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value | uuid stringlengths 3 36 ⌀ | chosen_generation_idx int64 0 5 |
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<|im_start|>system
You are a mathematical reasoning assistant. Solve the problem step by step, but be concise. Put the final answer exactly once in \boxed{ANSWER}. Do not write anything after the boxed answer.<|im_end|>
<|im_start|>user
Consider the sequence $ \left( x_n \right)_{n\ge 1} $ having $ x_1>1 $ and satisfyi... | <think>
Okay, so I need to show that the sequence \( (x_n)_{n \ge 1} \) is convergent and find its limit. The given conditions are \( x_1 > 1 \) and for every natural number \( n \), the sum of the first \( n+1 \) terms equals the product of those terms. That is,
\[
x_1 + x_2 + \cdots + x_{n+1} = x_1 x_2 \cdots x_{n+1... | <|im_start|>system
You are a mathematical reasoning assistant. Solve the problem step by step, but be concise. Put the final answer exactly once in \boxed{ANSWER}. Do not write anything after the boxed answer.<|im_end|>
<|im_start|>user
Consider the sequence $ \left( x_n \right)_{n\ge 1} $ having $ x_1>1 $ and satisfyi... | Consider the sequence $ \left( x_n \right)_{n\ge 1} $ having $ x_1>1 $ and satisfying the equation
$$ x_1+x_2+\cdots +x_{n+1} =x_1x_2\cdots x_{n+1} ,\quad\forall n\in\mathbb{N} . $$
Show that this sequence is convergent and find its limit. | 1 | aops_forum | Other | math-word-problem | 42c1b6e8-d366-5a02-95bc-2301fcf65388 | 1 |
<|im_start|>system
You are a mathematical reasoning assistant. Solve the problem step by step, but be concise. Put the final answer exactly once in \boxed{ANSWER}. Do not write anything after the boxed answer.<|im_end|>
<|im_start|>user
Let \(\triangle ABC\) have side lengths \(AB=30\), \(BC=32\), and \(AC=34\). Point ... | <think>
Okay, let me try to figure out this problem. So, we have triangle ABC with sides AB=30, BC=32, and AC=34. We need to find the minimum area of triangle AI₁I₂ where I₁ and I₂ are the incenters of triangles ABX and ACX respectively, as point X varies along BC.
First, maybe I should sketch the triangle ABC to visu... | <|im_start|>system
You are a mathematical reasoning assistant. Solve the problem step by step, but be concise. Put the final answer exactly once in \boxed{ANSWER}. Do not write anything after the boxed answer.<|im_end|>
<|im_start|>user
Let \(\triangle ABC\) have side lengths \(AB=30\), \(BC=32\), and \(AC=34\). Point ... | Let \(\triangle ABC\) have side lengths \(AB=30\), \(BC=32\), and \(AC=34\). Point \(X\) lies in the interior of \(\overline{BC}\), and points \(I_1\) and \(I_2\) are the incenters of \(\triangle ABX\) and \(\triangle ACX\), respectively. Find the minimum possible area of \(\triangle AI_1I_2\) as \( X\) varies along \(... | 126 | aops_forum | Geometry | math-word-problem | c06e7dce-3eec-58b1-b85d-45fdf7e59619 | 1 |
<|im_start|>system
You are a mathematical reasoning assistant. Solve the problem step by step, but be concise. Put the final answer exactly once in \boxed{ANSWER}. Do not write anything after the boxed answer.<|im_end|>
<|im_start|>user
(7 points) A refrigerator factory needs to produce 1590 refrigerators. It has alrea... | <think>
Okay, let's try to figure out this refrigerator factory problem. Hmm, so the factory needs to make a total of 1590 refrigerators. They've already been working for 12 days, producing 80 units each day. Now they need to finish the rest, but they're going to speed up production to 90 units per day. The question is... | <|im_start|>system
You are a mathematical reasoning assistant. Solve the problem step by step, but be concise. Put the final answer exactly once in \boxed{ANSWER}. Do not write anything after the boxed answer.<|im_end|>
<|im_start|>user
(7 points) A refrigerator factory needs to produce 1590 refrigerators. It has alrea... | (7 points) A refrigerator factory needs to produce 1590 refrigerators. It has already been producing for 12 days, with 80 units produced each day. The remaining units will be produced at a rate of 90 units per day. How many more days are needed to complete the task? | 7 | olympiads | Algebra | math-word-problem | bdfe16dc-9fab-59ba-93c9-26d21076a74e | 1 |
<|im_start|>system
You are a mathematical reasoning assistant. Solve the problem step by step, but be concise. Put the final answer exactly once in \boxed{ANSWER}. Do not write anything after the boxed answer.<|im_end|>
<|im_start|>user
Task 4. (20 points) A finite increasing sequence of natural numbers $a_{1}, a_{2}, ... | <think>
Okay, so I need to solve this problem about a sequence where each term from the third one onward is defined by a recurrence relation: \( a_{k+2} = 3a_{k+1} - 2a_k - 2 \). The sequence is finite, increasing, and consists of natural numbers. Also, it must include the term 2022. The goal is to find the maximum num... | <|im_start|>system
You are a mathematical reasoning assistant. Solve the problem step by step, but be concise. Put the final answer exactly once in \boxed{ANSWER}. Do not write anything after the boxed answer.<|im_end|>
<|im_start|>user
Task 4. (20 points) A finite increasing sequence of natural numbers $a_{1}, a_{2}, ... | Task 4. (20 points) A finite increasing sequence of natural numbers $a_{1}, a_{2}, \ldots, a_{n}(n \geq 3)$ is given, and for all $\kappa \leq n-2$ the equality $a_{k+2}=3 a_{k+1}-2 a_{k}-2$ holds. The sequence must contain $a_{k}=2022$. Determine the maximum number of three-digit numbers, divisible by 4, that this seq... | 225 | olympiads | Algebra | math-word-problem | f30852fc-0a43-5590-8eae-30b5a7cc91f9 | 0 |
<|im_start|>system
You are a mathematical reasoning assistant. Solve the problem step by step, but be concise. Put the final answer exactly once in \boxed{ANSWER}. Do not write anything after the boxed answer.<|im_end|>
<|im_start|>user
## Problem Statement
Write the decomposition of vector $x$ in terms of vectors $p,... | <think>
Okay, so I need to figure out how to decompose the vector x into a combination of vectors p, q, and r. That means I have to find scalars a, b, and c such that:
x = a*p + b*q + c*r
Given the vectors:
x = {-1 ; 7 ; -4}
p = {-1 ; 2 ; 1}
q = {2 ; 0 ; 3}
r = {1 ; 1 ; -1}
So, in component form, this equation b... | <|im_start|>system
You are a mathematical reasoning assistant. Solve the problem step by step, but be concise. Put the final answer exactly once in \boxed{ANSWER}. Do not write anything after the boxed answer.<|im_end|>
<|im_start|>user
## Problem Statement
Write the decomposition of vector $x$ in terms of vectors $p,... | ## Problem Statement
Write the decomposition of vector $x$ in terms of vectors $p, q, r$:
$x=\{-1 ; 7 ;-4\}$
$p=\{-1 ; 2 ; 1\}$
$q=\{2 ; 0 ; 3\}$
$r=\{1 ; 1 ;-1\}$ | 2p-q+3r | olympiads | Algebra | math-word-problem | 892dc56b-12d8-51e7-8e43-87ba3706cf21 | 1 |
<|im_start|>system
You are a mathematical reasoning assistant. Solve the problem step by step, but be concise. Put the final answer exactly once in \boxed{ANSWER}. Do not write anything after the boxed answer.<|im_end|>
<|im_start|>user
G2.2 $E$ is an interior point of the rectangle $A B C D$. Given that the lengths of... | <think>
Okay, so I have this geometry problem here. Let me read it again. "E is an interior point of the rectangle ABCD. Given that the lengths of EA, EB, EC, and ED are 2, sqrt(11), 4, and b respectively, find the value of b." Hmm, interesting. I need to find the value of b. Let me think about how to approach this.
F... | <|im_start|>system
You are a mathematical reasoning assistant. Solve the problem step by step, but be concise. Put the final answer exactly once in \boxed{ANSWER}. Do not write anything after the boxed answer.<|im_end|>
<|im_start|>user
G2.2 $E$ is an interior point of the rectangle $A B C D$. Given that the lengths of... | G2.2 $E$ is an interior point of the rectangle $A B C D$. Given that the lengths of $E A, E B, E C$ and $E D$ are $2, \sqrt{11}, 4$ and $b$ respectively, find the value of $b$. | 3 | olympiads | Geometry | math-word-problem | 8fc81590-665b-5b75-bab3-0c486c4baa33 | 0 |
"<|im_start|>system\nYou are a mathematical reasoning assistant. Solve the problem step by step, but(...TRUNCATED) | "<think>\nOkay, let me try to figure out how to solve this problem. So, the setup is similar to the (...TRUNCATED) | "<|im_start|>system\nYou are a mathematical reasoning assistant. Solve the problem step by step, but(...TRUNCATED) | "## 107.\n\nOn another island of knights, liars, and normal people, the king held opposite views and(...TRUNCATED) | 1 | olympiads | Logic and Puzzles | math-word-problem | 2e846f58-f39b-5f8b-a6c4-9ef56f172f36 | 0 |
"<|im_start|>system\nYou are a mathematical reasoning assistant. Solve the problem step by step, but(...TRUNCATED) | "<think>\nOkay, so I need to find the number of non-degenerate triangles formed by 4n points on the (...TRUNCATED) | "<|im_start|>system\nYou are a mathematical reasoning assistant. Solve the problem step by step, but(...TRUNCATED) | "## Task A-4.3.\n\nOn the edge of a square, a total of $4 n$ points are marked: all four vertices of(...TRUNCATED) | \binom{4n}{3}-4\binom{n+1}{3} | olympiads | Combinatorics | math-word-problem | 903432c0-2d4e-557e-9749-718978af0b82 | 1 |
"<|im_start|>system\nYou are a mathematical reasoning assistant. Solve the problem step by step, but(...TRUNCATED) | "<think>\nOkay, so I need to find the probability that a randomly chosen 5-digit telephone number in(...TRUNCATED) | "<|im_start|>system\nYou are a mathematical reasoning assistant. Solve the problem step by step, but(...TRUNCATED) | "The telephone numbers in a certain city consist of 5 digits, each of which can be any one of the te(...TRUNCATED) | \frac{189}{625} | cn_contest | Combinatorics | math-word-problem | 178b544f-2efc-59b5-a62a-8a1154aa3d55 | 1 |
"<|im_start|>system\nYou are a mathematical reasoning assistant. Solve the problem step by step, but(...TRUNCATED) | "<think>\nOkay, let me try to figure out this problem step by step. First, let's understand the give(...TRUNCATED) | "<|im_start|>system\nYou are a mathematical reasoning assistant. Solve the problem step by step, but(...TRUNCATED) | "Let $\\mathrm{AB}$ be the major axis of the ellipse $\\frac{x^{2}}{16}+\\frac{y^{2}}{6}=1$. The mov(...TRUNCATED) | 8 | olympiads | Geometry | math-word-problem | cb151148-8952-5061-a9bc-6877014a49c9 | 1 |
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