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 three-digit numbers $\\overline{МГУ}$, comprised of different digits $M$, $\\Gamma$, and $Y$, for which the equality $\\overline{\\text{МГУ}} = (M + \\Gamma + Y) \\times (M + \\Gamma + Y - 2)$ holds."
}
] | math | {
"style": "rule",
"ground_truth": "195"
} | {
"index": 9400,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all three-digit numbers \\( \\overline{\\mathrm{MGU}} \\) consisting of distinct digits \\( M, \\Gamma, \\) and \\( U \\) for which the equality \\( \\overline{\\mathrm{MGU}} = (M + \\Gamma + U) \\times (M + \\Gamma + U - 2) \\) holds."
}
] | math | {
"style": "rule",
"ground_truth": "195"
} | {
"index": 9401,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all three-digit numbers that are equal to the sum of all their digits plus twice the square of the sum of their digits. List all possible numbers in ascending order without spaces and enter the resulting concatenated multi-digit number."
}
] | math | {
"style": "rule",
"ground_truth": "171465666"
} | {
"index": 9402,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all triples $(\\alpha, \\beta, \\gamma)$ of positive real numbers for which the expression $$ K = \\frac{\\alpha+3 \\gamma}{\\alpha + 2\\beta + \\gamma} + \\frac{4\\beta}{\\alpha+\\beta+2\\gamma} - \\frac{8 \\gamma}{\\alpha+ \\beta + 3\\gamma} $$ obtains its minimum va... | math | {
"style": "rule",
"ground_truth": "2/5"
} | {
"index": 9403,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations\r\n\\[ \\begin{cases}x^3 \\equal{} 3x\\minus{}12y\\plus{}50, \\\\ y^3 \\equal{} 12y\\plus{}3z\\minus{}2, \\\\ z^3 \\equal{} 27z \\plus{} 27x. \\end{cases}\\]\r\n\r\n[i]Razvan Gelca.[/i]"
}
] | math | {
"style": "rule",
"ground_truth": "(2, 4, 6)"
} | {
"index": 9404,
"solution": "\nWe are given the system of equations:\n\\[\n\\begin{cases}\nx^3 = 3x - 12y + 50, \\\\\ny^3 = 12y + 3z - 2, \\\\\nz^3 = 27z + 27x.\n\\end{cases}\n\\]\n\nTo find all triples \\((x, y, z)\\) of real numbers that satisfy these equations, we analyze the behavior of the functions involved.... |
DeepScaleR | [
{
"role": "user",
"content": "Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$."
}
] | math | {
"style": "rule",
"ground_truth": "(3, 2, 1), (3, 1, 2), (2, 3, 1), (2, 1, 3), (1, 3, 2), (1, 2, 3)"
} | {
"index": 9405,
"solution": "\nWe are required to find all triples \\((a, b, c)\\) of positive integers such that:\n\n\\[\na^3 + b^3 + c^3 = (abc)^2\n\\]\n\n### Step 1: Analyze the Equation\n\nWe start by analyzing the given equation:\n\n\\[\na^3 + b^3 + c^3 = (abc)^2\n\\]\n\nWe will assume, without loss of genera... |
DeepScaleR | [
{
"role": "user",
"content": "Find all triples $(a, b, c)$ of real numbers such that\n$$ a^2 + ab + c = 0, $$\n$$b^2 + bc + a = 0, $$\n$$c^2 + ca + b = 0.$$"
}
] | math | {
"style": "rule",
"ground_truth": "(0, 0, 0)\\left(-\\frac{1}{2}, -\\frac{1}{2}, -\\frac{1}{2}\\right)"
} | {
"index": 9406,
"solution": "\nTo find all triples \\((a, b, c)\\) of real numbers satisfying the system of equations:\n\n\\[\n\\begin{align*}\n1. \\quad & a^2 + ab + c = 0, \\\\\n2. \\quad & b^2 + bc + a = 0, \\\\\n3. \\quad & c^2 + ca + b = 0, \n\\end{align*}\n\\]\n\nwe proceed by considering possible symmetric ... |
DeepScaleR | [
{
"role": "user",
"content": "Find all triples $(x, y, z)$ of nonnegative integers such that\n$$ x^5+x^4+1=3^y7^z $$"
}
] | math | {
"style": "rule",
"ground_truth": "(0, 0, 0), (1, 1, 0), (2, 0, 2)"
} | {
"index": 9407,
"solution": "\nGiven the equation:\n\n\\[\nx^5 + x^4 + 1 = 3^y 7^z\n\\]\n\nwe are tasked with finding all triples \\((x, y, z)\\) of nonnegative integers that satisfy this equation.\n\n### Step 1: Analyze Small Values of \\(x\\)\n\nStart by checking small values of \\(x\\) to find possible solution... |
DeepScaleR | [
{
"role": "user",
"content": "Find all triples $(x; y; p)$ of two non-negative integers $x, y$ and a prime number p such that $ p^x-y^p=1 $"
}
] | math | {
"style": "rule",
"ground_truth": "(0, 0, 2), (1, 1, 2), (2, 2, 3)"
} | {
"index": 9408,
"solution": "\nThe problem requires us to find all triples \\((x, y, p)\\) consisting of two non-negative integers \\(x\\) and \\(y\\), and a prime number \\(p\\), such that:\n\n\\[\np^x - y^p = 1\n\\]\n\nTo solve this problem, we'll analyze it case by case, beginning with small values for \\(x\\) ... |
DeepScaleR | [
{
"role": "user",
"content": "Find all triples of positive integers $(x, y, z)$ such that $x^{2}+y-z=100$ and $x+y^{2}-z=124$."
}
] | math | {
"style": "rule",
"ground_truth": "(12,13,57)"
} | {
"index": 9409,
"solution": "Cancel $z$ to get $24=(y-x)(y+x-1)$. Since $x, y$ are positive, we have $y+x-1 \\geq 1+1-1>0$, so $0<y-x<y+x-1$. But $y-x$ and $y+x-1$ have opposite parity, so $(y-x, y+x-1) \\in\\{(1,24),(3,8)\\}$ yields $(y, x) \\in\\{(13,12),(6,3)\\}$. Finally, $0<z=x^{2}+y-100$ forces $(x, y, z)=(1... |
DeepScaleR | [
{
"role": "user",
"content": "Find all triples of positive integers $(x,y,z)$ that satisfy the equation \n$$2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023.$$"
}
] | math | {
"style": "rule",
"ground_truth": "(2, 3, 3)"
} | {
"index": 9410,
"solution": "\nTo solve the given equation for triples \\((x, y, z)\\) of positive integers:\n\n\\[\n2(x + y + z + 2xyz)^2 = (2xy + 2yz + 2zx + 1)^2 + 2023,\n\\]\n\nwe start by analyzing the structure of the equation. The equation can be seen as comparing the square of two polynomials with an addit... |
DeepScaleR | [
{
"role": "user",
"content": "Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$."
}
] | math | {
"style": "rule",
"ground_truth": "(5, 3, 19)"
} | {
"index": 9411,
"solution": "\nWe are tasked with finding all triples of primes \\((p, q, r)\\) that satisfy the equation:\n\n\\[\n3p^4 - 5q^4 - 4r^2 = 26.\n\\]\n\nWe begin by analyzing the equation with respect to the properties of prime numbers.\n\n1. **Testing Small Primes**:\n - Since \\( p^4 \\), \\( q^4 \\... |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of $a$ for which the points $(0,0,0),$ $(1,a,0),$ $(0,1,a),$ and $(a,0,1)$ are coplanar."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 9412,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of $b$ such that $b+\\lceil b\\rceil=17.8$."
}
] | math | {
"style": "rule",
"ground_truth": "8.8"
} | {
"index": 9413,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of $k$ for which the positive difference between the solutions of\n\\[5x^2 + 4x + k = 0\\]equals the sum of the squares of these solutions. Enter all possible values of $k,$ separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}, -\\frac{12}{5}"
} | {
"index": 9414,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of $k$ so that\n\\[x^2 - (k - 3) x - k + 6 > 0\\]for all $x.$"
}
] | math | {
"style": "rule",
"ground_truth": "(-3,5)"
} | {
"index": 9415,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of $r$ such that $\\lfloor r \\rfloor + r = 12.2$."
}
] | math | {
"style": "rule",
"ground_truth": "r=6.2"
} | {
"index": 9416,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of $r$ such that $\\lfloor r \\rfloor + r = 16.5$."
}
] | math | {
"style": "rule",
"ground_truth": "8.5"
} | {
"index": 9417,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of $x$ so that $\\arccos x > \\arcsin x.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\left[ -1, \\frac{1}{\\sqrt{2}} \\right)"
} | {
"index": 9418,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of $x$ such that\n\\[3^x + 4^x + 5^x = 6^x.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 9419,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of $x$ that satisfy\n\\[5x - 1 < (x + 1)^2 < 7x - 3.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "(2,4)"
} | {
"index": 9420,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of $x$ that satisfy $x=1-x+x^{2}-x^{3}+x^{4}-x^{5}+\\cdots$ (be careful; this is tricky)."
}
] | math | {
"style": "rule",
"ground_truth": "x=\\frac{-1+\\sqrt{5}}{2}"
} | {
"index": 9421,
"solution": "Multiplying both sides by $1+x$ gives $(1+x) x=1$, or $x=\\frac{-1 \\pm \\sqrt{5}}{2}$. However, the series only converges for $|x|<1$, so only the answer $x=\\frac{-1+\\sqrt{5}}{2}$ makes sense."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of $x$ that satisfy the equation $|x-3|=2x+4$. Express your answers in simplest fractional form."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac13"
} | {
"index": 9422,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of $x$ which satisfy\n\\[\\frac{6}{\\sqrt{x - 8} - 9} + \\frac{1}{\\sqrt{x - 8} - 4} + \\frac{7}{\\sqrt{x - 8} + 4} + \\frac{12}{\\sqrt{x - 8} + 9} = 0.\\]Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "17,44"
} | {
"index": 9423,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of $x$ with $0 \\le x < 2 \\pi$ that satisfy $\\sin x + \\cos x = \\sqrt{2}.$ Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 9424,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of $x$ with $0 \\le x < \\pi$ that satisfy $\\sin x - \\cos x = 1$. Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"index": 9425,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of $z$ such that $z^4 - 4z^2 + 3 = 0$. Enter all the solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{3},-1,1,\\sqrt{3}"
} | {
"index": 9426,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of \\( a \\) for which the equation \\( x^{2} + 2ax = 8a \\) has two distinct integer roots. In your answer, record the product of all such values of \\( a \\), rounding to two decimal places if necessary."
}
] | math | {
"style": "rule",
"ground_truth": "506.25"
} | {
"index": 9427,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of \\( a \\) for which the equation \\( x^{2} + 2ax = 8a \\) has two distinct integer roots. Record the product of all such \\( a \\), rounding to the nearest hundredth if necessary."
}
] | math | {
"style": "rule",
"ground_truth": "506.25"
} | {
"index": 9428,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of \\( a \\) for which the quadratic trinomials \\( x^{2} - 6x + 4a \\) and \\( x^{2} + ax + 6 \\) each have two roots, with the sum of the squares of the roots of the first trinomial equal to the sum of the squares of the roots of the second trinomial."
}
] | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"index": 9429,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of \\( a \\) for which the system \n\n\\[\n\\left\\{\n\\begin{array}{l}\nx^{2} + 4y^{2} = 1 \\\\\nx + 2y = a\n\\end{array}\n\\right.\n\\]\n\nhas a unique solution. If necessary, round your answer to two decimal places. If there are no solutions, answer with 0."
... | math | {
"style": "rule",
"ground_truth": "-1.41"
} | {
"index": 9430,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of \\( a \\) such that the roots \\( x_1, x_2, x_3 \\) of the polynomial\n\\[ x^3 - 6x^2 + ax + a \\]\nsatisfy\n\\[ \\left(x_1 - 3\\right)^3 + \\left(x_2 - 3\\right)^3 + \\left(x_3 - 3\\right)^3 = 0. \\]"
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"index": 9431,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of \\( n \\in \\mathbf{N} \\) for which there exist a number \\( m \\in \\mathbf{N} \\), a triangle \\( ABC \\) with sides \\( AB = 33 \\), \\( AC = 21 \\), \\( BC = n \\), and points \\( D \\), \\( E \\) on sides \\( AB \\), \\( AC \\) respectively, satisfying ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 9432,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of \\( n \\in \\mathbf{N} \\) possessing the following property: if you write the numbers \\( n^3 \\) and \\( n^4 \\) next to each other (in decimal system), then in the resulting sequence, each of the 10 digits \\( 0, 1, \\ldots, 9 \\) appears exactly once."
... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 9433,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of \\( x \\) for which the greater of the numbers \\( \\sqrt{\\frac{x}{2}} \\) and \\( \\operatorname{tg} x \\) is not greater than 1. Provide the total length of the intervals on the number line that satisfy this condition, rounding the result to the nearest hu... | math | {
"style": "rule",
"ground_truth": "1.21"
} | {
"index": 9434,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of \\( x \\) for which the smaller of the numbers \\( \\frac{1}{x} \\) and \\( \\sin x \\) is greater than \\( \\frac{1}{2} \\). In the answer, provide the total length of the resulting intervals on the number line, rounding to the nearest hundredth if necessary... | math | {
"style": "rule",
"ground_truth": "2.09"
} | {
"index": 9435,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of \\(a\\) such that the roots \\(x_1, x_2, x_3\\) of the polynomial \\(x^3 - 6x^2 + ax + a\\) satisfy \\((x_1 - 3)^2 + (x_2 - 3)^3 + (x_3 - 3)^3 = 0\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"index": 9436,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of the parameter \\(a\\) for which the quadratic trinomial \\(\\frac{1}{3} x^2 + \\left(a+\\frac{1}{2}\\right) x + \\left(a^2 + a\\right)\\) has two roots, the sum of the cubes of which is exactly 3 times their product. In your answer, specify the largest of suc... | math | {
"style": "rule",
"ground_truth": "-1/4"
} | {
"index": 9437,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all values of the real number $a$ so that the four complex roots of\n\\[z^4 - 6z^3 + 11az^2 - 3(2a^2 + 3a - 3) z + 1 = 0\\]form the vertices of a parallelogram in the complex plane. Enter all the values, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 9438,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find an $n$ such that $n!-(n-1)!+(n-2)!-(n-3)!+\\cdots \\pm 1$ ! is prime. Be prepared to justify your answer for $\\left\\{\\begin{array}{c}n, \\\\ {\\left[\\frac{n+225}{10}\\right],}\\end{array} n \\leq 25\\right.$ points, where $[N]$ is the greatest integer less than $N$."
... | math | {
"style": "rule",
"ground_truth": "3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160"
} | {
"index": 9439,
"solution": "$3,4,5,6,7,8,10,15,19,41$ (26 points), 59, 61 (28 points), 105 (33 points), 160 (38 points) are the only ones less than or equal to 335. If anyone produces an answer larger than 335, then we ask for justification to call their bluff. It is not known whether or not there are infinitely ... |
DeepScaleR | [
{
"role": "user",
"content": "Find an approximate value of $0.998^6$ such that the error is less than $0.001$."
}
] | math | {
"style": "rule",
"ground_truth": "0.988"
} | {
"index": 9440,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find an axis of symmetry for the function $f(x) = \\cos(2x + \\frac{\\pi}{6})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{12}"
} | {
"index": 9441,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find an eight-digit palindrome that is a multiple of three, composed of the digits 0 and 1, given that all its prime divisors only use the digits 1, 3, and %. (Palindromes read the same forwards and backwards, for example, 11011)."
}
] | math | {
"style": "rule",
"ground_truth": "10111101"
} | {
"index": 9442,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find an integer $n$ such that the decimal representation of the number $5^{n}$ contains at least 1968 consecutive zeros."
}
] | math | {
"style": "rule",
"ground_truth": "1968"
} | {
"index": 9443,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find an integer $n$, where $100 \\leq n \\leq 1997$, such that \n\\[ \\frac{2^n+2}{n} \\] \nis also an integer."
}
] | math | {
"style": "rule",
"ground_truth": "946"
} | {
"index": 9444,
"solution": "To find an integer \\( n \\) such that \\( 100 \\leq n \\leq 1997 \\) and \n\\[\n\\frac{2^n + 2}{n}\n\\] \nis an integer, we need to ensure that \\( n \\mid (2^n + 2) \\). This means that the expression can be rewritten using divisibility:\n\n\\[\n2^n + 2 \\equiv 0 \\pmod{n}.\n\\]\n\nT... |
DeepScaleR | [
{
"role": "user",
"content": "Find an ordered pair $(a, b)$ of real numbers for which $x^{2}+a x+b$ has a non-real root whose cube is 343."
}
] | math | {
"style": "rule",
"ground_truth": "(7, 49)"
} | {
"index": 9445,
"solution": "The cube roots of 343 are the roots of $x^{3}-343$, which is $(x-7)\\left(x^{2}+7 x+49\\right)$. Therefore the ordered pair we want is $(\\mathbf{7, 49})$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find an ordered pair $(u,v)$ that solves the system: \\begin{align*} 5u &= -7 - 2v,\\\\ 3u &= 4v - 25 \\end{align*}"
}
] | math | {
"style": "rule",
"ground_truth": "(-3,4)"
} | {
"index": 9446,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find an ordered pair $(x,y)$ that satisfies both of the equations below: \\begin{align*} 2x - 3y &= -5,\\\\ 5x - 2y &= 4. \\end{align*}"
}
] | math | {
"style": "rule",
"ground_truth": "(2,3)"
} | {
"index": 9447,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find and describe the pattern by which the sequence of numbers is formed. Determine the next number in this sequence.\n\n$$\n112, 224, 448, 8816, 6612\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "224"
} | {
"index": 9448,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find any quadruple of positive integers $(a, b, c, d)$ satisfying $a^{3}+b^{4}+c^{5}=d^{11}$ and $a b c<10^{5}$."
}
] | math | {
"style": "rule",
"ground_truth": "(128,32,16,4) \\text{ or } (160,16,8,4)"
} | {
"index": 9449,
"solution": "It's easy to guess that there are solutions such that $a, b, c, d$ are in the form of $n^{x}$, where $n$ is a rather small number. After a few attempts, we can see that we obtain simple equations when $n=2$ or $n=3$ : for $n=2$, the equation becomes in the form of $2^{t}+2^{t}+2^{t+1}=... |
DeepScaleR | [
{
"role": "user",
"content": "Find any solution to the rebus\n\n$$\n\\overline{A B C A}=182 \\cdot \\overline{C D}\n$$\n\nwhere \\( A, B, C, D \\) are four distinct non-zero digits (the notation \\(\\overline{X Y \\ldots Z}\\) denotes the decimal representation of a number).\n\nAs an answer, write the four-... | math | {
"style": "rule",
"ground_truth": "2916"
} | {
"index": 9450,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find at the point \\( M_{0}(1,1,1) \\) the direction of the greatest change of the scalar field \\( u = xy + yz + xz \\) and the magnitude of this greatest change at this point."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 9451,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find both the sum and the product of the coordinates of the midpoint of the segment with endpoints $(8, 15)$ and $(-2, -3)$."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 9452,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find constants $A$, $B$, and $C$, such that\n$$\\frac{-x^2+3x-4}{x^3+x}= \\frac{A}{x} +\\frac{Bx+C}{x^2+1} $$Enter your answer as the ordered triplet $(A,B,C)$."
}
] | math | {
"style": "rule",
"ground_truth": "(-4,3,3)"
} | {
"index": 9453,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find constants $A,$ $B,$ and $C$ so that\n\\[\\frac{4x}{(x - 5)(x - 3)^2} = \\frac{A}{x - 5} + \\frac{B}{x - 3} + \\frac{C}{(x - 3)^2}.\\]Enter the ordered triple $(A,B,C).$"
}
] | math | {
"style": "rule",
"ground_truth": "(5,-5,-6)"
} | {
"index": 9454,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find constants $A,$ $B,$ and $C$ so that\n\\[\\frac{x^2 - 7}{(x - 2)(x - 3)(x - 5)} = \\frac{A}{x - 2} + \\frac{B}{x - 3} + \\frac{C}{x - 5}.\\]Enter the ordered triple $(A,B,C).$"
}
] | math | {
"style": "rule",
"ground_truth": "(-1,-1,3)"
} | {
"index": 9455,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find constants $b_1, b_2, b_3, b_4, b_5, b_6, b_7$ such that\n\\[\n\\cos^7 \\theta = b_1 \\cos \\theta + b_2 \\cos 2 \\theta + b_3 \\cos 3 \\theta + b_4 \\cos 4 \\theta + b_5 \\cos 5 \\theta + b_6 \\cos 6 \\theta + b_7 \\cos 7 \\theta\n\\]\nfor all angles $\\theta$, and compute... | math | {
"style": "rule",
"ground_truth": "\\frac{1716}{4096}"
} | {
"index": 9456,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find distinct digits to replace the letters \\(A, B, C, D\\) such that the following division in the decimal system holds:\n\n$$\n\\frac{ABC}{BBBB} = 0,\\overline{BCDB \\, BCDB \\, \\ldots}\n$$\n\n(in other words, the quotient should be a repeating decimal)."
}
] | math | {
"style": "rule",
"ground_truth": "219"
} | {
"index": 9457,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find four consecutive odd numbers, none of which are divisible by 3, such that their sum is divisible by 5. What is the smallest possible value of this sum?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 9458,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find four distinct positive integers $a, b, c, d$ less than $15$ which are invertible modulo $15$. Calculate the remainder when $(abc + abd + acd + bcd)(abcd)^{-1}$ is divided by $15$."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 9459,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find four positive integers that are divisors of each number in the list $$45, 90, -15, 135, 180.$$ Calculate the sum of these four integers."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 9460,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find four-thirds of $\\frac{9}{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 9461,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find how many even natural-number factors does $n = 2^3 \\cdot 3^2 \\cdot 5^2$ have, where the sum of the exponents in any factor does not exceed 4?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 9462,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find how many integer values of \\( x \\) are there such that \\( \\lceil{\\sqrt{x}}\\rceil=18 \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 9463,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find how many positive integers $k\\in\\{1,2,\\ldots,2022\\}$ have the following property: if $2022$ real numbers are written on a circle so that the sum of any $k$ consecutive numbers is equal to $2022$ then all of the $2022$ numbers are equal."
}
] | math | {
"style": "rule",
"ground_truth": "672"
} | {
"index": 9464,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find in explicit form all ordered pairs of positive integers $(m, n)$ such that $mn-1$ divides $m^2 + n^2$."
}
] | math | {
"style": "rule",
"ground_truth": "(2, 1), (3, 1), (1, 2), (1, 3)"
} | {
"index": 9465,
"solution": "\nTo find all ordered pairs of positive integers \\((m, n)\\) such that \\(mn-1\\) divides \\(m^2 + n^2\\), we start by considering the condition:\n\\[\n\\frac{m^2 + n^2}{mn - 1} = c \\quad \\text{where} \\quad c \\in \\mathbb{Z}.\n\\]\nThis implies:\n\\[\nm^2 + n^2 = c(mn - 1).\n\\]\n... |
DeepScaleR | [
{
"role": "user",
"content": "Find maximal positive integer $p$ such that $5^7$ is sum of $p$ consecutive positive integers"
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 9466,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find maximum value of number $a$ such that for any arrangement of numbers $1,2,\\ldots ,10$ on a circle, we can find three consecutive numbers such their sum bigger or equal than $a$ ."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 9467,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find minimum of $x+y+z$ where $x$ , $y$ and $z$ are real numbers such that $x \\geq 4$ , $y \\geq 5$ , $z \\geq 6$ and $x^2+y^2+z^2 \\geq 90$"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 9468,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find one fourth of 12.8, expressed as a simplified improper fraction and also as a mixed number."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\frac{1}{5}"
} | {
"index": 9469,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find one third of 5.4, expressed as a simplified fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{5}"
} | {
"index": 9470,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find one third of 7.2, expressed as a simplified fraction or a mixed number."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\frac{2}{5}"
} | {
"index": 9471,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find one-fourth of 6 times 8."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 9472,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find out how many positive integers $n$ not larger than $2009$ exist such that the last digit of $n^{20}$ is $1$ ."
}
] | math | {
"style": "rule",
"ground_truth": "804"
} | {
"index": 9473,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge $AB$ of the regular octahedron from point $P$ outside if and only if the intersection of non degenerate tria... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 9474,
"solution": "\nTo determine the maximum number of edges of a regular octahedron that can be seen from a point outside the octahedron, we start by considering the geometric properties of the octahedron and the visibility conditions.\n\nA regular octahedron has 12 edges. The visibility of an edge fro... |
DeepScaleR | [
{
"role": "user",
"content": "Find real numbers \\( x, y, z \\) greater than 1 that satisfy the equation\n\n\\[ x + y + z + \\frac{3}{x - 1} + \\frac{3}{y - 1} + \\frac{3}{z - 1} = 2(\\sqrt{x + 2} + \\sqrt{y + 2} + \\sqrt{z + 2}). \\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 + \\sqrt{13}}{2}"
} | {
"index": 9475,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find some four different natural numbers with the following property: if you add to the product of any two of them the product of the two remaining numbers. you get a prime number."
}
] | math | {
"style": "rule",
"ground_truth": "1, 2, 3, 5"
} | {
"index": 9476,
"solution": "\nTo solve this problem, let's denote the four different natural numbers as \\( a \\), \\( b \\), \\( c \\), and \\( d \\). The condition given is that for any pair of these numbers, say \\( (a, b) \\), the expression:\n\n\\[\nab + cd\n\\]\n\nmust be a prime number. Similarly, for the ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the $2019$ th strictly positive integer $n$ such that $\\binom{2n}{n}$ is not divisible by $5$ ."
}
] | math | {
"style": "rule",
"ground_truth": "37805"
} | {
"index": 9477,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the $1314^{\\text{th}}$ digit past the decimal point in the decimal expansion of $\\dfrac{5}{14}$."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 9478,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the $2 \\times 2$ matrix $\\mathbf{M}$ such that $\\mathbf{M} \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix} = \\begin{pmatrix} -4 \\\\ 4 \\end{pmatrix}$ and $\\mathbf{M} \\begin{pmatrix} -3 \\\\ 1 \\end{pmatrix} = \\begin{pmatrix} -23 \\\\ 2 \\end{pmatrix}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 6 & -5 \\\\ 0 & 2 \\end{pmatrix}"
} | {
"index": 9479,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the $2 \\times 2$ matrix $\\mathbf{M}$ such that $\\mathbf{M} \\begin{pmatrix} 3 \\\\ 0 \\end{pmatrix} = \\begin{pmatrix} 6 \\\\ 21 \\end{pmatrix}$ and $\\mathbf{M} \\begin{pmatrix} -1 \\\\ 5 \\end{pmatrix} = \\begin{pmatrix} 3 \\\\ -17 \\end{pmatrix}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 2 & 1 \\\\ 7 & -2 \\end{pmatrix}"
} | {
"index": 9480,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the $3 \\times 3$ matrix $\\mathbf{M}$ such that\n\\[\\mathbf{M} \\mathbf{v} = -4 \\mathbf{v}\\]for all three-dimensional vectors $\\mathbf{v}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -4 & 0 & 0 \\\\ 0 & -4 & 0 \\\\ 0 & 0 & -4 \\end{pmatrix}"
} | {
"index": 9481,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the $3 \\times 3$ matrix $\\mathbf{M}$ such that for a $3 \\times 3$ matrix $\\mathbf{N},$ $\\mathbf{M} \\mathbf{N}$ is the result of swapping the first row and second row of $\\mathbf{N},$ and doubling the third row of $\\mathbf{N}.$ In other words,\n\\[\\mathbf{M} \\beg... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 0 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 2 \\end{pmatrix}"
} | {
"index": 9482,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the $\\emph{positive}$ real number(s) $x$ such that $\\frac{1}{2}\\left( 3x^2-1\\right) = \\left( x^2-50x-10\\right)\\left( x^2+25x+5\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "25 + 2\\sqrt{159}"
} | {
"index": 9483,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the $r$ that satisfies $\\log_{16} (r+16) = \\frac{5}{4}$."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 9484,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the $x$-intercept point of the line $3x+5y=20$. Provide your answer as an ordered pair. Express the $x$ and $y$ coordinates as common fractions as needed."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(\\frac{20}{3},0\\right)"
} | {
"index": 9485,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the $y$-intercept point of the line $3x+5y=20$. Provide your answer as an ordered pair."
}
] | math | {
"style": "rule",
"ground_truth": "(0,4)"
} | {
"index": 9486,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the $y$-intercepts of the following system of equations:\n1. $2x - 3y = 6$\n2. $x + 4y = -8$"
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 9487,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the 150th term of the sequence that consists of all those positive integers which are either powers of 3 or sums of distinct powers of 3."
}
] | math | {
"style": "rule",
"ground_truth": "2280"
} | {
"index": 9488,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the 20th term in the sequence: $\\frac{1}{2}, \\frac{1}{3}, \\frac{2}{3}, \\frac{1}{4}, \\frac{2}{4}, \\frac{3}{4}, \\ldots, \\frac{1}{m+1}, \\frac{2}{m+1}, \\ldots, \\frac{m}{m+1}, \\ldots$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{7}"
} | {
"index": 9489,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the 6-digit number beginning and ending in the digit 2 that is the product of three consecutive even integers."
}
] | math | {
"style": "rule",
"ground_truth": "287232"
} | {
"index": 9490,
"solution": "Because the last digit of the product is 2, none of the three consecutive even integers end in 0. Thus they must end in $2,4,6$ or $4,6,8$, so they must end in $4,6,8$ since $2 \\cdot 4 \\cdot 6$ does not end in 2. Call the middle integer $n$. Then the product is $(n-2) n(n+2)=n^{3}-4 ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the 6-digit repetend in the decimal representation of $\\frac 3{13}.$"
}
] | math | {
"style": "rule",
"ground_truth": "230769"
} | {
"index": 9491,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the 6-digit repetend in the decimal representation of $\\frac{7}{29}$."
}
] | math | {
"style": "rule",
"ground_truth": "241379"
} | {
"index": 9492,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the GCD (Greatest Common Divisor) of all the numbers of the form \\( n^{13} - n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2730"
} | {
"index": 9493,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the absolute value of the difference of single-digit integers $A$ and $B$ such that $$ \\begin{array}{c@{}c@{\\;}c@{}c@{}c@{}c} & & & B& B & A_6\\\\ & & & \\mathbf{4} & \\mathbf{1} & B_6\\\\& & + & A & \\mathbf{1} & \\mathbf{5_6}\\\\ \\cline{2-6} & & A & \\mathbf{1} & \\m... | math | {
"style": "rule",
"ground_truth": "1_6"
} | {
"index": 9494,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the absolute value of the difference of single-digit integers $C$ and $D$ such that in base 8:\n$$ \\begin{array}{c@{}c@{\\;}c@{}c@{}c@{}c}\n & & & D & D & C_8 \\\\\n-& & & \\mathbf{6} & \\mathbf{3} & D_8 \\\\\n\\cline{2-6}\n& & C & \\mathbf{3} & \\mathbf{1} & \\mathbf{5_8... | math | {
"style": "rule",
"ground_truth": "5_8"
} | {
"index": 9495,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the absolute value of the difference of single-digit integers \\( C \\) and \\( D \\) such that in base \\( 5 \\):\n$$ \\begin{array}{c@{}c@{\\;}c@{}c@{}c@{}c}\n& & & D & D & C_5 \\\\\n& & & \\mathbf{3} & \\mathbf{2} & D_5 \\\\\n& & + & C & \\mathbf{2} & \\mathbf{4_5} \\\\... | math | {
"style": "rule",
"ground_truth": "1_5"
} | {
"index": 9496,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the absolute value of the difference of the solutions of $x^2-5x+5=0$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 9497,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the angle $D A C$ given that $A B = B C$ and $A C = C D$, and the lines on which points $A, B, C, D$ lie are parallel with equal distances between adjacent lines. Point $A$ is to the left of $B$, $C$ is to the left of $B$, and $D$ is to the right of $C$."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 9498,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the angle between the vectors $\\begin{pmatrix} 2 \\\\ -1 \\\\ 1 \\end{pmatrix}$ and $\\begin{pmatrix} -1 \\\\ 1 \\\\ 0 \\end{pmatrix},$ in degrees."
}
] | math | {
"style": "rule",
"ground_truth": "150^\\circ"
} | {
"index": 9499,
"solution": ""
} |
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