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[{"content":"Find the largest positive integer $n$ such that $n!$ can be expressed as the product of(...TRUNCATED) | "Find the largest positive integer $n$ such that $n!$ can be expressed as the product of exactly $n-(...TRUNCATED) | "Find the largest positive integer $n$ such that $n!$ can be expressed as the product of exactly $n-(...TRUNCATED) |
ai2-adapt-dev/openmath-2-math
| "**Problem Restatement:**\nFind the largest positive integer \\( n \\) such that \\( n! \\) can be e(...TRUNCATED) | 51,361 | 12,586 | 1,942 | ["Okay, so I need to find the largest positive integer n such that n! can be expressed as the produc(...TRUNCATED) | ["To find the largest positive integer \\( n \\) such that \\( n! \\) can be expressed as the produc(...TRUNCATED) | ["<think>\nOkay, so I need to find the largest positive integer n such that n! can be expressed as t(...TRUNCATED) |
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The response contains only English.
| ["<think>\nOkay, so I need to find the largest positive integer n such that n! can be expressed as t(...TRUNCATED) | "<think>\nOkay, so I need to find the largest positive integer \\( n \\) such that \\( n! \\) can be(...TRUNCATED) | [{"from":"human","value":"Find the largest positive integer $n$ such that $n!$ can be expressed as t(...TRUNCATED) |
[{"content":"Find all values of $x$ that satisfy\n\\[\\frac{x+1}{x-1} < \\sqrt{x^2+1} < \\frac{x+2}{(...TRUNCATED) |
Find all values of $x$ that satisfy
\[\frac{x+1}{x-1} < \sqrt{x^2+1} < \frac{x+2}{x-2}.\]
|
Find all values of $x$ that satisfy
\[\frac{x+1}{x-1} < \sqrt{x^2+1} < \frac{x+2}{x-2}.\]
|
ai2-adapt-dev/openmath-2-math
| "Given the inequality\n\\[\n\\frac{x+1}{x-1} < \\sqrt{x^2+1} < \\frac{x+2}{x-2},\n\\]\nwe want to fi(...TRUNCATED) | 3,308 | 13,768 | 259 | ["Okay, so I need to solve this inequality: (x + 1)/(x - 1) < sqrt(x² + 1) < (x + 2)/(x - 2). Hmm, (...TRUNCATED) | ["To solve the inequality \\(\\frac{x+1}{x-1} < \\sqrt{x^2+1} < \\frac{x+2}{x-2}\\), we break it dow(...TRUNCATED) | ["<think>\nOkay, so I need to solve this inequality: (x + 1)/(x - 1) < sqrt(x² + 1) < (x + 2)/(x - (...TRUNCATED) |
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] | "The response contains multiple instances of English, making it clear that the answer is given in on(...TRUNCATED) | ["<think>\nOkay, so I need to solve this inequality: (x + 1)/(x - 1) < sqrt(x² + 1) < (x + 2)/(x - (...TRUNCATED) | "<think>\nOkay, so I need to solve this inequality: (x + 1)/(x - 1) < sqrt(x² + 1) < (x + 2)/(x - 2(...TRUNCATED) | [{"from":"human","value":"Find all values of $x$ that satisfy\n\\[\\frac{x+1}{x-1} < \\sqrt{x^2+1} <(...TRUNCATED) |
[{"content":"Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and (...TRUNCATED) | "Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and the product (...TRUNCATED) | "Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and the product (...TRUNCATED) |
ai2-adapt-dev/openmath-2-math
| "**Problem:** \nFind the number of ordered pairs of positive integers \\((a,b)\\) such that \\(a + (...TRUNCATED) | 16,612 | 15,946 | 1,319 | ["Okay, so I need to find the number of ordered pairs of positive integers (a, b) such that a + b = (...TRUNCATED) | ["To find the number of ordered pairs of positive integers \\((a, b)\\) such that \\(a + b = 1000\\)(...TRUNCATED) | ["<think>\nOkay, so I need to find the number of ordered pairs of positive integers (a, b) such that(...TRUNCATED) |
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] | "The provided answer only contains one language, which is English. Each explanation, calculation, an(...TRUNCATED) | ["<think>\nOkay, so I need to find the number of ordered pairs of positive integers (a, b) such that(...TRUNCATED) | "<think>\nOkay, so I need to find the number of ordered pairs of positive integers (a, b) such that (...TRUNCATED) | [{"from":"human","value":"Find the number of ordered pairs of positive integers $(a,b)$ such that $a(...TRUNCATED) |
[{"content":"Let $h(x) = x^4 + 2x^3 + 3x^2 + 4x + 5.$ What is the remainder when the polynomial $h((...TRUNCATED) | "Let $h(x) = x^4 + 2x^3 + 3x^2 + 4x + 5.$ What is the remainder when the polynomial $h(x^{10})$ is (...TRUNCATED) | "Let $h(x) = x^4 + 2x^3 + 3x^2 + 4x + 5.$ What is the remainder when the polynomial $h(x^{10})$ is (...TRUNCATED) |
ai2-adapt-dev/openmath-2-math
| "Given: \n\\[\nh(x) = x^4 + 2x^3 + 3x^2 + 4x + 5.\n\\]\n\nWe want to find the remainder when \n\\[\n(...TRUNCATED) | 17,552 | 16,106 | 410 | ["Okay, so I need to find the remainder when h(x¹⁰) is divided by h(x). The polynomial h(x) is gi(...TRUNCATED) | ["To find the remainder when \\( h(x^{10}) \\) is divided by \\( h(x) \\), we start with the polynom(...TRUNCATED) | ["<think>\nOkay, so I need to find the remainder when h(x¹⁰) is divided by h(x). The polynomial h(...TRUNCATED) |
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] | "The response contains a mix of English and mathematical notation (using LaTeX), which does not adhe(...TRUNCATED) | ["<think>\nOkay, so I need to find the remainder when h(x¹⁰) is divided by h(x). Hmm, let me star(...TRUNCATED) | "<think>\nOkay, so I need to find the remainder when the polynomial h(x¹⁰) is divided by h(x). Le(...TRUNCATED) | [{"from":"human","value":"Let $h(x) = x^4 + 2x^3 + 3x^2 + 4x + 5.$ What is the remainder when the p(...TRUNCATED) |
[{"content":"In the complex plane, let $A$ and $B$ be two points represented by complex numbers $a$ (...TRUNCATED) | "In the complex plane, let $A$ and $B$ be two points represented by complex numbers $a$ and $b$, res(...TRUNCATED) | "In the complex plane, let $A$ and $B$ be two points represented by complex numbers $a$ and $b$, res(...TRUNCATED) |
ai2-adapt-dev/openmath-2-math
| "**Given:**\n- Points \\( A \\) and \\( B \\) in the complex plane with complex numbers \\( a \\) an(...TRUNCATED) | 25,229 | 11,674 | 2,282 | ["Okay, let's see. So we have two points A and B in the complex plane, represented by complex number(...TRUNCATED) | ["Given two points \\( A \\) and \\( B \\) in the complex plane represented by complex numbers \\( a(...TRUNCATED) | ["<think>\nOkay, let's see. So we have two points A and B in the complex plane, represented by compl(...TRUNCATED) |
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The response consists entirely of English, with no words or phrases from any other language.
| ["<think>\nOkay, let's see. So we have two points A and B in the complex plane, represented by compl(...TRUNCATED) | "<think>\nOkay, let me try to work through this problem step by step. Hmm, we have two points A and (...TRUNCATED) | [{"from":"human","value":"In the complex plane, let $A$ and $B$ be two points represented by complex(...TRUNCATED) |
[{"content":"In the trapezoid shown, the length of the segment $AB$ is fixed. For $\\angle A$ to be (...TRUNCATED) | "In the trapezoid shown, the length of the segment $AB$ is fixed. For $\\angle A$ to be the largest (...TRUNCATED) | "In the trapezoid shown, the length of the segment $AB$ is fixed. For $\\angle A$ to be the largest (...TRUNCATED) |
ai2-adapt-dev/openmath-2-math
| "**Problem Restatement:**\n\nWe have a trapezoid \\(ABCD\\) with \\(AB \\parallel DC\\) and \\(AB\\)(...TRUNCATED) | 37,992 | 11,403 | 2,980 | ["Alright, let's try to figure out this trapezoid problem. So, we have a trapezoid with points A, B,(...TRUNCATED) | ["To solve the problem, we need to determine the range of \\( x \\) such that angle \\( A \\) is the(...TRUNCATED) | ["<think>\nAlright, let's try to figure out this trapezoid problem. So, we have a trapezoid with poi(...TRUNCATED) |
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] | "The provided answer contains solely English language elements, with no indication of any other lang(...TRUNCATED) | ["<think>\nAlright, so I need to figure out the largest possible value of n/m where m and n are the (...TRUNCATED) | "<think>\nOkay, so I need to figure out the largest possible value of n/m, where m and n are the bou(...TRUNCATED) | [{"from":"human","value":"In the trapezoid shown, the length of the segment $AB$ is fixed. For $\\an(...TRUNCATED) |
[{"content":"Three couples (six people) are seated around a circular table. Each person is seated ne(...TRUNCATED) | "Three couples (six people) are seated around a circular table. Each person is seated next to someon(...TRUNCATED) | "Three couples (six people) are seated around a circular table. Each person is seated next to someon(...TRUNCATED) |
ai2-adapt-dev/openmath-2-math
| "Let's analyze the problem step-by-step.\n\n---\n\n### Problem Summary:\n- There are 3 couples (6 pe(...TRUNCATED) | 40,939 | 11,033 | 1,818 | ["Okay, so we have three couples, six people total, sitting around a circular table. The conditions (...TRUNCATED) | ["Given three couples (six people) seated around a circular table, where each person is seated next (...TRUNCATED) | ["<think>\nOkay, so we have three couples, six people total, sitting around a circular table. The co(...TRUNCATED) |
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The provided answer is written in English and does not contain any other language.
| ["<think>\nOkay, so we have this problem here about three couples (six people) sitting around a circ(...TRUNCATED) | "<think>\nAlright, so I have this problem here: Three couples (six people) are seated around a circu(...TRUNCATED) | [{"from":"human","value":"Three couples (six people) are seated around a circular table. Each person(...TRUNCATED) |
[{"content":"Find the value of $d$ if $x(2x-3) < d$ if and only when $x \\in \\left(-\\frac{5}{2}, 3(...TRUNCATED) |
Find the value of $d$ if $x(2x-3) < d$ if and only when $x \in \left(-\frac{5}{2}, 3\right)$.
|
Find the value of $d$ if $x(2x-3) < d$ if and only when $x \in \left(-\frac{5}{2}, 3\right)$.
|
ai2-adapt-dev/openmath-2-math
| "Given the inequality:\n\\[\nx(2x - 3) < d\n\\]\nholds **if and only if**\n\\[\nx \\in \\left(-\\fra(...TRUNCATED) | 31,995 | 18,816 | 585 | ["Okay, so I need to find the value of d such that the inequality x(2x - 3) < d holds if and only if(...TRUNCATED) | ["To find the value of \\( d \\) such that \\( x(2x - 3) < d \\) if and only if \\( x \\in \\left( -(...TRUNCATED) | ["<think>\nOkay, so I need to find the value of d such that the inequality x(2x - 3) < d holds if an(...TRUNCATED) |
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The provided answer only contains one language: English.
| ["<think>\nOkay, so I need to find the value of d such that the inequality x(2x - 3) < d holds if an(...TRUNCATED) | "<think>\nOkay, let me try to figure out this problem. So, we need to find the value of d such that (...TRUNCATED) | [{"from":"human","value":"Find the value of $d$ if $x(2x-3) < d$ if and only when $x \\in \\left(-\\(...TRUNCATED) |
[{"content":"Let $f(x)=ax^2+bx+c$ be a quadratic function defined for all real numbers $x$. If a new(...TRUNCATED) | "Let $f(x)=ax^2+bx+c$ be a quadratic function defined for all real numbers $x$. If a new function $g(...TRUNCATED) | "Let $f(x)=ax^2+bx+c$ be a quadratic function defined for all real numbers $x$. If a new function $g(...TRUNCATED) |
ai2-adapt-dev/openmath-2-math
| "Given:\n- \\( f(x) = ax^2 + bx + c \\) is a quadratic function defined for all real \\( x \\).\n- A(...TRUNCATED) | 18,159 | 11,279 | 861 | ["Okay, so I need to figure out the value of \\( b \\) in the quadratic function \\( f(x) = ax^2 + b(...TRUNCATED) | ["Given the quadratic function \\( f(x) = ax^2 + bx + c \\) and the new function \\( g(x) = f\\left((...TRUNCATED) | ["<think>\nOkay, so I need to figure out the value of \\( b \\) in the quadratic function \\( f(x) =(...TRUNCATED) |
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The provided answer contains only one language, which is English.
| ["<think>\nOkay, so I need to figure out the value of \\( b \\) in the quadratic function \\( f(x) =(...TRUNCATED) | "<think>\nOkay, let's see here. The problem is about a quadratic function f(x) = ax² + bx + c. Then(...TRUNCATED) | [{"from":"human","value":"Let $f(x)=ax^2+bx+c$ be a quadratic function defined for all real numbers (...TRUNCATED) |
[{"content":"Consider the set of all triangles $ORQ$ where $O$ is the origin and $R$ and $Q$ are dis(...TRUNCATED) | "Consider the set of all triangles $ORQ$ where $O$ is the origin and $R$ and $Q$ are distinct points(...TRUNCATED) | "Consider the set of all triangles $ORQ$ where $O$ is the origin and $R$ and $Q$ are distinct points(...TRUNCATED) |
ai2-adapt-dev/openmath-2-math
| "**Problem restatement:** \nGiven points \\( R, Q \\) with nonnegative integer coordinates \\((x, y(...TRUNCATED) | 35,371 | 19,731 | 2,194 | ["Okay, so I need to solve this problem about triangles ORQ where O is the origin, and R and Q are d(...TRUNCATED) | ["To solve the problem, we need to find the number of distinct triangles \\(ORQ\\) where \\(O\\) is (...TRUNCATED) | ["<think>\nOkay, so I need to solve this problem about triangles ORQ where O is the origin, and R an(...TRUNCATED) |
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The provided answer is in only one language, which is English.
| ["<think>\nOkay, so I need to find the number of distinct triangles ORQ where O is the origin, and R(...TRUNCATED) | "<think>\nOkay, so I need to find the number of distinct triangles ORQ where O is the origin, and R (...TRUNCATED) | [{"from":"human","value":"Consider the set of all triangles $ORQ$ where $O$ is the origin and $R$ an(...TRUNCATED) |
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