Datasets:
source stringclasses 7
values | problem stringlengths 19 6.18k | generation stringlengths 345 68.1k |
|---|---|---|
olympiads | 11. Given that the internal angles $A, B, C$ of $\triangle ABC$ have opposite sides $a, b, c$ respectively, and $\sqrt{3} b \cos \frac{A+B}{2}=c \sin B$.
(1) Find the size of $\angle C$;
(2) If $a+b=\sqrt{3} c$, find $\sin A$. | <think>
Okay, let's tackle this problem step by step. First, the problem is about a triangle ABC with angles A, B, C opposite sides a, b, c respectively. The given equation is √3 * b * cos[(A + B)/2] = c * sin B. We need to find the measure of angle C in part (1), and if a + b = √3 * c, find sin A in part (2).
Startin... |
olympiads | 2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation. | <think>
Okay, let's tackle this problem. So, N is an integer represented as 777 in base b, and we need to find the smallest positive integer b such that N is a fourth power of an integer in decimal. Hmm, interesting. Let me break this down step by step.
First, I need to recall how numbers are represented in different ... |
olympiads | 7.1. Solve the equation $\frac{n!}{2}=k!+l!$ in natural numbers, where $n!=1 \cdot 2 \cdot \ldots n$. If there are no solutions, write 0; if there is one solution, write $n$; if there are multiple solutions, write the sum of the values of $n$ for all solutions. Recall that a solution is a triplet $(n, k, l)$; if soluti... | <think>
Okay, let's try to solve this equation: n! / 2 = k! + l! where n, k, l are natural numbers. The problem asks to find all possible solutions, and if there are multiple, sum the n values. If none, write 0. Alright, let's start by understanding the equation.
First, n! divided by 2 equals the sum of two factorials... |
olympiads | 1. A line $l$ intersects a hyperbola $c$, then the maximum number of intersection points is ( ).
A. 1
B. 2
C. 3
D. 4 | <think>
Okay, so I need to figure out the maximum number of points where a line can intersect a hyperbola. The options are 1, 2, 3, or 4. Let me start by recalling what a hyperbola is.
A hyperbola is a type of conic section, right? It's defined as the set of all points where the difference of the distances from two f... |
olympiads | 2. How many integers $b$ exist such that the equation $x^{2}+b x-9600=0$ has an integer solution that is a multiple of both 10 and 12? Specify the largest possible $b$. | <think>
Okay, let's see. The problem is asking how many integers \( b \) exist such that the quadratic equation \( x^2 + b x - 9600 = 0 \) has an integer solution that is a multiple of both 10 and 12. Also, we need to specify the largest possible \( b \).
First, let's parse the question. We need to find integers \( b ... |
olympiads | 10.319. The diagonals of an isosceles trapezoid are perpendicular to each other, and its area is $a^{2}$. Determine the height of the trapezoid. | <think>
Okay, let's tackle this problem step by step. So, we have an isosceles trapezoid with perpendicular diagonals, and the area is given as \( a^2 \). We need to find the height of the trapezoid. Hmm, alright, let's start by recalling some properties of isosceles trapezoids and then see how the diagonals being perp... |
olympiads | 8. Find the last four digits of $7^{7^{-7}}$ (100 sevens).
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | <think>
Okay, let's see. The problem is to find the last four digits of 7 raised to the power of 7^(-7), but there are 100 sevens mentioned. Wait, maybe I misread that. The original problem says "8. Find the last four digits of \(7^{7^{-7}}\) (100 sevens)." Hmm. So the notation here is a bit confusing. The user mention... |
olympiads | 9.27 In the metro train at the initial stop, 100 passengers entered. How many ways are there to distribute the exit of all these passengers at the next 16 stops of the train? | <think>
Okay, so there's this problem about a metro train where 100 passengers get on at the initial stop. The question is asking how many ways there are to distribute the exits of all these passengers at the next 16 stops. Hmm, let me think about how to approach this.
First, let me paraphrase the problem to make sure... |
olympiads | 8. If $n$ is a natural number less than 50, find all values of $n$ such that the values of the algebraic expressions $4n+5$ and $7n+6$ have a common divisor greater than 1. | <think>
Okay, so I need to find all natural numbers n less than 50 where the expressions 4n + 5 and 7n + 6 share a common divisor greater than 1. Hmm, let's break this down step by step.
First, let me recall that if two numbers have a common divisor greater than 1, then their greatest common divisor (GCD) must be grea... |
olympiads | Example 5. Expand the function
$$
f(z)=\frac{z}{z^{2}-2 z-3}
$$
into a Taylor series in the neighborhood of the point $z_{0}=0$ using expansion (12), and find the radius of convergence of the series. | <think>
Okay, so I need to expand the function f(z) = z/(z² - 2z - 3) into a Taylor series around z₀ = 0 and find the radius of convergence. Hmm, let me think. First, I remember that Taylor series expansions around 0 are like Maclaurin series. The general approach for rational functions like this is probably partial fr... |
aops_forum | Let $a_1,a_2,a_3,a_4,a_5$ be distinct real numbers. Consider all sums of the form $a_i + a_j$ where $i,j \in \{1,2,3,4,5\}$ and $i \neq j$. Let $m$ be the number of distinct numbers among these sums. What is the smallest possible value of $m$? | <think>
Okay, so I need to figure out the smallest possible number of distinct sums you can get by adding two distinct numbers from a set of five distinct real numbers. Hmm, the problem is asking for the minimal number of distinct sums when you consider all possible pairs. Let me think step by step.
First, let me unde... |
olympiads | 149. Two equally matched opponents are playing chess. Find the most probable number of wins for any chess player if $2 N$ decisive (without draws) games will be played. | <think>
Okay, so we have two equally matched opponents playing chess, and they're going to play 2N decisive games. We need to find the most probable number of wins for any player. Hmm, let me think.
First, since they're equally matched, the probability of either player winning a game should be the same, right? So, li... |
cn_contest | 7. A circle with a radius of 1 has six points, these six points divide the circle into six equal parts. Take three of these points as vertices to form a triangle. If the triangle is neither equilateral nor isosceles, then the area of this triangle is ( ).
(A) $\frac{\sqrt{3}}{3}$
(B) $\frac{\sqrt{3}}{2}$
(C) 1
(D) $\sq... | <think>
Okay, let's see. There's a circle with radius 1, divided into six equal parts by six points. So, those points are equally spaced around the circumference, right? Each arc between consecutive points must be 60 degrees because 360 divided by 6 is 60. That makes sense. So, the circle is divided into six 60-degree ... |
End of preview. Expand in Data Studio
dp_removed_OpenR1-Math-220k
Derived from open-r1/OpenR1-Math-220k, default/train at e4e141ec9dea9f8326f4d347be56105859b2bd68. Original source card: source_card.md.
15172 problems, 27739 retained traces. One row per problem; generations contains ALL retained traces. Parallel arrays refer to those traces; generation_indices records original positions. Text is unchanged. Only math-verify-correct traces are included. Explicit length stops and incomplete reasoning are excluded. Missing finish reason policy: drop; fallback does not prove the final answer was untruncated and is identified in completion_evidence.
Length <= 20480 using Qwen/Qwen2.5-3B at 3aab1f1954e9cc14eb9509a215f9e5ca08227a9b: tokens(problem) + tokens(generation), without special tokens, chat formatting, padding or truncation. Training templates add overhead.
Train/eval overlap removal: normalization v2, distinct 8-grams, coverage >= 0.7 per evaluation problem. Short eval problems use normalized exact substring matching. References: AIME 2024/2025/2026, AMC23, MATH-500; exact pins in provenance.json. No within/between-training deduplication or blanket source-label exclusions. See filter_report.json and decontamination_report.json for counts, matches and near misses.
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