step by step. The curve is given by the equation \\( x^2 + y^2 - 2x - 2y + 1 = 0 \\). First, maybe I should rewrite this equation to recognize what kind of curve it is. Let me complete the squares for bo... | {
"ground_truth": "2\\sqrt{2}+3",
"style": "rule"
} | {
"index": -1,
"split": "default"
} | |
olympiads | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "3. How many noncongruent triangles are there with one side of length 20 , one side of length 17 , and one $60^{\\circ}$ angle?",
"role": "user"
}
] | [
{
"content": "<think>\nOkay, so I need to figure out how many noncongruent triangles can be formed with sides 20 and 17, and a 60-degree angle. Hmm, let me start by recalling some triangle properties. \n\nFirst, when dealing with triangles with given sides and angles, the Law of Cosines and the Law of Sines are... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": -1,
"split": "default"
} | |
cn_contest | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "2. Given that when $0 \\leqslant \\theta \\leqslant \\frac{\\pi}{2}$, it always holds that $\\cos ^{2} \\theta+2 m \\sin \\theta-2 m-2<0$. Then the range of real number $m$ is $\... | [
{
"content": "<think>\nAlright, so I have this problem here: Given that when 0 ≤ θ ≤ π/2, the inequality cos²θ + 2m sinθ - 2m - 2 < 0 always holds. I need to find the range of the real number m that satisfies this condition. Hmm, okay. Let me try to work through this step by step.\n\nFirst, let's parse the ineq... | {
"ground_truth": "m>-\\frac{1}{2}",
"style": "rule"
} | {
"index": -1,
"split": "default"
} | |
olympiads | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "[ Tangent Circles [ Equilateral (Equiangular) Triangle ]\n\nIn a given circle with a radius of 3, six equal circles are inscribed, each touching the given circle; moreover, each ... | [
{
"content": "<think>\nOkay, let's see. The problem is about an equilateral triangle with six equal circles inscribed inside a larger circle of radius 3. Each of the six smaller circles touches the large circle and two adjacent ones. We need to find the radii of these six smaller circles. Hmm, let me visualize ... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": -1,
"split": "default"
} | |
olympiads | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "6. (8 points) By expanding the expression $(1+\\sqrt{5})^{206}$ using the binomial theorem, we obtain terms of the form $C_{206}^{k}(\\sqrt{5})^{k}$. Find the value of $k$ for wh... | [
{
"content": "<think>\nOkay, so I need to find the value of k where the term C(206, k) * (√5)^k is maximized in the expansion of (1 + √5)^206. Hmm, let's think about how to approach this. \n\nFirst, I remember that in the binomial expansion, the term with the maximum value is usually around a specific k value. ... | {
"ground_truth": "143",
"style": "rule"
} | {
"index": -1,
"split": "default"
} | |
olympiads | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Problem 6.5. In the park, paths are laid out as shown in the figure. Two workers started to asphalt them, starting simultaneously from point $A$. They lay asphalt at constant spe... | [
{
"content": "<think>\nOkay, let's see. So there are two workers starting from point A at the same time. They're asphalting different paths. The first worker goes from A to B to C, and the second worker goes from A to D to E to F to C. Both finish at the same time after 9 hours. Also, the second worker is 1.2 t... | {
"ground_truth": "45",
"style": "rule"
} | {
"index": -1,
"split": "default"
} |
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