data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Determine how many integers $n$ between 1 and 15 (inclusive) result in a fraction $\\frac{n}{30}$ that has a repeating decimal."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 7600,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many integers are in the list containing the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them."
}
] | math | {
"style": "rule",
"ground_truth": "871"
} | {
"index": 7601,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many integers are in the list when Eleanor writes down the smallest positive multiple of 15 that is a perfect square, the smallest positive multiple of 15 that is a perfect cube, and all the multiples of 15 between them."
}
] | math | {
"style": "rule",
"ground_truth": "211"
} | {
"index": 7602,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many non-similar regular 500-pointed stars exist, given that a regular $n$-pointed star adheres to the rules set in the original problem description."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 7603,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many ordered pairs $(a, b)$, where $a$ is a positive real number and $b$ is an integer between $1$ and $210$, inclusive, satisfy the equation $(\\log_b a)^{2023} = \\log_b(a^{2023})$."
}
] | math | {
"style": "rule",
"ground_truth": "630"
} | {
"index": 7604,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many ordered pairs of positive integers $(x, y)$ where $x < y$, such that the harmonic mean of $x$ and $y$ is equal to $24^{10}$."
}
] | math | {
"style": "rule",
"ground_truth": "619"
} | {
"index": 7605,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many ordered pairs of positive integers $(x, y)$, where $x < y$, have a harmonic mean of $5^{20}$."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 7606,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many perfect cubes exist between \\(3^6 + 1\\) and \\(3^{12} + 1\\), inclusive."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 7607,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many positive integer multiples of $2002$ can be represented in the form $10^{j} - 10^{i}$, where $i$ and $j$ are integers and $0 \\leq i < j \\leq 150$."
}
] | math | {
"style": "rule",
"ground_truth": "1825"
} | {
"index": 7608,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many prime dates occurred in 2008, a leap year. A \"prime date\" is when both the month and the day are prime numbers."
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"index": 7609,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many prime dates occurred in a leap year, such as 2012, where both the month and the day are prime numbers."
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"index": 7610,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many regions of space are divided by:\na) The six planes of a cube's faces.\nb) The four planes of a tetrahedron's faces."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 7611,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many roots of the equation\n\n$$\n4 \\sin 2x + 3 \\cos 2x - 2 \\sin x - 4 \\cos x + 1 = 0\n$$\n\nare located in the interval \\(\\left[10^{2014!} \\pi, 10^{2014!+2015} \\pi\\right]\\). In the answer, write down the sum of all the digits of the found number."
}
] | math | {
"style": "rule",
"ground_truth": "18135"
} | {
"index": 7612,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many solutions the following equation has:\n\\[\n\\frac{(x-1)(x-2)(x-3)\\dotsm(x-50)}{(x-2^2)(x-4^2)(x-6^2)\\dotsm(x-24^2)} = 0\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 7613,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many triangles can be formed using the vertices of a regular hexadecagon (a 16-sided polygon)."
}
] | math | {
"style": "rule",
"ground_truth": "560"
} | {
"index": 7614,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many two-digit numbers satisfy the following property: when the number is added to the number obtained by reversing its digits, the sum is $132.$"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 7615,
"solution": "\n#### Solution 1 - Detailed Analysis\n\n1. **Represent the number and its reverse:** \n Let the two-digit number be represented as $10a + b$, where $a$ is the tens digit and $b$ is the units digit. The reverse of this number would be $10b + a$.\n\n2. **Formulate the equation:** \n... |
DeepScaleR | [
{
"role": "user",
"content": "Determine how much money the Romanov family will save by using a multi-tariff meter over three years.\n\nThe cost of the meter is 3500 rubles. The installation cost is 1100 rubles. On average, the family's electricity consumption is 300 kWh per month, with 230 kWh used from 23:... | math | {
"style": "rule",
"ground_truth": "3824"
} | {
"index": 7616,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine if there are other composite numbers smaller than 20 that are also abundant besides number 12. If so, list them. If not, confirm 12 remains the smallest abundant number."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 7617,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine if there exists a positive integer \\( m \\) such that the equation\n\\[\n\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}+\\frac{1}{abc}=\\frac{m}{a+b+c}\n\\]\nhas infinitely many solutions in positive integers \\( (a, b, c) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 7618,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine into how many parts a regular tetrahedron is divided by six planes, each passing through one edge and the midpoint of the opposite edge of the tetrahedron. Find the volume of each part if the volume of the tetrahedron is 1."
}
] | math | {
"style": "rule",
"ground_truth": "1/24"
} | {
"index": 7619,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine one of the symmetry axes of the function $y = \\cos 2x - \\sin 2x$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\pi}{8}"
} | {
"index": 7620,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the absolute value of the difference between single-digit integers $C$ and $D$ such that:\n$$ \\begin{array}{c@{}c@{\\;}c@{}c@{}c@{}c}\n & & & D& D & C_6\\\\\n & & & \\mathbf{5} & \\mathbf{2} & D_6\\\\\n & & + & C & \\mathbf{2} & \\mathbf{4_6}\\\\\n \\cline{2-6}\n & &... | math | {
"style": "rule",
"ground_truth": "1_6"
} | {
"index": 7621,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the appropriate value of $h$ so that the following equation in base $h$ is accurate:\n$$\\begin{array}{c@{}c@{}c@{}c@{}c@{}c}\n&&8&6&7&4_h \\\\\n&+&4&3&2&9_h \\\\\n\\cline{2-6}\n&1&3&0&0&3_h.\n\\end{array}$$"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 7622,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the area and the circumference of a circle with the center at the point \\( R(2, -1) \\) and passing through the point \\( S(7, 4) \\). Express your answer in terms of \\( \\pi \\)."
}
] | math | {
"style": "rule",
"ground_truth": "10\\pi \\sqrt{2}"
} | {
"index": 7623,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the area enclosed by the curve of $y = \\arccos(\\cos x)$ and the $x$-axis over the interval $\\frac{\\pi}{4} \\le x \\le \\frac{9\\pi}{4}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi^2}{2}"
} | {
"index": 7624,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the area enclosed by the curves \\( y = \\sin x \\) and \\( y = \\left(\\frac{4}{\\pi}\\right)^{2} \\sin \\left(\\frac{\\pi}{4}\\right) x^{2} \\) (the latter is a quadratic function)."
}
] | math | {
"style": "rule",
"ground_truth": "1 - \\frac{\\sqrt{2}}{2}\\left(1 + \\frac{\\pi}{12}\\right)"
} | {
"index": 7625,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the area enclosed by the parabola $y = x^{2} - 5x + 6$ and the coordinate axes (and adjacent to both axes)."
}
] | math | {
"style": "rule",
"ground_truth": "4.666666666666667"
} | {
"index": 7626,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the area of a triangle with side lengths 7, 7, and 5."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{42.75}}{2}"
} | {
"index": 7627,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the area of the Crescent Gemini."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17\\pi}{4}"
} | {
"index": 7628,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the area of the circle described by the equation \\(3x^2 + 3y^2 - 15x + 9y + 27 = 0\\) in terms of \\(\\pi\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"index": 7629,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the area of the circle described by the graph of the equation\n\\[r = 4 \\cos \\theta - 3 \\sin \\theta.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25\\pi}{4}"
} | {
"index": 7630,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the area of the region bounded by the graph of \\[x^2+y^2 = 6|x-y| + 6|x+y|\\].\nA) 54\nB) 63\nC) 72\nD) 81\nE) 90"
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 7631,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the area of the region of the circle defined by $x^2 + y^2 - 8x + 16 = 0$ that lies below the $x$-axis and to the left of the line $y = x - 4$."
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"index": 7632,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the area of the smallest square that can contain a circle with a radius of 6."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 7633,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the area of the triangle bounded by the axes and the curve $y = (x-5)^2 (x+3)$."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 7634,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the area, in square units, of triangle $PQR$, where the coordinates of the vertices are $P(-3, 4)$, $Q(4, 9)$, and $R(5, -3)$."
}
] | math | {
"style": "rule",
"ground_truth": "44.5"
} | {
"index": 7635,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the average daily high temperature in Addington from September 15th, 2008, through September 21st, 2008, inclusive. The high temperatures for the days are as follows: 51, 64, 60, 59, 48, 55, 57 degrees Fahrenheit. Express your answer as a decimal to the nearest tenth.... | math | {
"style": "rule",
"ground_truth": "56.3"
} | {
"index": 7636,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the base seven product of the numbers $321_7$ and $13_7$."
}
] | math | {
"style": "rule",
"ground_truth": "4503_7"
} | {
"index": 7637,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the binomial coefficient and the coefficient of the 4th term in the expansion of $\\left( \\left. x^{2}- \\frac{1}{2x} \\right. \\right)^{9}$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{21}{2}"
} | {
"index": 7638,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the coefficient of $x^{8}$ in the expansion of \\\\((x^{3}+ \\frac{1}{2 \\sqrt {x}})^{5}\\\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 7639,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the coefficient of \\(x^{29}\\) in the expansion of \\(\\left(1 + x^{5} + x^{7} + x^{9}\\right)^{16}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "65520"
} | {
"index": 7640,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the coefficient of the $x^5$ term in the expansion of $(x+1)(x^2-x-2)^3$."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"index": 7641,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the coefficient of the $x^{3}$ term in the expansion of $(2x+1)(x-1)^{5}$."
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"index": 7642,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the coefficient of the term containing $x^3$ in the expansion of ${(1+2x)}^{5}$. (The result should be represented as a number.)"
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 7643,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the common rational root \\( k \\) of the following polynomial equations which is not integral:\n\\[45x^4 + ax^3 + bx^2 + cx + 8 = 0\\]\n\\[8x^5 + dx^4 + ex^3 + fx^2 + gx + 45 = 0\\]\nThis root \\( k \\) is assumed to be a negative non-integer."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"index": 7644,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the complex number $z$ satisfying the equation $2z-3\\bar{z}=-2-30i$. Note that $\\bar{z}$ denotes the conjugate of $z$."
}
] | math | {
"style": "rule",
"ground_truth": "2-6i"
} | {
"index": 7645,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the complex number $z$ satisfying the equation $2z-3i\\bar{z}=-7+3i$. Note that $\\bar{z}$ denotes the conjugate of $z$."
}
] | math | {
"style": "rule",
"ground_truth": "1+3i"
} | {
"index": 7646,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the constant $d$ such that\n$$\\left(3x^3 - 2x^2 + x - \\frac{5}{4}\\right)(ex^3 + dx^2 + cx + f) = 9x^6 - 5x^5 - x^4 + 20x^3 - \\frac{25}{4}x^2 + \\frac{15}{4}x - \\frac{5}{2}$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 7647,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the constant term in the expansion of $(x-2+ \\frac {1}{x})^{4}$."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 7648,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the constants $\\alpha$ and $\\beta$ such that $\\frac{x-\\alpha}{x+\\beta} = \\frac{x^2 - 64x + 975}{x^2 + 99x - 2200}$. What is $\\alpha+\\beta$?"
}
] | math | {
"style": "rule",
"ground_truth": "138"
} | {
"index": 7649,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the degree of the polynomial resulting from $(5x^6 - 4x^5 + x^2 - 18)(2x^{12} + 6x^9 - 11x^6 + 10) - (x^3 + 4)^6$ when this expression is expanded and simplified."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 7650,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the difference between the sum of the first one hundred positive even integers and the sum of the first one hundred positive multiples of 3."
}
] | math | {
"style": "rule",
"ground_truth": "-5050"
} | {
"index": 7651,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the digits $a, b, c, d, e$ such that the two five-digit numbers formed with them satisfy the equation $\\overline{a b c d e} \\cdot 9 = \\overline{e d c b a}$."
}
] | math | {
"style": "rule",
"ground_truth": "10989"
} | {
"index": 7652,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the distance in feet between the 5th red light and the 23rd red light, where the lights are hung on a string 8 inches apart in the pattern of 3 red lights followed by 4 green lights. Recall that 1 foot is equal to 12 inches."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 7653,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the distance that the origin $O(0,0)$ moves under the dilation transformation that sends the circle of radius $4$ centered at $B(3,1)$ to the circle of radius $6$ centered at $B'(7,9)$."
}
] | math | {
"style": "rule",
"ground_truth": "0.5\\sqrt{10}"
} | {
"index": 7654,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the exact value of\n\\[\n\\sqrt{\\left( 2 - \\sin^2 \\frac{\\pi}{9} \\right) \\left( 2 - \\sin^2 \\frac{2 \\pi}{9} \\right) \\left( 2 - \\sin^2 \\frac{4 \\pi}{9} \\right)}.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{619}}{16}"
} | {
"index": 7655,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the exact value of\n\\[\\sqrt{\\left( 2 - \\sin^2 \\frac{\\pi}{7} \\right) \\left( 2 - \\sin^2 \\frac{2 \\pi}{7} \\right) \\left( 2 - \\sin^2 \\frac{3 \\pi}{7} \\right)}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{8}"
} | {
"index": 7656,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the exact value of the series\n\\[\n\\frac{1}{3 + 1} + \\frac{2}{3^2 + 1} + \\frac{4}{3^4 + 1} + \\frac{8}{3^8 + 1} + \\frac{16}{3^{16} + 1} + \\dotsb.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 7657,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the exact value of the series\n\\[\\frac{1}{3 + 1} + \\frac{2}{3^2 + 1} + \\frac{4}{3^4 + 1} + \\frac{8}{3^8 + 1} + \\frac{16}{3^{16} + 1} + \\dotsb.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 7658,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the exact value of the series\n\\[\\frac{1}{5 + 1} + \\frac{2}{5^2 + 1} + \\frac{4}{5^4 + 1} + \\frac{8}{5^8 + 1} + \\frac{16}{5^{16} + 1} + \\dotsb.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 7659,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the fifth-largest divisor of 2,500,000,000."
}
] | math | {
"style": "rule",
"ground_truth": "156,250,000"
} | {
"index": 7660,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the following number:\n\n\\[\n\\frac{12346 \\cdot 24689 \\cdot 37033 + 12347 \\cdot 37034}{12345^{2}}\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "74072"
} | {
"index": 7661,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the form of $n$ such that $2^n + 2$ is divisible by $n$ where $n$ is less than 100."
}
] | math | {
"style": "rule",
"ground_truth": "n=6, 66, 946"
} | {
"index": 7662,
"solution": "Note that $2^n+2=2(2^{n-1}+1)$ so that $n$ is of the form $2r$ with $r$ odd. We will consider two cases. i) $n=2p$ with $p$ prime. $2p \\mid 2^{2p}+2$, implies that $p \\mid 2^{2n-1}+1$ and hence $p \\mid 2^{4p-2}-1$. On the other hand Fermat's little theorem guarantees that $p \\mid 2... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the fourth-largest divisor of $1,234,560,000$."
}
] | math | {
"style": "rule",
"ground_truth": "154,320,000"
} | {
"index": 7663,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the greatest common divisor (gcd) of all the numbers of the form\n\n$$\n(a-b)(b-c)(c-d)(d-a)(a-c)(b-d)\n$$\n\nwhere $a, b, c, d$ are integers."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 7664,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the greatest common divisor of all nine-digit integers formed by repeating a three-digit integer three times. For example, 256,256,256 or 691,691,691 are integers of this form."
}
] | math | {
"style": "rule",
"ground_truth": "1001001"
} | {
"index": 7665,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the greatest integer less than or equal to \\[\\frac{5^{50} + 3^{50}}{5^{45} + 3^{45}}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "3124"
} | {
"index": 7666,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the greatest number m such that the\nsystem \u001a $x^2$ + $y^2$ = 1; | $x^3$ - $y^3$ |+|x-y|= $m^3$ has a solution."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{2}"
} | {
"index": 7667,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the greatest positive integer $ n$ such that in three-dimensional space, there exist n points $ P_{1},P_{2},\\cdots,P_{n},$ among $ n$ points no three points are collinear, and for arbitary $ 1\\leq i < j < k\\leq n$, $ P_{i}P_{j}P_{k}$ isn't obtuse triangle."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 7668,
"solution": "\nTo determine the greatest positive integer \\( n \\) such that in three-dimensional space, there exist \\( n \\) points \\( P_{1}, P_{2}, \\cdots, P_{n} \\) where no three points are collinear and for any \\( 1 \\leq i < j < k \\leq n \\), the triangle \\( P_{i}P_{j}P_{k} \\) is not ... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the greatest possible value of \\(\\sum_{i=1}^{10} \\cos(3x_i)\\) for real numbers $x_1,x_2,\\dots,x_{10}$ satisfying \\(\\sum_{i=1}^{10} \\cos(x_i) = 0\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{480}{49}"
} | {
"index": 7669,
"solution": "The maximum value is $480/49$. Since $\\cos(3x_i) = 4 \\cos(x_i)^3 - 3 \\cos(x_i)$, it is equivalent to maximize $4 \\sum_{i=1}^{10} y_i^3$ for $y_1,\\dots,y_{10} \\in [-1,1]$ with $\\sum_{i=1}^{10} y_i = 0$; note that this domain is compact, so the maximum value is guaranteed to exist... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the greatest real number $ C $, such that for every positive integer $ n\\ge 2 $, there exists $ x_1, x_2,..., x_n \\in [-1,1]$, so that\n$$\\prod_{1\\le i<j\\le n}(x_i-x_j) \\ge C^{\\frac{n(n-1)}{2}}$$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 7670,
"solution": "\nTo determine the greatest real number \\( C \\) such that for every positive integer \\( n \\geq 2 \\), there exist \\( x_1, x_2, \\ldots, x_n \\in [-1, 1] \\) satisfying\n\\[\n\\prod_{1 \\le i < j \\le n} (x_i - x_j) \\geq C^{\\frac{n(n-1)}{2}},\n\\]\nwe consider the example where \... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the heaviest weight that can be obtained using a combination of 2 lb and 4 lb, and 12 lb weights with a maximum of two weights used at a time."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 7671,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the height of a tower from a 20-meter distant building, given that the angle of elevation to the top of the tower is 30° and the angle of depression to the base of the tower is 45°."
}
] | math | {
"style": "rule",
"ground_truth": "20 \\left(1 + \\frac {\\sqrt {3}}{3}\\right)"
} | {
"index": 7672,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the increment of the argument and the function \\( y = x^2 \\) if the argument \\( x \\) changes from 2 to 2.5."
}
] | math | {
"style": "rule",
"ground_truth": "2.25"
} | {
"index": 7673,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the integer $x$ that satisfies the following set of congruences:\n\\begin{align*}\n4+x &\\equiv 3^2 \\pmod{2^3} \\\\\n6+x &\\equiv 4^2 \\pmod{3^3} \\\\\n8+x &\\equiv 6^2 \\pmod{5^3}\n\\end{align*}\nFind the remainder when $x$ is divided by $120$."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 7674,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the integer \\( m \\), \\( -180 \\leq m \\leq 180 \\), such that \\(\\sin m^\\circ = \\sin 945^\\circ.\\)"
}
] | math | {
"style": "rule",
"ground_truth": "-135"
} | {
"index": 7675,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the integer \\( x \\) for which the following equation holds true:\n\\[ 1 \\cdot 2023 + 2 \\cdot 2022 + 3 \\cdot 2021 + \\dots + 2022 \\cdot 2 + 2023 \\cdot 1 = 2023 \\cdot 1012 \\cdot x. \\]"
}
] | math | {
"style": "rule",
"ground_truth": "675"
} | {
"index": 7676,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the integer root of the polynomial \\[2x^3 + ax^2 + bx + c = 0,\\] where $a, b$, and $c$ are rational numbers. The equation has $4-2\\sqrt{3}$ as a root and another root whose sum with $4-2\\sqrt{3}$ is 8."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"index": 7677,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest and smallest fractions $F = \\frac{y-x}{x+4y}$ \nif the real numbers $x$ and $y$ satisfy the equation $x^2y^2 + xy + 1 = 3y^2$."
}
] | math | {
"style": "rule",
"ground_truth": "$0 \\leq \\frac{y-x}{x+4y} \\leq 4$"
} | {
"index": 7678,
"solution": "\nGiven the equation \\( x^2y^2 + xy + 1 = 3y^2 \\), we need to determine the largest and smallest values of the fraction \\( F = \\frac{y-x}{x+4y} \\).\n\n### Step 1: Analyze the Given Equation\nTo simplify the problem, we first explore the given equation:\n\\[\nx^2y^2 + xy + 1 = 3y^2... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest constant $K\\geq 0$ such that $$ \\frac{a^a(b^2+c^2)}{(a^a-1)^2}+\\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\\frac{c^c(a^2+b^2)}{(c^c-1)^2}\\geq K\\left (\\frac{a+b+c}{abc-1}\\right)^2 $$ holds for all positive real numbers $a,b,c$ such that $ab+bc+ca=abc$ .... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 7679,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest integer $n$ such that $7^{2048}-1$ is divisible by $2^{n}$."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 7680,
"solution": "We have $$7^{2048}-1=(7-1)(7+1)\\left(7^{2}+1\\right)\\left(7^{4}+1\\right) \\cdots\\left(7^{1024}+1\\right)$$ In the expansion, the eleven terms other than $7+1$ are divisible by 2 exactly once, as can be checked easily with modulo 4."
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest integer $x$ for which $4^{27} + 4^{1010} + 4^{x}$ is a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "1992"
} | {
"index": 7681,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest multiple of 36 that consists of all even and distinct digits."
}
] | math | {
"style": "rule",
"ground_truth": "8640"
} | {
"index": 7682,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest natural number $r$ with the property that among any five subsets with $500$ elements of the set $\\{1,2,\\ldots,1000\\}$ there exist two of them which share at least $r$ elements."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 7683,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest natural number \\( n \\) so that\n\n\\[ 4^{995} + 4^{1500} + 4^{n} \\]\n\nis a square number."
}
] | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"index": 7684,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest natural number \\( n \\) such that\n\n\\[ 4^{995} + 4^{1500} + 4^{n} \\]\n\nis a square number."
}
] | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"index": 7685,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest odd positive integer $n$ such that every odd integer $k$ with $1<k<n$ and $\\gcd(k, n)=1$ is a prime."
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"index": 7686,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest of all integers $n$ with the property that $n$ is divisible by all positive integers that are less than $\\sqrt[3]{n}$."
}
] | math | {
"style": "rule",
"ground_truth": "420"
} | {
"index": 7687,
"solution": "Observation from that $\\operatorname{lcm}(2,3,4,5,6,7)=420$ is divisible by every integer less than or equal to $7=[\\sqrt[3]{420}]$ and that $\\operatorname{lcm}(2,3,4,5,6,7,8)=840$ is not divisible by $9=[\\sqrt[3]{840}]$. One may guess 420 is the required integer. Let $N$ be the re... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest positive integer $n$ for which there exists a set $S$ with exactly $n$ numbers such that \n- each member in $S$ is a positive integer not exceeding $2002$ ,\n- if $a,b\\in S$ (not necessarily different), then $ab\\not\\in S$ ."
}
] | math | {
"style": "rule",
"ground_truth": "1958"
} | {
"index": 7688,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest positive integer $n$ such that there exist positive integers $x, y, z$ so that \\[\nn^2 = x^2+y^2+z^2+2xy+2yz+2zx+3x+3y+3z-6\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 7689,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest positive integer \\( n \\) with \\( n < 500 \\) for which \\( 6048 \\cdot 28^n \\) is a perfect cube (that is, it is equal to \\( m^3 \\) for some positive integer \\( m \\) )."
}
] | math | {
"style": "rule",
"ground_truth": "497"
} | {
"index": 7690,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest prime factor of the sum \\(\\sum_{k=1}^{11} k^{5}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "263"
} | {
"index": 7691,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest prime number less than 5000 of the form \\( a^n - 1 \\), where \\( a \\) and \\( n \\) are positive integers, and \\( n \\) is greater than 1."
}
] | math | {
"style": "rule",
"ground_truth": "127"
} | {
"index": 7692,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest real number $c$ such that for any 2017 real numbers $x_{1}, x_{2}, \\ldots, x_{2017}$, the inequality $$\\sum_{i=1}^{2016} x_{i}\\left(x_{i}+x_{i+1}\\right) \\geq c \\cdot x_{2017}^{2}$$ holds."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1008}{2017}"
} | {
"index": 7693,
"solution": "Let $n=2016$. Define a sequence of real numbers \\left\\{p_{k}\\right\\} by $p_{1}=0$, and for all $k \\geq 1$, $$p_{k+1}=\\frac{1}{4\\left(1-p_{k}\\right)}$$ Note that, for every $i \\geq 1$, $$\\left(1-p_{i}\\right) \\cdot x_{i}^{2}+x_{i} x_{i+1}+p_{i+1} x_{i+1}^{2}=\\left(\\frac{x_{... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest square number that is not divisible by 100 and, when its last two digits are removed, is also a square number."
}
] | math | {
"style": "rule",
"ground_truth": "1681"
} | {
"index": 7694,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest value of $S$ such that any finite collection of small squares with a total area $S$ can always be placed inside a unit square $T$ in such a way that no two of the small squares share an interior point."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 7695,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest value of $m$ such that $5x^2 + mx + 45$ can be factored as the product of two linear factors with integer coefficients."
}
] | math | {
"style": "rule",
"ground_truth": "226"
} | {
"index": 7696,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest value of \\(x\\) that satisfies the equation \\(\\sqrt{3x} = 5x^2\\). Express your answer in simplest fractional form."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(\\frac{3}{25}\\right)^{\\frac{1}{3}}"
} | {
"index": 7697,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the largest value the expression $$ \\sum_{1\\le i<j\\le 4} \\left( x_i+x_j \\right)\\sqrt{x_ix_j} $$ may achieve, as $ x_1,x_2,x_3,x_4 $ run through the non-negative real numbers, and add up to $ 1. $ Find also the specific values of this numbers that make t... | math | {
"style": "rule",
"ground_truth": "3/4"
} | {
"index": 7698,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the least positive period $q$ of the functions $g$ such that $g(x+2) + g(x-2) = g(x)$ for all real $x$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 7699,
"solution": ""
} |
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