data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The sides opposite to the internal angles $A$, $B$, $C$ of $\\triangle ABC$ are $a$, $b$, $c$ respectively. Given $\\frac{a-b+c}{c}= \\frac{b}{a+b-c}$ and $a=2$, the maximum area of $\\triangle ABC$ is __________."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 34300,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sides opposite to the internal angles $A$, $B$, and $C$ of $\\triangle ABC$ are $a$, $b$, and $c$ respectively. It is given that $b\\sin C + c\\sin B = 4a\\sin B\\sin C$ and $b^2 + c^2 - a^2 = 8$. The area of $\\triangle ABC$ is __________."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"index": 34301,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The simple (i.e., non-intersecting) quadrilateral \\(ABCD\\) has sides \\(AB\\), \\(BC\\), and \\(CD\\) with lengths 4, 5, and 20, respectively. If the angles \\(B\\) and \\(C\\) are obtuse, and \\(\\sin C = -\\cos B = \\frac{3}{5}\\), then what is the length of the side \\(AD\... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 34302,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The simple quadrilateral \\(ABCD\\) has sides \\(AB\\), \\(BC\\), and \\(CD\\) with lengths 4, 5, and 20, respectively. If the angles \\(B\\) and \\(C\\) are obtuse, and \\(\\sin C = -\\cos B = \\frac{3}{5}\\), calculate the length of the side \\(AD\\)."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 34303,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The simplest form of $1 - \\frac{1}{1 + \\frac{a}{1 - a}}$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "a"
} | {
"index": 34304,
"solution": "We start by simplifying the expression inside the fraction:\n\\[ 1 - \\frac{1}{1 + \\frac{a}{1 - a}}. \\]\n\n1. **Simplify the innermost fraction**:\n \\[ \\frac{a}{1 - a}. \\]\n\n2. **Add this fraction to 1**:\n \\[ 1 + \\frac{a}{1 - a} = \\frac{1 - a + a}{1 - a} = \\frac{1}{1 - ... |
DeepScaleR | [
{
"role": "user",
"content": "The six edges of a tetrahedron $ABCD$ measure $7, 13, 18, 27, 36$ and $41$ units. If the length of edge $AB$ is $41$, then the length of edge $CD$ is\n$\\textbf{(A)}\\ 7\\qquad \\textbf{(B)}\\ 13\\qquad \\textbf{(C)}\\ 18\\qquad \\textbf{(D)}\\ 27\\qquad \\textbf{(E)}\\ 36$"
... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 34305,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The six faces of a three-inch wooden cube are each painted red. The cube is then cut into one-inch cubes along the lines shown in the diagram. How many of the one-inch cubes have red paint on at least two faces? [asy]\n\npair A,B,C,D,E,F,G;\n\npair a,c,d,f,g,i,j,l,m,o,p,r,s,u,v... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 34306,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The six-digit number $20210A$ is prime for only one digit $A.$ What is $A?$"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 34307,
"solution": "1. **Check divisibility by 5**: The number $\\underline{2}\\,\\underline{0}\\,\\underline{2}\\,\\underline{1}\\,\\underline{0}\\,\\underline{A}$ ends in digit $A$. If $A = 5$, the number ends in $5$ and is divisible by $5$. Therefore, option $\\textbf{(C)}\\ 5$ can be eliminated.\n\n2... |
DeepScaleR | [
{
"role": "user",
"content": "The six-digit number $M=\\overline{abc321}$, where $a, b, c$ are three different numbers, and all are greater than 3. If $M$ is a multiple of 7, what is the smallest value of $M$?"
}
] | math | {
"style": "rule",
"ground_truth": "468321"
} | {
"index": 34308,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The six-digit number \\( 2PQRST \\) is multiplied by 3, and the result is the six-digit number \\( PQRST2 \\). What is the sum of the digits of the original number?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 34309,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sixteenth and seventeenth terms of an arithmetic sequence are 8 and 10, respectively. What is the second term?"
}
] | math | {
"style": "rule",
"ground_truth": "-20"
} | {
"index": 34310,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sixth graders were discussing how old their principal is. Anya said, \"He is older than 38 years.\" Borya said, \"He is younger than 35 years.\" Vova: \"He is younger than 40 years.\" Galya: \"He is older than 40 years.\" Dima: \"Borya and Vova are right.\" Sasha: \"You are... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 34311,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sizes of circular pizzas are determined by their diameter. If Lana's initial pizza was 14 inches in diameter and she decides to order a larger pizza with a diameter of 18 inches instead, what is the percent increase in the area of her pizza?"
}
] | math | {
"style": "rule",
"ground_truth": "65.31\\%"
} | {
"index": 34312,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The skeletal structure of circumcircumcircumcoronene, a hydrocarbon with the chemical formula $\\mathrm{C}_{150} \\mathrm{H}_{30}$, is shown below. Each line segment between two atoms is at least a single bond. However, since each carbon (C) requires exactly four bonds connecte... | math | {
"style": "rule",
"ground_truth": "267227532"
} | {
"index": 34313,
"solution": "The problem is equivalent to the one in OEIS A008793, a.k.a. \"number of ways to tile hexagon of edge n with diamonds of side 1.\" Notice that there is a bjiection between such a tiling and the number of ways to stack some unit cubes alongside a corner of an $n \\times n \\times n$ bo... |
DeepScaleR | [
{
"role": "user",
"content": "The skeletal structure of coronene, a hydrocarbon with the chemical formula $\\mathrm{C}_{24} \\mathrm{H}_{12}$, is shown below. Each line segment between two atoms is at least a single bond. However, since each carbon (C) requires exactly four bonds connected to it and each hy... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 34314,
"solution": "Note that each carbon needs exactly one double bond. Label the six carbons in the center $1,2,3,4,5,6$ clockwise. We consider how these six carbons are double-bonded. If a carbon in the center is not double-bonded to another carbon in the center, it must double-bond to the correspondi... |
DeepScaleR | [
{
"role": "user",
"content": "The slant height of a cone forms an angle $\\alpha$ with the plane of its base, where $\\cos \\alpha = \\frac{1}{4}$. A sphere is inscribed in the cone, and a plane is drawn through the circle of tangency of the sphere and the lateral surface of the cone. The volume of the part... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 34315,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The slant height of a cone is 13 cm, and the height from the vertex to the center of the base is 12 cm. What is the number of cubic centimeters in the volume of the cone? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "100\\pi"
} | {
"index": 34316,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The slope angle of the line $x - y + 3 = 0$ is $\\tan^{-1}(1)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 34317,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The slope angle of the line passing through points M(-3, 2) and N(-2, 3) is equal to what angle, measured in radians."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 34318,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The slope angle of the tangent line to the curve $f\\left(x\\right)=- \\frac{ \\sqrt{3}}{3}{x}^{3}+2$ at $x=1$ is $\\tan^{-1}\\left( \\frac{f'\\left(1\\right)}{\\mid f'\\left(1\\right) \\mid} \\right)$, where $f'\\left(x\\right)$ is the derivative of $f\\left(x\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"index": 34319,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The slope angle of the tangent line to the curve $y= \\frac{1}{2}x^{2}$ at the point $(1, \\frac{1}{2})$ is"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 34320,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The slope angle of the tangent line to the curve $y= \\sqrt {x}$ at $x= \\frac {1}{4}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {\\pi}{4}"
} | {
"index": 34321,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The slope angle of the tangent line to the curve $y=\\frac{1}{2}x^2+2$ at the point $\\left(-1, \\frac{5}{2}\\right)$ is what?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{4}"
} | {
"index": 34322,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The slope angle of the tangent line to the curve $y=x\\cos x$ at $x=0$ is what angle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 34323,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The slope angle of the tangent line to the curve y=\\frac{1}{3}x^{3} at x=1 is what value?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 34324,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The slope angle of the tangent to the curve $y=x^{3}-4x$ at the point $(1,-3)$ is calculated in radians."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}\\pi"
} | {
"index": 34325,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The slope of a line is $-2$ and its $x$-intercept is $(5,0).$ What is the $y$-intercept point of the line? Express your answer as an ordered pair."
}
] | math | {
"style": "rule",
"ground_truth": "(0,10)"
} | {
"index": 34326,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The slope of the line $\\frac{x}{3} + \\frac{y}{2} = 1$ is"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{3}"
} | {
"index": 34327,
"solution": "1. Start with the given equation:\n \\[\n \\frac{x}{3} + \\frac{y}{2} = 1\n \\]\n\n2. To isolate $y$, subtract $\\frac{x}{3}$ from both sides:\n \\[\n \\frac{y}{2} = 1 - \\frac{x}{3}\n \\]\n\n3. Multiply both sides by $2$ to solve for $y$:\n \\[\n y = 2 \\left(1 - \\fr... |
DeepScaleR | [
{
"role": "user",
"content": "The slope of the tangent line to the curve $y=\\frac{1}{3}{x^3}-\\frac{2}{x}$ at $x=1$ is $\\alpha$. Find $\\frac{{sin\\alpha cos2\\alpha}}{{sin\\alpha+cos\\alpha}}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{5}"
} | {
"index": 34328,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The slopes of lines $l_1$ and $l_2$ are the two roots of the equation $6x^2+x-1=0$, respectively. The angle between lines $l_1$ and $l_2$ is __________."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 34329,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The smaller root of the equation $\\left(x- \\frac{3}{4} \\right) \\left(x - \\frac{3}{4} \\right) + \\left(x - \\frac{3}{4} \\right) \\left(x - \\frac{1}{2} \\right) = 0$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"index": 34330,
"solution": "1. **Identify the form of the equation**: The given equation is:\n \\[\n \\left(x-\\frac{3}{4}\\right)\\left(x-\\frac{3}{4}\\right)+\\left(x-\\frac{3}{4}\\right)\\left(x-\\frac{1}{2}\\right) = 0\n \\]\n This can be rewritten using $a = x - \\frac{3}{4}$ and $b = x - \\frac{1}{... |
DeepScaleR | [
{
"role": "user",
"content": "The smaller square has an area of 16 and the grey triangle has an area of 1. What is the area of the larger square? \nA) 17 \nB) 18 \nC) 19 \nD) 20 \nE) 21"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 34331,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The smaller square in the figure below has a perimeter of $4$ cm, and the larger square has an area of $16$ $\\text{cm}^2$. What is the distance from point $A$ to point $B$? Express your answer as a decimal to the nearest tenth.\n\n[asy]\ndraw((0,0)--(12,0));\ndraw((2,0)--(2,1... | math | {
"style": "rule",
"ground_truth": "5.8"
} | {
"index": 34332,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The smallest 9-digit integer that can be divided by 11 is ____."
}
] | math | {
"style": "rule",
"ground_truth": "100000010"
} | {
"index": 34333,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The smallest of nine consecutive integers is 2012. These nine integers are placed in the circles to the right. The sum of the three integers along each of the four lines is the same. If this sum is as small as possible, what is the value of $u$?"
}
] | math | {
"style": "rule",
"ground_truth": "2015"
} | {
"index": 34334,
"solution": "If we have a configuration of the numbers that has the required property, then we can add or subtract the same number from each of the numbers in the circles and maintain the property. (This is because there are the same number of circles in each line.) Therefore, we can subtract 2012... |
DeepScaleR | [
{
"role": "user",
"content": "The smallest positive integer $x$ for which $1260x=N^3$, where $N$ is an integer, is:"
}
] | math | {
"style": "rule",
"ground_truth": "7350"
} | {
"index": 34335,
"solution": "1. **Factorize 1260**: We start by factorizing 1260 into its prime factors:\n \\[\n 1260 = 2^2 \\cdot 3^2 \\cdot 5 \\cdot 7\n \\]\n\n2. **Condition for $N^3$**: For $N^3$ to be a perfect cube, each prime factor in its factorization must have an exponent that is a multiple of 3. ... |
DeepScaleR | [
{
"role": "user",
"content": "The smallest positive integer \\( n \\) that satisfies \\( \\sqrt{n} - \\sqrt{n-1} < 0.01 \\) is:\n(29th Annual American High School Mathematics Examination, 1978)"
}
] | math | {
"style": "rule",
"ground_truth": "2501"
} | {
"index": 34336,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The smallest possible value of $m$ for which Casper can buy exactly $10$ pieces of strawberry candy, $18$ pieces of lemon candy, and $20$ pieces of cherry candy, given that each piece of orange candy costs $15$ cents."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34337,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The smallest product one could obtain by multiplying two numbers in the set $\\{ -7,-5,-1,1,3 \\}$ is"
}
] | math | {
"style": "rule",
"ground_truth": "-21"
} | {
"index": 34338,
"solution": "To find the smallest product obtainable by multiplying two numbers from the set $\\{-7, -5, -1, 1, 3\\}$, we need to consider the following cases:\n\n1. **Multiplying two negative numbers**: This will yield a positive product. The possible products are:\n - $(-7) \\times (-5) = 35$\... |
DeepScaleR | [
{
"role": "user",
"content": "The smallest sum one could get by adding three different numbers from the set $\\{ 7,25,-1,12,-3 \\}$ is"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 34339,
"solution": "To find the smallest sum possible by adding three different numbers from the set $\\{7, 25, -1, 12, -3\\}$, we need to consider the smallest numbers in the set, as adding smaller numbers will result in a smaller sum.\n\n1. **Identify the smallest numbers in the set**: The three smalle... |
DeepScaleR | [
{
"role": "user",
"content": "The smallest three positive proper divisors of an integer n are $d_1 < d_2 < d_3$ and they satisfy $d_1 + d_2 + d_3 = 57$ . Find the sum of the possible values of $d_2$ ."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 34340,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solid $S$ consists of the set of all points $(x,y,z)$ such that $|x| + |y| \\le 1,$ $|x| + |z| \\le 1,$ and $|y| + |z| \\le 1.$ Find the volume of $S.$"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 34341,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solid \\( T \\) consists of all points \\((x,y,z)\\) such that \\( |x| + |y| \\le 2 \\), \\( |x| + |z| \\le 1 \\), and \\( |z| + |y| \\le 1 \\). Find the volume of \\( T \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 34342,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solid \\( T \\) consists of all points \\((x,y,z)\\) such that \\(|x| + |y| \\leq 2\\), \\(|x| + |z| \\leq 2\\), and \\(|y| + |z| \\leq 2\\). Find the volume of \\( T \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1664}{81}"
} | {
"index": 34343,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All other edges have length $s$. Given that $s=6\\sqrt{2}$, what is the volume of the solid?\n[asy] size(180); import three; pathpen = black+linewidth(0.65); pointp... | math | {
"style": "rule",
"ground_truth": "288"
} | {
"index": 34344,
"solution": "Draw an altitude from a vertex of the square base to the top edge. By using $30,60, 90$ triangle ratios, we obtain that the altitude has a length of $3 \\sqrt{6}$, and that little portion that hangs out has a length of $3\\sqrt2$. This is a triangular pyramid with a base of $3\\sqrt6,... |
DeepScaleR | [
{
"role": "user",
"content": "The solution of $8x+1\\equiv 5 \\pmod{12}$ is $x\\equiv a\\pmod{m}$ for some positive integers $m\\geq 2$ and $a<m$. Find $a+m$."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 34345,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solution of $\\sqrt{5x-1}+\\sqrt{x-1}=2$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "$x=1$"
} | {
"index": 34346,
"solution": "1. **Start by squaring both sides of the equation**:\n \\[\n \\left(\\sqrt{5x-1} + \\sqrt{x-1}\\right)^2 = 2^2\n \\]\n Expanding the left side using the identity $(a+b)^2 = a^2 + 2ab + b^2$, we get:\n \\[\n \\sqrt{5x-1}^2 + 2\\sqrt{5x-1}\\sqrt{x-1} + \\sqrt{x-1}^2 = 4\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The solution of the equation $7^{x+7} = 8^x$ can be expressed in the form $x = \\log_b 7^7$. What is $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{7}"
} | {
"index": 34347,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solution of the equation $\\dfrac{1+2^{x}}{1+x^{-x}}= \\dfrac{1}{4}$ is $x=$ ."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 34348,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solution set of $6x^2+5x<4$ is the set of all values of $x$ such that"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{3}<x<\\frac{1}{2}"
} | {
"index": 34349,
"solution": "1. **Rewrite the inequality**: Start by rewriting the given inequality:\n \\[\n 6x^2 + 5x < 4\n \\]\n Subtract 4 from both sides to set the inequality to zero:\n \\[\n 6x^2 + 5x - 4 < 0\n \\]\n\n2. **Factor the quadratic expression**: Next, factor the quadratic expressio... |
DeepScaleR | [
{
"role": "user",
"content": "The solution to the equation \\(\\arcsin x + \\arcsin 2x = \\arccos x + \\arccos 2x\\) is"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{5}"
} | {
"index": 34350,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solution to the inequality\n\n$$\n(x-1)^{[\\sqrt{1}]}(x-2)^{[\\sqrt{2}]} \\ldots(x-k)^{[\\sqrt{k}]} \\ldots(x-150)^{[\\sqrt{150}]}<0\n$$\n\nis a union of several non-overlapping intervals. Find the sum of their lengths. If necessary, round the answer to the nearest 0.01.\n\... | math | {
"style": "rule",
"ground_truth": "78.00"
} | {
"index": 34351,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solution to the inequality\n\\[y = -x^2 + ax + b \\le 0\\]is $(-\\infty,-3] \\cup [5,\\infty).$ Find the vertex of the parabola $y = -x^2 + ax + b.$"
}
] | math | {
"style": "rule",
"ground_truth": "(1,16)"
} | {
"index": 34352,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solutions of $x(3x-7)=-3$ may be expressed in the form $\\frac{m+\\sqrt{n}}{p}$ and $\\frac{m-\\sqrt{n}}{p}$, where $m$, $n$, and $p$ have a greatest common divisor of 1. Find $m+n+p$."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 34353,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solutions to $4x^2 + 3 = 3x - 9$ can be written in the form $x = a \\pm b i,$ where $a$ and $b$ are real numbers. What is $a + b^2$? Express your answer as a fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{207}{64}"
} | {
"index": 34354,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solutions to $z^4 = -16i$ can be expressed in the form\n\\begin{align*}\nz_1 &= r_1 (\\cos \\theta_1 + i \\sin \\theta_1), \\\\\nz_2 &= r_2 (\\cos \\theta_2 + i \\sin \\theta_2), \\\\\nz_3 &= r_3 (\\cos \\theta_3 + i \\sin \\theta_3), \\\\\nz_4 &= r_4 (\\cos \\theta_4 + i \... | math | {
"style": "rule",
"ground_truth": "810^\\circ"
} | {
"index": 34355,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solutions to the equation $(z+6)^8=81$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled $A,B,$ and $C$. What is the least possible area of triangle $ABC$?\n\nEnter your answer in the form $\\frac{a \\sqrt{b} - c}{d},$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{2} - 3}{2}"
} | {
"index": 34356,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solutions to the equation $(z-4)^6 = 64$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labelled $D, E,$ and $F$. What is the least possible area of triangle $DEF$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 34357,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solutions to the equation $x(5x+2) = 6(5x+2)$ are ___."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{5}"
} | {
"index": 34358,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solutions to the equation $x^2 - 3|x| - 2 = 0$ are."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-3 - \\sqrt{17}}{2}"
} | {
"index": 34359,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solutions to the equation $x^2 = x$ are $x=0$ and $x=1$."
}
] | math | {
"style": "rule",
"ground_truth": "0 or 1"
} | {
"index": 34360,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solutions to the equation \\( x^3 - 4 \\lfloor x \\rfloor = 5 \\), where \\( x \\) is a real number, are denoted by \\( x_1, x_2, x_3, \\ldots, x_k \\) for some positive integer \\( k \\). Find \\( \\sum_{i=1}^{k} x_{i}^{3} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34361,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solutions to the equations $z^2 = 9 + 9\\sqrt{7}i$ and $z^2 = 5 + 5\\sqrt{2}i$, where $i = \\sqrt{-1}$, form the vertices of a parallelogram in the complex plane. Determine the area of this parallelogram, which can be written in the form $p\\sqrt{q} - r\\sqrt{s}$, where $p,... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}\\sqrt{35} - \\frac{3}{4}\\sqrt{15}"
} | {
"index": 34362,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The solutions to the equations $z^2=4+4\\sqrt{15}i$ and $z^2=2+2\\sqrt 3i,$ where $i=\\sqrt{-1},$ form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form $p\\sqrt q-r\\sqrt s,$ where $p,$ $q,$ $r,$ and $s$ are positiv... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 34363,
"solution": "We solve each equation separately and find the vertices of the parallelogram formed by the solutions in the complex plane.\n\n#### Solving $z^2 = 4 + 4\\sqrt{15}i$:\nLet $z = r(\\cos\\theta + i\\sin\\theta) = r\\operatorname{cis}\\theta$, where $r$ is the magnitude of $z$ and $\\theta... |
DeepScaleR | [
{
"role": "user",
"content": "The solutions to the system of equations\n$\\log_{225}x+\\log_{64}y=4$\n$\\log_{x}225-\\log_{y}64=1$\n are $(x_1,y_1)$ and $(x_2,y_2)$. Find $\\log_{30}\\left(x_1y_1x_2y_2\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34364,
"solution": "Let $A=\\log_{225}x$ and let $B=\\log_{64}y$.\n From the first equation: $A+B=4 \\Rightarrow B = 4-A$.\n Plugging this into the second equation yields $\\frac{1}{A}-\\frac{1}{B}=\\frac{1}{A}-\\frac{1}{4-A}=1 \\Rightarrow A = 3\\pm\\sqrt{5}$ and thus, $B=1\\pm\\sqrt{5}$.\n So, $\\log_{... |
DeepScaleR | [
{
"role": "user",
"content": "The specific heat capacity of a body with mass $m=2$ kg depends on the temperature as follows: $c=c_{0}(1+\\alpha t)$, where $c_{0}=150$ J/kg$\\cdot^\\circ$C is the specific heat capacity at $0^\\circ$C, $\\alpha=0.05\\ ^\\circ$C$^{-1}$ is the temperature coefficient, and $t$ i... | math | {
"style": "rule",
"ground_truth": "96000"
} | {
"index": 34365,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The specific heat capacity of a body with mass \\( m = 3 \\) kg depends on the temperature in the following way: \\( c = c_{0}(1 + \\alpha t) \\), where \\( c_{0} = 200 \\) J/kg·°C is the specific heat capacity at \\( 0^{\\circ} \\mathrm{C} \\), \\( \\alpha = 0.05 \\,^{\\circ} ... | math | {
"style": "rule",
"ground_truth": "112.5"
} | {
"index": 34366,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sphere with radius 1 and center $(0,0,1)$ rests on the $xy$-plane. A light source is at $P = (0,-1,2).$ Then the boundary of the shadow of the sphere can be expressed in the form $y = f(x),$ for some function $f(x).$ Find the function $f(x).$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{x^2}{4} - 1"
} | {
"index": 34367,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The spikiness of a sequence $a_{1}, a_{2}, \\ldots, a_{n}$ of at least two real numbers is the sum $\\sum_{i=1}^{n-1}\\left|a_{i+1}-a_{i}\\right|$. Suppose $x_{1}, x_{2}, \\ldots, x_{9}$ are chosen uniformly and randomly from the interval $[0,1]$. Let $M$ be the largest possibl... | math | {
"style": "rule",
"ground_truth": "\\frac{79}{20}"
} | {
"index": 34368,
"solution": "Our job is to arrange the nine numbers in a way that maximizes the spikiness. Let an element be a peak if it is higher than its neighbor(s) and a valley if it is lower than its neighbor(s). It is not hard to show that an optimal arrangement has every element either a peak or a valley ... |
DeepScaleR | [
{
"role": "user",
"content": "The spinner shown is divided into 6 sections of equal size. Determine the probability of landing on a section that contains the letter Q using this spinner."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{6}"
} | {
"index": 34369,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The square\n\n$\\begin{tabular}{|c|c|c|} \\hline 50 & \\textit{b} & \\textit{c} \\\\ \\hline \\textit{d} & \\textit{e} & \\textit{f} \\\\ \\hline \\textit{g} & \\textit{h} & 2 \\\\ \\hline \\end{tabular}$\nis a multiplicative magic square. That is, the product of the numbers in... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 34370,
"solution": "1. **Identify the products of rows, columns, and diagonals:** Given the multiplicative magic square, we know that the product of the numbers in each row, column, and diagonal must be the same. Let's denote this common product by $P$. Thus, we have:\n - $50 \\cdot b \\cdot c = P$\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The square $BCDE$ is inscribed in circle $\\omega$ with center $O$ . Point $A$ is the reflection of $O$ over $B$ . A \"hook\" is drawn consisting of segment $AB$ and the major arc $\\widehat{BE}$ of $\\omega$ (passing through $C$ and $D$ ). Assume $BCDE$ h... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 34371,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The square $A B C D$ is enlarged from vertex $A$ resulting in the square $A B^{\\prime} C^{\\prime} D^{\\prime}$. The intersection point of the diagonals of the enlarged square is $M$. It is given that $M C = B B^{\\prime}$. What is the scale factor of the enlargement?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 34372,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The square \\( STUV \\) is formed by a square bounded by 4 equal rectangles. The perimeter of each rectangle is \\( 40 \\text{ cm} \\). What is the area, in \\( \\text{cm}^2 \\), of the square \\( STUV \\)?\n(a) 400 \n(b) 200 \n(c) 160 \n(d) 100 \n(e) 80"
}
] | math | {
"style": "rule",
"ground_truth": "400"
} | {
"index": 34373,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The square of $5-\\sqrt{y^2-25}$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "y^2-10\\sqrt{y^2-25}"
} | {
"index": 34374,
"solution": "We are given the expression $(5-\\sqrt{y^2-25})^2$ and need to simplify it. We will use the formula for the square of a binomial, which is $(a-b)^2 = a^2 - 2ab + b^2$.\n\n1. **Identify $a$ and $b$:**\n - Here, $a = 5$ and $b = \\sqrt{y^2-25}$.\n\n2. **Apply the binomial square formu... |
DeepScaleR | [
{
"role": "user",
"content": "The square of $a$ and the square root of $b$ vary inversely. If $a=2$ when $b=81$, then find $b$ when $ab=48$."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 34375,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The square of a natural number \\( a \\) gives a remainder of 8 when divided by a natural number \\( n \\). The cube of the number \\( a \\) gives a remainder of 25 when divided by \\( n \\). Find \\( n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "113"
} | {
"index": 34376,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The square of a three-digit number ends with three identical digits different from zero. Write the smallest such three-digit number."
}
] | math | {
"style": "rule",
"ground_truth": "462"
} | {
"index": 34377,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The square of an integer is 182 greater than the integer itself. What is the sum of all integers for which this is true?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 34378,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The square of an integer is called a perfect square. If $x$ is a perfect square, the next larger perfect square is"
}
] | math | {
"style": "rule",
"ground_truth": "$x+2\\sqrt{x}+1$"
} | {
"index": 34379,
"solution": "1. **Identify the nature of $x$**: Given that $x$ is a perfect square, we can express $x$ as $k^2$ for some integer $k$.\n\n2. **Find the next perfect square**: The next integer after $k$ is $k+1$. The square of $k+1$ is:\n \\[\n (k+1)^2 = k^2 + 2k + 1\n \\]\n\n3. **Express in t... |
DeepScaleR | [
{
"role": "user",
"content": "The square quilt block shown is made from 16 unit squares, four of which have been divided in half to form triangles. Additionally, two squares are completely filled while others are empty. What fraction of the square quilt is shaded? Express your answer as a common fraction."
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 34380,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The square quilt block shown is made from nine unit squares, some of which have been divided in half to form triangles. What fraction of the square quilt is shaded? Express your answer as a common fraction.\n\n[asy]size(75);\nfill(scale(3)*unitsquare,gray(.6));\npath[] interior... | math | {
"style": "rule",
"ground_truth": "\\frac13"
} | {
"index": 34381,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The square quilt block shown is made from sixteen unit squares, where eight of these squares have been divided in half diagonally to form triangles. Each triangle is shaded. What fraction of the square quilt is shaded? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 34382,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The square root of $2x$ is greater than 3 and less than 4. How many integer values of $x$ satisfy this condition?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 34383,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The square root of $x$ is greater than 2 and less than 4. How many integer values of $x$ satisfy this condition?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 34384,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The square root of $x$ is greater than 3 and less than 4. How many integer values of $x$ satisfy this condition?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 34385,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The square root of a two-digit number is expressed as an infinite decimal fraction, the first four digits of which (including the integer part) are the same. Find this number without using tables."
}
] | math | {
"style": "rule",
"ground_truth": "79"
} | {
"index": 34386,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The square was cut into 25 smaller squares, of which exactly one has a side length different from 1 (each of the others has a side length of 1).\n\nFind the area of the original square."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 34387,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The square with vertices $(-a, -a), (a, -a), (-a, a), (a, a)$ is cut by the line $y = x/2$ into congruent quadrilaterals. The perimeter of one of these congruent quadrilaterals divided by $a$ equals what? Express your answer in simplified radical form."
}
] | math | {
"style": "rule",
"ground_truth": "4+\\sqrt{5}"
} | {
"index": 34388,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The squares of a $3 \\times 3$ grid are filled with positive integers such that 1 is the label of the upperleftmost square, 2009 is the label of the lower-rightmost square, and the label of each square divides the one directly to the right of it and the one directly below it. H... | math | {
"style": "rule",
"ground_truth": "2448"
} | {
"index": 34389,
"solution": "We factor 2009 as $7^{2} \\cdot 41$ and place the 41 's and the 7 's in the squares separately. The number of ways to fill the grid with 1's and 41 's so that the divisibility property is satisfied is equal to the number of nondecreasing sequences $a_{1}, a_{2}, a_{3}$ where each $a_{... |
DeepScaleR | [
{
"role": "user",
"content": "The squares of a chessboard are labelled with numbers, as shown below.\n\n[asy]\nunitsize(0.8 cm);\n\nint i, j;\n\nfor (i = 0; i <= 8; ++i) {\n draw((i,0)--(i,8));\n draw((0,i)--(8,i));\n}\n\nfor (i = 0; i <= 7; ++i) {\nfor (j = 0; j <= 7; ++j) {\n label(\"$\\frac{1}{\" + st... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 34390,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The standard enthalpy of formation (ΔH_f°) of a substance is equal to the heat effect of the formation reaction of 1 mole of the substance from simple substances in their standard states (at 1 atm pressure and a given temperature). Therefore, it is necessary to find the heat ef... | math | {
"style": "rule",
"ground_truth": "-82.9"
} | {
"index": 34391,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The state income tax where Kristin lives is levied at the rate of $p\\%$ of the first $\\$28000$ of annual income plus $(p + 2)\\%$ of any amount above $\\$28000$. Kristin noticed that the state income tax she paid amounted to $(p + 0.25)\\%$ of her annual income. What was her ... | math | {
"style": "rule",
"ground_truth": "32000"
} | {
"index": 34392,
"solution": "Let $A$ represent Kristin's annual income and $T$ represent the total income tax she paid. The tax rate is $p\\%$ for the first $\\$28000$ and $(p+2)\\%$ for any amount above $\\$28000$. We are given that the total tax $T$ is also $(p+0.25)\\%$ of her annual income $A$.\n\n1. **Expres... |
DeepScaleR | [
{
"role": "user",
"content": "The statement $x^2 - x - 6 < 0$ is equivalent to the statement:"
}
] | math | {
"style": "rule",
"ground_truth": "-2 < x < 3"
} | {
"index": 34393,
"solution": "1. **Factorize the quadratic expression**: We start by factoring the quadratic expression given in the inequality $x^2 - x - 6 < 0$. We look for two numbers that multiply to $-6$ (the constant term) and add up to $-1$ (the coefficient of $x$). These numbers are $3$ and $-2$. Therefore... |
DeepScaleR | [
{
"role": "user",
"content": "The states of Sunshine and Prairie have adopted new license plate configurations. Sunshine license plates are formatted \"LDDLDL\" where L denotes a letter and D denotes a digit. Prairie license plates are formatted \"LDDLDD\". Assuming all 10 digits and 26 letters are equally ... | math | {
"style": "rule",
"ground_truth": "10816000"
} | {
"index": 34394,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The steamboat \"Rarity\" travels for three hours at a constant speed after leaving the city, then drifts with the current for an hour, then travels for three hours at the same speed, and so on. If the steamboat starts its journey in city A and goes to city B, it takes it 10 hou... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 34395,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The stem and leaf plot represents the heights, in inches, of the players on the Pine Ridge Middle School boys' basketball team. Calculate the mean height of the players on the team. (Note: $5|3$ represents 53 inches.)\n\nHeight of the Players on the Basketball Team (inches)\n\n... | math | {
"style": "rule",
"ground_truth": "61.44"
} | {
"index": 34396,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The stem and leaf plot represents the heights, in inches, of the players on the Spring Vale Middle School girls' basketball team. What is the mean height of the players on the team? (Note: $5|3$ represents 53 inches.)\n\nHeight of the Players on the Basketball Team (inches)\n\n... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 34397,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The stem and leaf plot shows the heights, in inches, of the players on the Westvale High School boys' basketball team. Calculate the mean height of the players on the team. (Note: $6|2$ represents 62 inches.)\n\nHeight of the Players on the Basketball Team (inches)\n\n$5|7$\n\n... | math | {
"style": "rule",
"ground_truth": "68.6"
} | {
"index": 34398,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The stem-and-leaf plot displays the lengths of songs on an album in minutes and seconds. There are 18 songs on the album. In the plot, $3\\ 45$ represents $3$ minutes, $45$ seconds, which is equivalent to $225$ seconds. What is the median length of the songs? Express your answe... | math | {
"style": "rule",
"ground_truth": "147.5"
} | {
"index": 34399,
"solution": ""
} |
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