data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Find all positive values of $c$ so that the inequality $x^2-6x+c<0$ has real solutions for $x$. Express your answer in interval notation."
}
] | math | {
"style": "rule",
"ground_truth": "(0,9)"
} | {
"index": 9300,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all possible positive integers represented in decimal as $13 x y 45 z$, which are divisible by 792, where $x, y, z$ are unknown digits."
}
] | math | {
"style": "rule",
"ground_truth": "1380456"
} | {
"index": 9301,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all possible three-digit numbers that can be obtained by removing three digits from the number 112277. Sum them and write the result as the answer."
}
] | math | {
"style": "rule",
"ground_truth": "1159"
} | {
"index": 9302,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all possible values of $\\frac{d}{a}$ where $a^{2}-6 a d+8 d^{2}=0, a \\neq 0$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}, \\frac{1}{4}"
} | {
"index": 9303,
"solution": "Dividing $a^{2}-6 a d+8 d^{2}=0$ by $a^{2}$, we get $1-6 \\frac{d}{a}+8\\left(\\frac{d}{a}\\right)^{2}=0$. The roots of this quadratic are $\\frac{1}{2}, \\frac{1}{4}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all prime numbers $ p,q$ less than 2005 and such that $ q|p^2 \\plus{} 4$, $ p|q^2 \\plus{} 4$."
}
] | math | {
"style": "rule",
"ground_truth": "(2, 2), (5, 29), (29, 5)"
} | {
"index": 9304,
"solution": "\nWe are given the problem to find all prime numbers \\( p \\) and \\( q \\) less than 2005 such that \\( q \\mid p^2 + 4 \\) and \\( p \\mid q^2 + 4 \\). Let's go through the solution step-by-step.\n\n1. **Understand the Divisibility Conditions:**\n\n We need to satisfy the conditio... |
DeepScaleR | [
{
"role": "user",
"content": "Find all prime numbers $ p,q,r$, such that $ \\frac{p}{q}\\minus{}\\frac{4}{r\\plus{}1}\\equal{}1$"
}
] | math | {
"style": "rule",
"ground_truth": "(7, 3, 2), (5, 3, 5), (3, 2, 7)"
} | {
"index": 9305,
"solution": "\nWe are tasked with finding all prime numbers \\( p, q, r \\) that satisfy the equation:\n\n\\[\n\\frac{p}{q} - \\frac{4}{r+1} = 1.\n\\]\n\nFirst, we rearrange the equation to find a common denominator:\n\n\\[\n\\frac{p}{q} - \\frac{4}{r+1} = 1 \\implies \\frac{p(r+1) - 4q}{q(r+1)} = ... |
DeepScaleR | [
{
"role": "user",
"content": "Find all prime numbers $a,b,c$ and positive integers $k$ satisfying the equation \\[a^2+b^2+16c^2 = 9k^2 + 1.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "(3, 3, 2, 3), (3, 37, 3, 13), (37, 3, 3, 13), (3, 17, 3, 7), (17, 3, 3, 7)"
} | {
"index": 9306,
"solution": "\nTo find all prime numbers \\(a, b, c\\) and positive integers \\(k\\) that satisfy the equation \n\\[\na^2 + b^2 + 16c^2 = 9k^2 + 1,\n\\]\nwe proceed as follows:\n\n1. **Modulo Consideration**: Observe the equation modulo 3. We have:\n \\[\n a^2 \\equiv 0 \\text{ or } 1 \\pmod{3}... |
DeepScaleR | [
{
"role": "user",
"content": "Find all prime numbers $p$ for which there exists a unique $a \\in\\{1,2, \\ldots, p\\}$ such that $a^{3}-3 a+1$ is divisible by $p$."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 9307,
"solution": "We show that $p=3$ is the only prime that satisfies the condition. Let $f(x)=x^{3}-3 x+1$. As preparation, let's compute the roots of $f(x)$. By Cardano's formula, it can be seen that the roots are $2 \\operatorname{Re} \\sqrt[3]{\\frac{-1}{2}+\\sqrt{\\left(\\frac{-1}{2}\\right)^{2}-\\... |
DeepScaleR | [
{
"role": "user",
"content": "Find all prime numbers $p$ such that $y^{2}=x^{3}+4x$ has exactly $p$ solutions in integers modulo $p$. In other words, determine all prime numbers $p$ with the following property: there exist exactly $p$ ordered pairs of integers $(x, y)$ such that $x, y \\in\\{0,1, \\ldots, p... | math | {
"style": "rule",
"ground_truth": "p=2 \\text{ and } p \\equiv 3(\\bmod 4)"
} | {
"index": 9308,
"solution": "Clearly $p=2$ works with solutions $(0,0)$ and $(1,1)$ and not $(0,1)$ or $(1,0)$. If $p \\equiv 3(\\bmod 4)$ then -1 is not a quadratic residue, so for $x^{3}+4x \\neq 0$, exactly one of $x^{3}+4x$ and $-x^{3}-4x$ is a square and gives two solutions (for positive and negative $y$), so... |
DeepScaleR | [
{
"role": "user",
"content": "Find all prime numbers $p$ such that there exists a unique $a \\in \\mathbb{Z}_p$ for which $a^3 - 3a + 1 = 0.$"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 9309,
"solution": "\nTo find all prime numbers \\( p \\) such that there exists a unique \\( a \\in \\mathbb{Z}_p \\) for which the equation \\( a^3 - 3a + 1 = 0 \\) holds, we proceed as follows:\n\nThe equation in question is \\( a^3 - 3a + 1 = 0 \\). We need to determine under what conditions (i.e., fo... |
DeepScaleR | [
{
"role": "user",
"content": "Find all prime numbers $p,q,r$ , such that $\\frac{p}{q}-\\frac{4}{r+1}=1$"
}
] | math | {
"style": "rule",
"ground_truth": "\\[\n(7, 3, 2), (3, 2, 7), (5, 3, 5)\n\\]"
} | {
"index": 9310,
"solution": "The given equation can be rearranged into the below form:\n$4q = (p-q)(r+1)$ \n \n$Case 1: 4|(p-q)$ \nthen we have\n$q = ((p-q)/4)(r+1)$ $=> (p-q)/4 = 1$ and $q = r + 1$ $=> r = 2, q = 3$ and $p = 7$ \n \n$Case 2: 4|(r+1)$ \nthen we have\n$q = (p-q)((r+1)/4)$ $=> (p-q) = 1$ and $q =... |
DeepScaleR | [
{
"role": "user",
"content": "Find all prime numbers whose representation in a base-14 numeral system has the form 101010...101 (alternating ones and zeros)."
}
] | math | {
"style": "rule",
"ground_truth": "197"
} | {
"index": 9311,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all primes $p$ and $q$ such that $3p^{q-1}+1$ divides $11^p+17^p$"
}
] | math | {
"style": "rule",
"ground_truth": "(3, 3)"
} | {
"index": 9312,
"solution": "\nTo solve the problem, we need to identify all pairs of primes \\( p \\) and \\( q \\) such that the expression \\( 3p^{q-1} + 1 \\) divides \\( 11^p + 17^p \\). The reference answer indicates that the only solution is the pair \\((3, 3)\\). Let's go through the process of verifying t... |
DeepScaleR | [
{
"role": "user",
"content": "Find all primes $p$ such that $p^2-p+1$ is a perfect cube."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 9313,
"solution": "\nTo solve the problem of finding all primes \\( p \\) such that \\( p^2 - p + 1 \\) is a perfect cube, we want \\( p^2 - p + 1 = n^3 \\) for some integer \\( n \\).\n\n1. **Case Analysis: Small Values of \\( p \\):**\n\n Start with small values of \\( p \\):\n \n - For \\( p = 2... |
DeepScaleR | [
{
"role": "user",
"content": "Find all primes $p$ such that there exist positive integers $x,y$ that satisfy $x(y^2-p)+y(x^2-p)=5p$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\[\n\\boxed{2, 3, 7}\n\\]"
} | {
"index": 9314,
"solution": "Rearrange the original equation to get \\begin{align*} 5p &= xy^2 + x^2 y - xp - yp \\\\ &= xy(x+y) -p(x+y) \\\\ &= (xy-p)(x+y) \\end{align*} Since $5p$ , $xy-p$ , and $x+y$ are all integers, $xy-p$ and $x+y$ must be a factor of $5p$ . Now there are four cases to consider.\nCase 1: and... |
DeepScaleR | [
{
"role": "user",
"content": "Find all quadruples of positive integers $(p, q, a, b)$, where $p$ and $q$ are prime numbers and $a > 1$, such that $$p^a = 1 + 5q^b.$$"
}
] | math | {
"style": "rule",
"ground_truth": "(2, 3, 4, 1) \\text{ and } (3, 2, 4, 4)"
} | {
"index": 9315,
"solution": "\nWe are tasked with finding all quadruples of positive integers \\((p, q, a, b)\\), where \\(p\\) and \\(q\\) are prime numbers, \\(a > 1\\), and they satisfy the equation:\n\n\\[\np^a = 1 + 5q^b.\n\\]\n\nGiven the integers involved, we will first check small values for \\(p\\), and t... |
DeepScaleR | [
{
"role": "user",
"content": "Find all rational roots of\n\\[4x^4 - 3x^3 - 13x^2 + 5x + 2 = 0\\]Enter all the rational roots, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "2,-\\frac{1}{4}"
} | {
"index": 9316,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real $ a$, such that there exist a function $ f: \\mathbb{R}\\rightarrow\\mathbb{R}$ satisfying the following inequality:\n\\[ x\\plus{}af(y)\\leq y\\plus{}f(f(x))\n\\]\nfor all $ x,y\\in\\mathbb{R}$"
}
] | math | {
"style": "rule",
"ground_truth": "a < 0 \\text{ or } a = 1"
} | {
"index": 9317,
"solution": "\nWe are tasked with finding all real values of \\( a \\) such that there exists a function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) satisfying the inequality:\n\\[\nx + af(y) \\leq y + f(f(x))\n\\]\nfor all \\( x, y \\in \\mathbb{R} \\).\n\n### Step-by-step Analysis\n\n1. **Cas... |
DeepScaleR | [
{
"role": "user",
"content": "Find all real $x$ such that \\[\\left\\lfloor x \\lfloor x \\rfloor\\right \\rfloor = 29.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "[5.8,6)"
} | {
"index": 9318,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $a$ for which there exists a non-constant function $f :\\Bbb R \\to \\Bbb R$ satisfying the following two equations for all $x\\in \\Bbb R:$\ni) $f(ax) = a^2f(x)$ and\nii) $f(f(x)) = a f(x).$"
}
] | math | {
"style": "rule",
"ground_truth": "0 \\text{ and } 1"
} | {
"index": 9319,
"solution": "\nTo solve this problem, we need to find all real numbers \\( a \\) for which there exists a non-constant function \\( f : \\mathbb{R} \\to \\mathbb{R} \\) satisfying the two conditions:\n\n1. \\( f(ax) = a^2f(x) \\) for all \\( x \\in \\mathbb{R} \\).\n2. \\( f(f(x)) = a f(x) \\) for ... |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $a$ such that the roots of the polynomial\n$$x^3 - 6x^2 + 21x + a$$form an arithmetic progression and are not all real."
}
] | math | {
"style": "rule",
"ground_truth": "-26"
} | {
"index": 9320,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $b$ such that the roots of the polynomial\n$$x^3 - 9x^2 + 39x + b$$ form an arithmetic progression, and one of the roots is real while the other two are complex conjugates."
}
] | math | {
"style": "rule",
"ground_truth": "-36"
} | {
"index": 9321,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $k$ for which there exists a nonzero, 2-dimensional vector $\\mathbf{v}$ such that\n\\[\\begin{pmatrix} 1 & 8 \\\\ 2 & 1 \\end{pmatrix} \\mathbf{v} = k \\mathbf{v}.\\]Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 9322,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $k$ for which there exists a nonzero, 2-dimensional vector $\\mathbf{v}$ such that\n\\[\\begin{pmatrix} 3 & 4 \\\\ 6 & 3 \\end{pmatrix} \\mathbf{v} = k \\mathbf{v}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "3 - 2\\sqrt{6}"
} | {
"index": 9323,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $k$ such that\n\\[\\left\\| k \\begin{pmatrix} 2 \\\\ -3 \\end{pmatrix} - \\begin{pmatrix} 4 \\\\ 7 \\end{pmatrix} \\right\\| = 2 \\sqrt{13}.\\]Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 9324,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $k$ such that $r^{4}+k r^{3}+r^{2}+4 k r+16=0$ is true for exactly one real number $r$."
}
] | math | {
"style": "rule",
"ground_truth": "\\pm \\frac{9}{4}"
} | {
"index": 9325,
"solution": "Any real quartic has an even number of real roots with multiplicity, so there exists real $r$ such that $x^{4}+k x^{3}+x^{2}+4 k x+16$ either takes the form $(x+r)^{4}$ (clearly impossible) or $(x+r)^{2}\\left(x^{2}+a x+b\\right)$ for some real $a, b$ with $a^{2}<4 b$. Clearly $r \\neq... |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $p$ such that the cubic equation\n$$\n5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p\n$$\nhas two roots that are both natural numbers."
}
] | math | {
"style": "rule",
"ground_truth": "76"
} | {
"index": 9326,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $x$ satisfying $$x^{9}+\\frac{9}{8} x^{6}+\\frac{27}{64} x^{3}-x+\\frac{219}{512}=0$$"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{2}, \\frac{-1 \\pm \\sqrt{13}}{4}$"
} | {
"index": 9327,
"solution": "Note that we can re-write the given equation as $$\\sqrt[3]{x-\\frac{3}{8}}=x^{3}+\\frac{3}{8}$$ Furthermore, the functions of $x$ on either side, we see, are inverses of each other and increasing. Let $f(x)=\\sqrt[3]{x-\\frac{3}{8}}$. Suppose that $f(x)=y=f^{-1}(x)$. Then, $f(y)=x$. H... |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $x$ satisfying the equation $x^{3}-8=16 \\sqrt[3]{x+1}$."
}
] | math | {
"style": "rule",
"ground_truth": "-2,1 \\pm \\sqrt{5}"
} | {
"index": 9328,
"solution": "Let $f(x)=\\frac{x^{3}-8}{8}$. Then $f^{-1}(x)=\\sqrt[3]{8x+8}=2\\sqrt[3]{x+1}$, and so the given equation is equivalent to $f(x)=f^{-1}(x)$. This implies $f(f(x))=x$. However, as $f$ is monotonically increasing, this implies that $f(x)=x$. As a result, we have $\\frac{x^{3}-8}{8}=x \\... |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $x$ so that the product $(x + i)((x + 1) + i)((x + 2) + i)$ is pure imaginary. Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "-3,-1,1"
} | {
"index": 9329,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $x$ such that\n\\[\\frac{8^x + 27^x}{12^x + 18^x} = \\frac{7}{6}.\\]Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "-1,1"
} | {
"index": 9330,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $x$ such that $$x^{2}+\\left\\lfloor\\frac{x}{2}\\right\\rfloor+\\left\\lfloor\\frac{x}{3}\\right\\rfloor=10$$"
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{14}"
} | {
"index": 9331,
"solution": "Evidently $x^{2}$ must be an integer. Well, there aren't that many things to check, are there? Among positive $x, \\sqrt{8}$ is too small and $\\sqrt{9}$ is too big; among negative $x,-\\sqrt{15}$ is too small and $-\\sqrt{13}$ is too big."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $x$ such that \\[3 \\le \\frac{x}{2x-5} < 8.\\](Give your answer in interval notation.)"
}
] | math | {
"style": "rule",
"ground_truth": "(\\tfrac83, 3]"
} | {
"index": 9332,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $x$ such that \\[\\frac{x^3+2x^2}{x^2+3x+2} + x = -6.\\]Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "-\\tfrac32"
} | {
"index": 9333,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $x$ such that the product $(x + 2i)((x + 1) + 2i)((x + 2) + 2i)((x + 3) + 2i)$ is purely imaginary."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 9334,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $x$ which satisfy \\[\\frac{x-1}{x-3} \\ge 2.\\](Give your answer in interval notation.)"
}
] | math | {
"style": "rule",
"ground_truth": "(3, 5]"
} | {
"index": 9335,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $x,y,z$ so that\n\\begin{align*}\n x^2 y + y^2 z + z^2 &= 0 \\\\\n z^3 + z^2 y + z y^3 + x^2 y &= \\frac{1}{4}(x^4 + y^4).\n\\end{align*}"
}
] | math | {
"style": "rule",
"ground_truth": "(0, 0, 0)"
} | {
"index": 9336,
"solution": "\nTo solve the system of equations for real numbers \\( x, y, \\) and \\( z \\):\n\n\\[\nx^2 y + y^2 z + z^2 = 0\n\\]\n\n\\[\nz^3 + z^2 y + z y^3 + x^2 y = \\frac{1}{4}(x^4 + y^4),\n\\]\n\nwe proceed with the following approach:\n\n### Step 1: Analyze the First Equation\nThe first equa... |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers $x,y,z\\geq 1$ satisfying \\[\\min(\\sqrt{x+xyz},\\sqrt{y+xyz},\\sqrt{z+xyz})=\\sqrt{x-1}+\\sqrt{y-1}+\\sqrt{z-1}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\[\n\\boxed{\\left(\\frac{c^2+c-1}{c^2}, \\frac{c}{c-1}, c\\right)}\n\\]"
} | {
"index": 9337,
"solution": "The key Lemma is: \\[\\sqrt{a-1}+\\sqrt{b-1} \\le \\sqrt{ab}\\] for all $a,b \\ge 1$ . Equality holds when $(a-1)(b-1)=1$ .\nThis is proven easily. \\[\\sqrt{a-1}+\\sqrt{b-1} = \\sqrt{a-1}\\sqrt{1}+\\sqrt{1}\\sqrt{b-1} \\le \\sqrt{(a-1+1)(b-1+1)} = \\sqrt{ab}\\] by Cauchy.\nEquality th... |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers \\( k \\) such that the inequality\n$$\na^{3}+b^{3}+c^{3}+d^{3}+1 \\geqslant k(a+b+c+d)\n$$\nholds for any \\( a, b, c, d \\in [-1, +\\infty) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 9338,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers \\( p \\) such that the cubic equation \\( 5x^3 - 5(p+1)x^2 + (71p-1)x + 1 = 66p \\) has two roots that are natural numbers."
}
] | math | {
"style": "rule",
"ground_truth": "76"
} | {
"index": 9339,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers \\( p \\) such that the cubic equation \\( 5x^{3} - 5(p+1)x^{2} + (71p - 1)x + 1 = 66p \\) has three roots, all of which are positive integers."
}
] | math | {
"style": "rule",
"ground_truth": "76"
} | {
"index": 9340,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers \\( x \\) such that\n\\[\n\\frac{16^x + 25^x}{20^x + 32^x} = \\frac{9}{8}.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "x = 0"
} | {
"index": 9341,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers \\( x \\) that satisfy the equation\n\n\\[\n\\frac{x-2020}{1}+\\frac{x-2019}{2}+\\cdots+\\frac{x-2000}{21}=\\frac{x-1}{2020}+\\frac{x-2}{2019}+\\cdots+\\frac{x-21}{2000},\n\\]\n\nand simplify your answer(s) as much as possible. Justify your solution."
}
... | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"index": 9342,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers \\( x \\) that satisfy the equation\n$$\n\\frac{x-2020}{1} + \\frac{x-2019}{2} + \\cdots + \\frac{x-2000}{21} = \\frac{x-1}{2020} + \\frac{x-2}{2019} + \\cdots + \\frac{x-21}{2000},\n$$\nand simplify your answer(s) as much as possible. Justify your solutio... | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"index": 9343,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real numbers \\(k\\) such that the inequality\n\\[ \na^{3} + b^{3} + c^{3} + d^{3} + 1 \\geq k(a + b + c + d)\n\\]\nholds for any \\(a, b, c, d \\in [-1, +\\infty)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 9344,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real parameters $a$ for which the equation $x^8 +ax^4 +1 = 0$ has four real roots forming an arithmetic progression."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{82}{9}"
} | {
"index": 9345,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real solutions $(x, y)$ of the system $x^{2}+y=12=y^{2}+x$."
}
] | math | {
"style": "rule",
"ground_truth": "(3,3),(-4,-4),\\left(\\frac{1+3 \\sqrt{5}}{2}, \\frac{1-3 \\sqrt{5}}{2}\\right),\\left(\\frac{1-3 \\sqrt{5}}{2}, \\frac{1+3 \\sqrt{5}}{2}\\right)"
} | {
"index": 9346,
"solution": "We have $x^{2}+y=y^{2}+x$ which can be written as $(x-y)(x+y-1)=0$. The case $x=y$ yields $x^{2}+x-12=0$, hence $(x, y)=(3,3)$ or $(-4,-4)$. The case $y=1-x$ yields $x^{2}+1-x-12=x^{2}-x-11=0$ which has solutions $x=\\frac{1 \\pm \\sqrt{1+44}}{2}=\\frac{1 \\pm 3 \\sqrt{5}}{2}$. The oth... |
DeepScaleR | [
{
"role": "user",
"content": "Find all real solutions to\n\\[\\frac{1}{(x - 1)(x - 2)} + \\frac{1}{(x - 2)(x - 3)} + \\frac{1}{(x - 3)(x - 4)} = \\frac{1}{6}.\\]Enter all solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "7,-2"
} | {
"index": 9347,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real solutions to $x^3+(x+1)^3+(x+2)^3=(x+3)^3$. Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 9348,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real solutions to $x^4+(2-x)^4=34$. Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "1 + \\sqrt{2}, 1 - \\sqrt{2}"
} | {
"index": 9349,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real solutions to $x^{4}+(2-x)^{4}=34$."
}
] | math | {
"style": "rule",
"ground_truth": "1 \\pm \\sqrt{2}"
} | {
"index": 9350,
"solution": "Let $y=2-x$, so $x+y=2$ and $x^{4}+y^{4}=34$. We know $$(x+y)^{4}=x^{4}+4 x^{3} y+6 x^{2} y^{2}+4 x y^{3}+y^{4}=x^{4}+y^{4}+2 x y(2 x^{2}+2 y^{2}+3 x y) .$$ Moreover, $x^{2}+y^{2}=(x+y)^{2}-2 x y$, so the preceding equation becomes $2^{4}=34+2 x y(2. 2^{2}-x y)$, or $(x y)^{2}-8 x y-9=... |
DeepScaleR | [
{
"role": "user",
"content": "Find all real values of $x$ for which $$\\frac{1}{\\sqrt{x}+\\sqrt{x-2}}+\\frac{1}{\\sqrt{x+2}+\\sqrt{x}}=\\frac{1}{4}$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{257}{16}"
} | {
"index": 9351,
"solution": "We note that $$\\begin{aligned} \\frac{1}{4} & =\\frac{1}{\\sqrt{x}+\\sqrt{x-2}}+\\frac{1}{\\sqrt{x+2}+\\sqrt{x}} \\\\ & =\\frac{\\sqrt{x}-\\sqrt{x-2}}{(\\sqrt{x}+\\sqrt{x-2})(\\sqrt{x}-\\sqrt{x-2})}+\\frac{\\sqrt{x+2}-\\sqrt{x}}{(\\sqrt{x+2}+\\sqrt{x})(\\sqrt{x+2}-\\sqrt{x})} \\\\ & =... |
DeepScaleR | [
{
"role": "user",
"content": "Find all real values of $x$ that satisfy $\\frac{1}{x+1} + \\frac{3}{x+7} \\ge \\frac23.$ (Give your answer in interval notation.)"
}
] | math | {
"style": "rule",
"ground_truth": "(-7, -4] \\cup (-1, 2]"
} | {
"index": 9352,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real values of $x$ that satisfy $\\frac{x(x+1)}{(x-4)^2} \\ge 12.$ (Give your answer in interval notation.)"
}
] | math | {
"style": "rule",
"ground_truth": "[3, 4) \\cup \\left(4, \\frac{64}{11}\\right]"
} | {
"index": 9353,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real values of $x$ that satisfy $x + \\frac{45}{x-4} = -10.$\n\nEnter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "-1, \\; -5"
} | {
"index": 9354,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real values of $x$ that satisfy \\[\\frac{1}{x^2+1} > \\frac{3}{x} + \\frac{17}{10}.\\](Give your answer in interval notation.)"
}
] | math | {
"style": "rule",
"ground_truth": "(-2, 0)"
} | {
"index": 9355,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all real values of $x$ which satisfy\n\\[\\frac{1}{x + 1} + \\frac{6}{x + 5} \\ge 1.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "(-5,-2] \\cup (-1,3]"
} | {
"index": 9356,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all reals $ k$ such that\r\n\\[ a^3 \\plus{} b^3 \\plus{} c^3 \\plus{} d^3 \\plus{} 1\\geq k(a \\plus{} b \\plus{} c \\plus{} d)\r\n\\]\r\nholds for all $ a,b,c,d\\geq \\minus{} 1$ .\r\n\r\n*Edited by orl.*"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 9357,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all roots of the equation \\((x-a)(x-b)=(x-c)(x-d)\\), given that \\(a+d=b+c=2016\\) and \\(a \\neq c\\) (numerical values are not given)."
}
] | math | {
"style": "rule",
"ground_truth": "1008"
} | {
"index": 9358,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all roots of the polynomial $x^3+x^2-4x-4$. Enter your answer as a list of numbers separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "-1,2,-2"
} | {
"index": 9359,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all roots of the polynomial $x^3-5x^2+3x+9$. Enter your answer as a list of numbers separated by commas. If a root occurs more than once, enter it as many times as its multiplicity."
}
] | math | {
"style": "rule",
"ground_truth": "-1,3,3"
} | {
"index": 9360,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all solutions $(x, y) \\in \\mathbb Z^2$ of the equation\n\\[x^3 - y^3 = 2xy + 8.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "(2,0),(0,-2)"
} | {
"index": 9361,
"solution": "\nConsider the equation \\(x^3 - y^3 = 2xy + 8\\). We are tasked with finding all integer solutions \\((x, y) \\in \\mathbb{Z}^2\\).\n\n### Step 1: Rewrite the Equation\n\nFirst, rewrite the given equation:\n\n\\[\nx^3 - y^3 = (x - y)(x^2 + xy + y^2).\n\\]\n\nSet this equal to the righ... |
DeepScaleR | [
{
"role": "user",
"content": "Find all solutions $x$ (real and otherwise) to the equation\n\\[x^4+64=0.\\]Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "2+2i,\\,-2-2i,\\,-2+2i,\\,2-2i"
} | {
"index": 9362,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all solutions $x$ of the inequality $$\\frac{5}{24} + \\left|x-\\frac{11}{48}\\right| < \\frac{5}{16}.$$Express your answer in interval notation, simplifying all fractions in your answer."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(\\frac{1}{8},\\frac{1}{3}\\right)"
} | {
"index": 9363,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all solutions to\n\\[\\sqrt[4]{47 - 2x} + \\sqrt[4]{35 + 2x} = 4.\\]Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "23,-17"
} | {
"index": 9364,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all solutions to\n\\[\\sqrt{x + 2 - 2 \\sqrt{x - 4}} + \\sqrt{x + 12 - 8 \\sqrt{x - 4}} = 4.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "[13]"
} | {
"index": 9365,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all solutions to\n\\[\\sqrt{x + 3 - 4 \\sqrt{x - 1}} + \\sqrt{x + 8 - 6 \\sqrt{x - 1}} = 1.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "[5,10]"
} | {
"index": 9366,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all solutions to\n\\[x^2 + 4x + 4x \\sqrt{x + 3} = 13.\\]Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 9367,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all solutions to $aabb=n^4-6n^3$ , where $a$ and $b$ are non-zero digits, and $n$ is an integer. ( $a$ and $b$ are not necessarily distinct.)"
}
] | math | {
"style": "rule",
"ground_truth": "6655"
} | {
"index": 9368,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all solutions to $(m^2+n)(m + n^2)= (m - n)^3$ , where m and n are non-zero integers.\nDo it"
}
] | math | {
"style": "rule",
"ground_truth": "\\[\n\\{(-1,-1), (8,-10), (9,-6), (9,-21)\\}\n\\]"
} | {
"index": 9369,
"solution": "Expanding both sides, \\[m^3+mn+m^2n^2+n^3=m^3-3m^2n+3mn^2-n^3\\] Note that $m^3$ can be canceled and as $n \\neq 0$ , $n$ can be factored out.\nWriting this as a quadratic equation in $n$ : \\[2n^2+(m^2-3m)n+(3m^2+m)=0\\] .\nThe discriminant $b^2-4ac$ equals \\[(m^2-3m)^2-8(3m^2+m)\\]... |
DeepScaleR | [
{
"role": "user",
"content": "Find all solutions to $x^{4}+2 x^{3}+2 x^{2}+2 x+1=0$ (including non-real solutions)."
}
] | math | {
"style": "rule",
"ground_truth": "-1, i, -i"
} | {
"index": 9370,
"solution": "We can factor the polynomial as $(x+1)^{2}(x^{2}+1)$. Therefore, the solutions are $-1, i, -i$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all solutions to the equation $\\!\\sqrt{2-3z} = 9$. Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{79}{3}"
} | {
"index": 9371,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all solutions to the equation $\\displaystyle\\sqrt[3]{2 - \\frac{x}{2}} = -3$."
}
] | math | {
"style": "rule",
"ground_truth": "58"
} | {
"index": 9372,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all solutions to the equation $\\displaystyle\\sqrt[3]{3 - x} = -\\frac{3}{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{51}{8}"
} | {
"index": 9373,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all solutions to the equation $\\sqrt{5+2z} = 11$."
}
] | math | {
"style": "rule",
"ground_truth": "58"
} | {
"index": 9374,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all solutions to the equation\\[ \\sqrt[4]{x} = \\frac{12}{7 - \\sqrt[4]{x}}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "81, 256"
} | {
"index": 9375,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all solutions to the inequality \\[\\frac{x}{x-1} + \\frac{x+2}{2x} \\ge 3.\\](Give your answer in interval notation.)"
}
] | math | {
"style": "rule",
"ground_truth": "(0, \\tfrac13] \\cup (1, 2]"
} | {
"index": 9376,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all square numbers $ S_1$ and $ S_2$ such that $ S_1 \\minus{} S_2 \\equal{} 1989.$"
}
] | math | {
"style": "rule",
"ground_truth": "$ (S_1,S_2)\\in \\{ (995^2,994^2), (333^2,330^2), (115^2,106^2), (83^2, 70^2), (67^2,50^2), (45^2, 6^2)\\}$"
} | {
"index": 9377,
"solution": "\nGiven the equation \\( S_1 - S_2 = 1989 \\), where \\( S_1 \\) and \\( S_2 \\) are square numbers, we seek to find all such pairs \\((S_1, S_2)\\). Let \\( S_1 = a^2 \\) and \\( S_2 = b^2 \\), where \\( a > b \\) are integers. Thus, we have:\n\n\\[\na^2 - b^2 = 1989.\n\\]\n\nThis can... |
DeepScaleR | [
{
"role": "user",
"content": "Find all the functions $f: \\mathbb{R} \\to\\mathbb{R}$ such that\n\\[f(x-f(y))=f(f(y))+xf(y)+f(x)-1\\]\nfor all $x,y \\in \\mathbb{R} $."
}
] | math | {
"style": "rule",
"ground_truth": "f(x)=1-\\dfrac{x^2}{2}"
} | {
"index": 9378,
"solution": "\nWe are given a functional equation for functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\):\n\n\\[\nf(x - f(y)) = f(f(y)) + x f(y) + f(x) - 1\n\\]\n\nfor all \\( x, y \\in \\mathbb{R} \\). We seek to find all possible functions \\( f \\) that satisfy this equation.\n\n### Step 1: Notic... |
DeepScaleR | [
{
"role": "user",
"content": "Find all the integer roots of\n\\[x^3 - 3x^2 - 13x + 15 = 0.\\]Enter all the integer roots, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "-3,1,5"
} | {
"index": 9379,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all the integers $n>1$ with the following property: the numbers $1,2, \\ldots, n$ can be arranged in a line so that, of any two adjacent numbers, one is divisible by the other."
}
] | math | {
"style": "rule",
"ground_truth": "2, 3, 4, 6"
} | {
"index": 9380,
"solution": "$2,3,4,6$ The values $n=2,3,4,6$ work, as shown by respective examples 1,$2 ; 2,1,3 ; 2,4,1,3 ; 3,6,2,4,1,5$. We shall show that there are no other possibilities. If $n=2 k+1$ is odd, then none of the numbers $k+1, k+2, \\ldots, 2 k+1$ can divide any other, so no two of these numbers a... |
DeepScaleR | [
{
"role": "user",
"content": "Find all the integers $x, y$ and $z$ greater than or equal to $0$ such that $2^x + 9 \\cdot 7^y = z^3$"
}
] | math | {
"style": "rule",
"ground_truth": "(0, 1, 4)"
} | {
"index": 9381,
"solution": "\nWe are tasked with finding all non-negative integer solutions \\((x, y, z)\\) to the equation:\n\n\\[\n2^x + 9 \\cdot 7^y = z^3\n\\]\n\nGiven that our solution must satisfy integer constraints and each variable is greater than or equal to zero, we will systematically explore potentia... |
DeepScaleR | [
{
"role": "user",
"content": "Find all the integers written as $\\overline{abcd}$ in decimal representation and $\\overline{dcba}$ in base $7$ ."
}
] | math | {
"style": "rule",
"ground_truth": "2116"
} | {
"index": 9382,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all the pairs of prime numbers $ (p,q)$ such that $ pq|5^p\\plus{}5^q.$"
}
] | math | {
"style": "rule",
"ground_truth": "(2, 3), (2, 5), (3, 2), (5, 2), (5, 5), (5, 313), (313, 5)"
} | {
"index": 9383,
"solution": "\nTo find all pairs of prime numbers \\((p, q)\\) such that \\(pq \\mid 5^p + 5^q\\), we consider several cases:\n\n1. **Case 1: Either \\(p\\) or \\(q\\) is 2.**\n - Assume \\(p = 2\\). Then \\(2q \\mid 5^2 + 5^q\\).\n - This simplifies to \\(2q \\mid 25 + 5^q\\).\n - Since \\(2... |
DeepScaleR | [
{
"role": "user",
"content": "Find all the positive integers less than 1000 such that the cube of the sum of its digits is equal to the square of such integer."
}
] | math | {
"style": "rule",
"ground_truth": "1 \\text{ and } 27"
} | {
"index": 9384,
"solution": "\nWe are tasked with finding all positive integers \\( n \\) less than 1000 such that the cube of the sum of its digits equals the square of the integer itself. Let \\( n \\) be a three-digit number with digits \\( a, b, \\) and \\( c \\) such that \\( n = 100a + 10b + c \\).\n\nThe pr... |
DeepScaleR | [
{
"role": "user",
"content": "Find all the positive perfect cubes that are not divisible by $10$ such that the number obtained by erasing the last three digits is also a perfect cube."
}
] | math | {
"style": "rule",
"ground_truth": "1331 \\text{ and } 1728"
} | {
"index": 9385,
"solution": "\nTo solve the given problem, we need to find all positive perfect cubes that are not divisible by \\(10\\) and have the property that when the last three digits are erased, the resulting number is also a perfect cube.\n\n1. **Understanding the Cube Condition**: Let \\( n^3 \\) be a pe... |
DeepScaleR | [
{
"role": "user",
"content": "Find all the real solutions to\n\\[\\frac{(x - 1)(x - 2)(x - 3)(x - 4)(x - 3)(x - 2)(x - 1)}{(x - 2)(x - 4)(x - 2)} = 1.\\]Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{2}, 2 - \\sqrt{2}"
} | {
"index": 9386,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all the roots of $\\left(x^{2}+3 x+2\\right)\\left(x^{2}-7 x+12\\right)\\left(x^{2}-2 x-1\\right)+24=0$."
}
] | math | {
"style": "rule",
"ground_truth": "0, 2, 1 \\pm \\sqrt{6}, 1 \\pm 2 \\sqrt{2}"
} | {
"index": 9387,
"solution": "We re-factor as $(x+1)(x-3)(x+2)(x-4)\\left(x^{2}-2 x-1\\right)+24$, or $\\left(x^{2}-2 x-3\\right)\\left(x^{2}-2 x-8\\right)\\left(x^{2}-2 x-1\\right)+24$, and this becomes $(y-4)(y-9)(y-2)+24$ where $y=(x-1)^{2}$. Now, $(y-4)(y-9)(y-2)+24=(y-8)(y-6)(y-1)$, so $y$ is 1, 6, or 8. Thus ... |
DeepScaleR | [
{
"role": "user",
"content": "Find all the roots of the equation\n\n\\[ 1 - \\frac{x}{1} + \\frac{x(x-1)}{2!} - \\frac{x(x-1)(x-2)}{3!} + \\frac{x(x-1)(x-2)(x-3)}{4!} - \\frac{x(x-1)(x-2)(x-3)(x-4)}{5!} + \\frac{x(x-1)(x-2)(x-3)(x-4)(x-5)}{6!} = 0 \\]\n\n(Where \\( n! = 1 \\cdot 2 \\cdot 3 \\cdots n \\))\n\... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 9388,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all the roots of the polynomial $x^{5}-5 x^{4}+11 x^{3}-13 x^{2}+9 x-3$."
}
] | math | {
"style": "rule",
"ground_truth": "1, \\frac{3+\\sqrt{3} i}{2}, \\frac{1-\\sqrt{3} i}{2}, \\frac{3-\\sqrt{3} i}{2}, \\frac{1+\\sqrt{3} i}{2}"
} | {
"index": 9389,
"solution": "The $x^{5}-5 x^{4}$ at the beginning of the polynomial motivates us to write it as $(x-1)^{5}+x^{3}-3 x^{2}+4 x-2$ and again the presence of the $x^{3}-3 x^{2}$ motivates writing the polynomial in the form $(x-1)^{5}+(x-1)^{3}+(x-1)$. Let $a$ and $b$ be the roots of the polynomial $x^{... |
DeepScaleR | [
{
"role": "user",
"content": "Find all the solutions to\n\\[\\frac{1}{x^2 + 11x - 8} + \\frac{1}{x^2 + 2x - 8} + \\frac{1}{x^2 - 13x - 8} = 0.\\]Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "8,1,-1,-8"
} | {
"index": 9390,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all the solutions to\n\\[\\sqrt[3]{15x - 1} + \\sqrt[3]{13x + 1} = 4 \\sqrt[3]{x}.\\]Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "0, \\frac{1}{14}, -\\frac{1}{12}"
} | {
"index": 9391,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all the solutions to\n\\[\\sqrt[3]{2 - x} + \\sqrt{x - 1} = 1.\\]Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "1,2,10"
} | {
"index": 9392,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all the solutions to\n\\[\\sqrt{(2 + \\sqrt{3})^x} + \\sqrt{(2 - \\sqrt{3})^x} = 4.\\]Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "2,-2"
} | {
"index": 9393,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all the solutions to\n\\[\\sqrt{x} + 2 \\sqrt{x^2 + 7x} + \\sqrt{x + 7} = 35 - 2x.\\]Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{841}{144}"
} | {
"index": 9394,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all the triples of positive integers $(a,b,c)$ for which the number\n\\[\\frac{(a+b)^4}{c}+\\frac{(b+c)^4}{a}+\\frac{(c+a)^4}{b}\\]\nis an integer and $a+b+c$ is a prime."
}
] | math | {
"style": "rule",
"ground_truth": "(1, 1, 1), (2, 2, 1), (6, 3, 2)"
} | {
"index": 9395,
"solution": "\nTo solve this problem, we are tasked with finding all triples of positive integers \\((a, b, c)\\) such that the expression \n\n\\[\n\\frac{(a+b)^4}{c} + \\frac{(b+c)^4}{a} + \\frac{(c+a)^4}{b}\n\\]\n\nis an integer and the sum \\(a + b + c\\) is a prime number.\n\n### Step-by-step S... |
DeepScaleR | [
{
"role": "user",
"content": "Find all the values of $m$ for which the zeros of $2 x^{2}-m x-8$ differ by $m-1$."
}
] | math | {
"style": "rule",
"ground_truth": "6,-\\frac{10}{3}"
} | {
"index": 9396,
"solution": "6,-\\frac{10}{3}."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all the ways in which the number 1987 can be written in another base as a three-digit number where the sum of the digits is 25."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 9397,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all three-digit integers \\( abc = n \\) such that \\( \\frac{2n}{3} = a! \\cdot b! \\cdot c! \\)."
}
] | math | {
"style": "rule",
"ground_truth": "432"
} | {
"index": 9398,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all three-digit numbers $\\overline{\\Pi B \\Gamma}$, consisting of distinct digits $\\Pi, B$, and $\\Gamma$, for which the following equality holds: $\\overline{\\Pi B \\Gamma} = (\\Pi + B + \\Gamma) \\times (\\Pi + B + \\Gamma + 1)$."
}
] | math | {
"style": "rule",
"ground_truth": "156"
} | {
"index": 9399,
"solution": ""
} |
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