data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given positive real numbers \\( a, b, c, d \\) that satisfy the equalities\n\n\\[ a^{2}+d^{2}-ad = b^{2}+c^{2}+bc \\quad \\text{and} \\quad a^{2}+b^{2} = c^{2}+d^{2}, \\]\n\nfind all possible values of the expression \\( \\frac{ab+cd}{ad+bc} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 14300,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive real numbers \\(a, b, c\\) satisfy \\(2(a+b)=ab\\) and \\(a+b+c=abc\\), find the maximum value of \\(c\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{15}"
} | {
"index": 14301,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given positive real numbers \\(a\\) and \\(b\\) that satisfy \\(ab(a+b) = 4\\), find the minimum value of \\(2a + b\\)."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 14302,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given quadrilateral $ABCD$ where $AC \\perp BD$ and $AC=2$, $BD=3$, find the minimum value of $\\overrightarrow{AB} \\cdot \\overrightarrow{CD}$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\dfrac{13}{4}"
} | {
"index": 14303,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given quadrilateral $ABCD$ with diagonals $AC$ and $BD$ intersecting at $O$, $BO=4$, $OD=5$, $AO=9$, $OC=2$, and $AB=7$, find the length of $AD$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{166}"
} | {
"index": 14304,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given quadrilateral $ABCD,$ side $\\overline{AB}$ is extended past $B$ to $A'$ so that $A'B = AB.$ Points $B',$ $C',$ and $D'$ are similarly constructed.\n\n[asy]\nunitsize(1 cm);\n\npair[] A, B, C, D;\n\nA[0] = (0,0);\nB[0] = (2,0);\nC[0] = (1.5,2);\nD[0] = (0.2,1.5);\nA[1] =... | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{1}{15}, \\frac{2}{15}, \\frac{4}{15}, \\frac{8}{15} \\right)"
} | {
"index": 14305,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given quadrilateral $\\Box FRDS$ with $\\triangle FDR$ being a right-angled triangle at point $D$, with side lengths $FD = 3$ inches, $DR = 4$ inches, $FR = 5$ inches, and $FS = 8$ inches, and $\\angle RFS = \\angle FDR$, find the length of RS."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{89}"
} | {
"index": 14306,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given quadrilateral ABCD, ∠A = 120∘, and ∠B and ∠D are right angles. Given AB = 13 and AD = 46, find the length of AC."
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"index": 14307,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given rational numbers $x$ and $y$ satisfy $|x|=9$, $|y|=5$. \n$(1)$ If $x \\lt 0$, $y \\gt 0$, find the value of $x+y$; \n$(2)$ If $|x+y|=x+y$, find the value of $x-y$."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 14308,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $a$ and $b \\gt 0$, if $a+2b=1$, then the minimum value of $\\frac{3}{b}+\\frac{1}{a}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "7 + 2\\sqrt{6}"
} | {
"index": 14309,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $a$ and $b$ satisfying $a^{2}-4\\ln a-b=0$, find the minimum value of $\\left(a-c\\right)^{2}+\\left(b+2c\\right)^{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{5}"
} | {
"index": 14310,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $a$ and $b$ satisfying $a^{2}b^{2}+2ab+2a+1=0$, calculate the minimum value of $ab\\left(ab+2\\right)+\\left(b+1\\right)^{2}+2a$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{4}"
} | {
"index": 14311,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $a$ and $b$ satisfying $ab=1$, and $a>b\\geq \\frac{2}{3}$, the maximum value of $\\frac{a-b}{a^{2}+b^{2}}$ is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{30}{97}"
} | {
"index": 14312,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $a$ and $b$ satisfying the equation $\\sqrt{(a-1)^2} + \\sqrt{(a-6)^2} = 10 - |b+3| - |b-2|$, find the maximum value of $a^2 + b^2$."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 14313,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $a$ and $b$, satisfying $e^{2-a}=a$ and $b\\left(\\ln b-1\\right)=e^{3}$, where $e$ is the base of natural logarithm, the value of $ab$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "e^{3}"
} | {
"index": 14314,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $a$, $b$, $c$, and $d$ satisfy $(b + 2a^2 - 6\\ln a)^2 + |2c - d + 6| = 0$, find the minimum value of $(a - c)^2 + (b - d)^2$."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 14315,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $a_1$, $a_2$, $a_3$ are not all zero, and positive numbers $x$, $y$ satisfy $x+y=2$. Let the maximum value of $$\\frac {xa_{1}a_{2}+ya_{2}a_{3}}{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$$ be $M=f(x,y)$, then the minimum value of $M$ is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {2}}{2}"
} | {
"index": 14316,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $b$ and $c$, and the function $f(x) = x^2 + bx + c$, the equation $f(f(x)) = 0$ has exactly three different real roots. Find the maximum value of the sum of the roots of $f(x)$."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 14317,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $b_0, b_1, \\dots, b_{2019}$ with $b_{2019} \\neq 0$, let $z_1,z_2,\\dots,z_{2019}$ be the roots in the complex plane of the polynomial \\[ P(z) = \\sum_{k=0}^{2019} b_k z^k. \\] Let $\\mu = (|z_1| + \\cdots + |z_{2019}|)/2019$ be the average of the distances... | math | {
"style": "rule",
"ground_truth": "2019^{-1/2019}"
} | {
"index": 14318,
"solution": "The answer is $M = 2019^{-1/2019}$. For any choices of $b_0,\\ldots,b_{2019}$ as specified, AM-GM gives \\[ \\mu \\geq |z_1\\cdots z_{2019}|^{1/2019} = |b_0/b_{2019}|^{1/2019} \\geq 2019^{-1/2019}. \\] To see that this is best possible, consider $b_0,\\ldots,b_{2019}$ given by $b_k = ... |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $x \\gt 0 \\gt y$, and $\\frac{1}{x+2}+\\frac{1}{1-y}=\\frac{1}{6}$, find the minimum value of $x-y$."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 14319,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ satisfy that three of the four numbers $x+y$, $x-y$, $\\frac{x}{y}$, and $xy$ are equal, determine the value of $|y|-|x|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 14320,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ satisfy the equation $x^2+y^2-4x+1=0$.\n(1) Find the maximum and minimum value of $\\frac {y}{x}$.\n(2) Find the maximum and minimum value of $y-x$.\n(3) Find the maximum and minimum value of $x^2+y^2$."
}
] | math | {
"style": "rule",
"ground_truth": "7-4\\sqrt{3}"
} | {
"index": 14321,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ satisfy the equation $x^{2}+y^{2}-4x+1=0$.\n$(1)$ Find the maximum and minimum values of $\\dfrac {y}{x}$;\n$(2)$ Find the maximum and minimum values of $y-x$;\n$(3)$ Find the maximum and minimum values of $x^{2}+y^{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "7-4 \\sqrt {3}"
} | {
"index": 14322,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ satisfying $\\sqrt{x-3}+y^{2}-4y+4=0$, find the value of the algebraic expression $\\frac{{x}^{2}-{y}^{2}}{xy}•\\frac{1}{{x}^{2}-2xy+{y}^{2}}÷\\frac{x}{{x}^{2}y-x{y}^{2}}-1$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 14323,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ satisfying $x^2+4y^2=4$, find the maximum value of $\\frac {xy}{x+2y-2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1+ \\sqrt {2}}{2}"
} | {
"index": 14324,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ satisfying $x^{2}+2y^{2}-2xy=4$, find the maximum value of $xy$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} + 2"
} | {
"index": 14325,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ satisfying $x^{2}+4y^{2}\\leqslant 4$, find the maximum value of $|x+2y-4|+|3-x-y|$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 14326,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, calculate the maximum value of $x-y$."
}
] | math | {
"style": "rule",
"ground_truth": "1+3\\sqrt{2}"
} | {
"index": 14327,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, find the maximum value of $x-y$."
}
] | math | {
"style": "rule",
"ground_truth": "1+3\\sqrt{2}"
} | {
"index": 14328,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ that satisfy the equation $x^{2}+y^{2}-4x+6y+12=0$, find the minimum value of $|2x-y-2|$."
}
] | math | {
"style": "rule",
"ground_truth": "5-\\sqrt{5}"
} | {
"index": 14329,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ that satisfy the system of inequalities $\\begin{cases} x - 2y - 2 \\leqslant 0 \\\\ x + y - 2 \\leqslant 0 \\\\ 2x - y + 2 \\geqslant 0 \\end{cases}$, if the minimum value of the objective function $z = ax + by + 5 (a > 0, b > 0)$ is $2$, determi... | math | {
"style": "rule",
"ground_truth": "\\frac{10 + 4\\sqrt{6}}{3}"
} | {
"index": 14330,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $x$, $y$ satisfying $x > y > 0$, and $x + y \\leqslant 2$, the minimum value of $\\dfrac{2}{x+3y}+\\dfrac{1}{x-y}$ is"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {3+2 \\sqrt {2}}{4}"
} | {
"index": 14331,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $x$, $y$, $z$ satisfying $2x-y-2z-6=0$, and $x^2+y^2+z^2\\leq4$, calculate the value of $2x+y+z$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 14332,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $x$, $y$, $z$ satisfying $\\begin{cases} xy+2z=1 \\\\ x^{2}+y^{2}+z^{2}=5 \\end{cases}$, the minimum value of $xyz$ is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "9 \\sqrt {11}-32"
} | {
"index": 14333,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $x$, $y$, and $z$ are chosen independently and at random from the interval $[0, m]$ for some positive integer $m$. The probability that no two of $x$, $y$, and $z$ are within 2 units of each other is greater than $\\frac{1}{2}$. Determine the smallest possibl... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 14334,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers $x$, $y$, and $z$ satisfy $x^2+y^2+z^2=4$, find the maximum value of $(2x-y)^2+(2y-z)^2+(2z-x)^2$."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 14335,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers \\( a, b, c \\) and a positive number \\( \\lambda \\) such that the polynomial \\( f(x) = x^3 + a x^2 + b x + c \\) has three real roots \\( x_1, x_2, x_3 \\), and the conditions \\( x_2 - x_1 = \\lambda \\) and \\( x_3 > \\frac{1}{2}(x_1 + x_2) \\) are sati... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{2}"
} | {
"index": 14336,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers \\( a, b, c \\) satisfy\n\\[\na^{2}+b^{2}-4a \\leqslant 1, \\quad b^{2}+c^{2}-8b \\leqslant -3, \\quad c^{2}+a^{2}-12c \\leqslant -26,\n\\]\nwhat is the value of \\( (a+b)^{c} \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 14337,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers \\( a, b, c \\) satisfy the system of inequalities\n\\[\na^{2}+b^{2}-4a \\leqslant 1, \\quad b^{2}+c^{2}-8b \\leqslant -3, \\quad c^{2}+a^{2}-12c \\leqslant -26,\n\\]\ncalculate the value of \\( (a+b)^{c} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 14338,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers \\( x \\) and \\( y \\) satisfy\n\\[\n\\left\\{\n\\begin{array}{l}\nx - y \\leq 0, \\\\\nx + y - 5 \\geq 0, \\\\\ny - 3 \\leq 0\n\\end{array}\n\\right.\n\\]\n\nIf the inequality \\( a(x^2 + y^2) \\leq (x + y)^2 \\) always holds, then the maximum value of the ... | math | {
"style": "rule",
"ground_truth": "25/13"
} | {
"index": 14339,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers \\( x \\) and \\( y \\) that satisfy \\( x^{2} + y^{2} \\leq 5 \\), find the maximum and minimum values of the function \\( f(x, y) = 3|x+y| + |4y+9| + |7y-3x-18| \\)."
}
] | math | {
"style": "rule",
"ground_truth": "27 + 6\\sqrt{5}"
} | {
"index": 14340,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers \\( x \\) and \\( y \\in (1, +\\infty) \\), and \\( x y - 2 x - y + 1 = 0 \\). Find the minimum value of \\( \\frac{3}{2} x^{2} + y^{2} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 14341,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers \\( x, y \\in (1,+\\infty) \\) such that \\( xy - 2x - y + 1 = 0 \\), find the minimum value of \\( \\frac{3}{2} x^{2} + y^{2} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 14342,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers \\( x, y, z, w \\) satisfying \\( x + y + z + w = 1 \\), find the maximum value of \\( M = xw + 2yw + 3xy + 3zw + 4xz + 5yz \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 14343,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers \\( x, y, z, w \\) such that \\( x + y + z + w = 1 \\), find the maximum value of \\( M = xw + 2yw + 3xy + 3zw + 4xz + 5yz \\)."
}
] | math | {
"style": "rule",
"ground_truth": "3/2"
} | {
"index": 14344,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers \\( x_1, x_2, \\ldots, x_{2021} \\) satisfy \\( \\sum_{i=1}^{2021} x_i^2 = 1 \\), find the maximum value of \\( \\sum_{i=1}^{2020} x_i^3 x_{i+1}^3 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 14345,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers \\( x_{1}, x_{2}, \\cdots, x_{1991} \\) that satisfy the condition\n\\[\n\\sum_{i=1}^{1990} \\left| x_{i} - x_{i+1} \\right| = 1991,\n\\]\nand \\( y_{k} = \\frac{1}{k} \\sum_{i=1}^{k} x_{i} \\) for \\( k = 1, 2, \\cdots, 1991 \\), determine the maximum value ... | math | {
"style": "rule",
"ground_truth": "1990"
} | {
"index": 14346,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers \\(a, b, c\\), the polynomial \n$$\ng(x) = x^{3} + a x^{2} + x + 10\n$$ \nhas three distinct roots, and these three roots are also roots of the polynomial\n$$\nf(x) = x^{4} + x^{3} + b x^{2} + 100 x + c.\n$$\n\nEvaluate the value of \\(f(1)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-7007"
} | {
"index": 14347,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers \\(a, b, c\\), the polynomial \n$$\ng(x) = x^{3} + a x^{2} + x + 10\n$$ \nhas three distinct roots, and these three roots are also roots of the polynomial\n$$\nf(x) = x^{4} + x^{3} + b x^{2} + 100 x + c.\n$$\n\nFind the value of \\(f(1)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-7007"
} | {
"index": 14348,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers \\(a\\) and \\(b\\) that satisfy \\(0 \\leqslant a, b \\leqslant 8\\) and \\(b^2 = 16 + a^2\\), find the sum of the maximum and minimum values of \\(b - a\\)."
}
] | math | {
"style": "rule",
"ground_truth": "12 - 4\\sqrt{3}"
} | {
"index": 14349,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers \\(a_{1}, a_{2}, \\cdots, a_{n}\\) which are all greater than 0 (where \\(n\\) is a natural number no less than 4) and satisfy the equation \\(a_{1} + a_{2} + \\cdots + a_{n} = 1\\), find the maximum value of the sum \\(S = \\sum_{k=1}^{n} \\frac{a_{k}^{2}}{a... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 14350,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given real numbers \\\\(x\\\\) and \\\\(y\\\\) satisfy the equation \\\\((x-3)^{2}+y^{2}=9\\\\), find the minimum value of \\\\(-2y-3x\\\\) \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "-3\\sqrt{13}-9"
} | {
"index": 14351,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given rectangle $ABCD$ with $E$ the midpoint of diagonal $BD$. Point $E$ is connected to a point $F$ on $DA$ so that $DF=\\frac{1}{4}DA$. Find the ratio of the area of $\\triangle DFE$ to the area of quadrilateral $ABEF$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"index": 14352,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given rectangle $R_1$ with one side $2$ inches and area $12$ square inches. Rectangle $R_2$ with diagonal $15$ inches is similar to $R_1$. Expressed in square inches the area of $R_2$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{135}{2}"
} | {
"index": 14353,
"solution": "1. **Identify the dimensions of $R_1$:**\n Given that one side of rectangle $R_1$ is $2$ inches and its area is $12$ square inches, we can find the other side by dividing the area by the given side:\n \\[\n \\text{Other side} = \\frac{\\text{Area}}{\\text{Given side}} = \\frac{1... |
DeepScaleR | [
{
"role": "user",
"content": "Given rectangle $R_1$ with one side $4$ inches and area $24$ square inches. Rectangle $R_2$ with diagonal $17$ inches is similar to $R_1$. Find the area of $R_2$ in square inches."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{433.5}{3.25}"
} | {
"index": 14354,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given rectangle ABCD where E is the midpoint of diagonal BD, point E is connected to point F on segment DA such that DF = 1/4 DA. Find the ratio of the area of triangle DFE to the area of quadrilateral ABEF."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"index": 14355,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given rectangle ABCD, AB=4, BC=8, points K and L are midpoints of BC and AD, respectively, and point M is the midpoint of KL. What is the area of the quadrilateral formed by the rectangle diagonals and segments KM and LM."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 14356,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given regular hexagon $ABCDEF$ , compute the probability that a randomly chosen point inside the hexagon is inside triangle $PQR$ , where $P$ is the midpoint of $AB$ , $Q$ is the midpoint of $CD$ , and $R$ is the midpoint of $EF$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"index": 14357,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given right $\\triangle ABC$ with legs $BC=3, AC=4$. Find the length of the shorter angle trisector from $C$ to the hypotenuse:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{12\\sqrt{3}-9}{13}"
} | {
"index": 14358,
"solution": "1. **Identify the triangle and its properties**: We are given a right triangle $\\triangle ABC$ with legs $BC = 3$ and $AC = 4$. Since it is a right triangle, we can use the Pythagorean theorem to find the hypotenuse $AB$:\n \\[\n AB = \\sqrt{BC^2 + AC^2} = \\sqrt{3^2 + 4^2} = \\s... |
DeepScaleR | [
{
"role": "user",
"content": "Given right triangle $ABC$ with $\\angle A = 90^\\circ$, $AC = 3$, $AB = 4$, and $BC = 5$, point $D$ is on side $BC$. If the perimeters of $\\triangle ACD$ and $\\triangle ABD$ are equal, calculate the area of $\\triangle ABD$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{12}{5}"
} | {
"index": 14359,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given right triangle $ABC$ with a right angle at vertex $C$ and $AB = 2BC$, calculate the value of $\\cos A$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 14360,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given right triangle $ABC$, with $AB=4, BC=3$, and $CA=5$. Circle $\\omega$ passes through $A$ and is tangent to $BC$ at $C$. What is the radius of $\\omega$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{8}"
} | {
"index": 14361,
"solution": "Let $O$ be the center of $\\omega$, and let $M$ be the midpoint of $AC$. Since $OA=OC$, $OM \\perp AC$. Also, $\\angle OCM=\\angle BAC$, and so triangles $ABC$ and $CMO$ are similar. Then, $CO/CM=AC/AB$, from which we obtain that the radius of $\\omega$ is $CO=\\frac{25}{8}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Given right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b+2$, where $b<150$, determine the number of possible triangles."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 14362,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sandwiches cost $4 each, sodas cost $1.50 each, and fries cost $2.50 each, and you buy 4 sandwiches, 6 sodas, and 3 orders of fries, calculate the total cost before the $5 discount, and then subtract the total cost after applying the discount."
}
] | math | {
"style": "rule",
"ground_truth": "27.50"
} | {
"index": 14363,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given set $A=\\{a-2, 12, 2a^2+5a\\}$, and $-3$ belongs to $A$, find the value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"index": 14364,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given set \\( A = \\{0, 1, 2, 3, 4, 5, 9\\} \\), and \\( a, b \\in A \\) where \\( a \\neq b \\). The number of functions of the form \\( y = -a x^2 + (4 - b)x \\) whose vertex lies in the first quadrant is ___."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 14365,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given set \\( S \\) such that \\( |S| = 10 \\). Let \\( A_1, A_2, \\cdots, A_k \\) be non-empty subsets of \\( S \\), and the intersection of any two subsets contains at most two elements. Find the maximum value of \\( k \\)."
}
] | math | {
"style": "rule",
"ground_truth": "175"
} | {
"index": 14366,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sets $A=\\{-1,1,2\\}$ and $B=\\{-2,1,2\\}$, a number $k$ is randomly selected from set $A$ and a number $b$ is randomly selected from set $B$. The probability that the line $y=kx+b$ does not pass through the third quadrant is $\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "P = \\frac{2}{9}"
} | {
"index": 14367,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sets $A=\\{0, |x|\\}$ and $B=\\{1, 0, -1\\}$. If $A \\subseteq B$, then find $A \\cap B$, $A \\cup B$, and the complement of $A$ in $B$, denoted by $\\complement_B A$."
}
] | math | {
"style": "rule",
"ground_truth": "\\{-1\\}"
} | {
"index": 14368,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sets $A=\\{1, 2, 3, 4\\}$, $B=\\{5, 6, 7\\}$, and $C=\\{8, 9\\}$. Now, take any two sets from these three sets, and then pick one element from each of the two selected sets to form a new set with two elements. The total number of distinct sets that can be formed is $\\_\\... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 14369,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sets $A=\\{1, a, b\\}$ and $B=\\{a, a^2, ab\\}$. If $A=B$, find the value of $a+b$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14370,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sets $A=\\{1,2,3,4,5\\}$, $B=\\{0,1,2,3,4\\}$, and a point $P$ with coordinates $(m,n)$, where $m\\in A$ and $n\\in B$, find the probability that point $P$ lies below the line $x+y=5$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{2}{5}"
} | {
"index": 14371,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sets $A=\\{1,3,5,7,9\\}$ and $B=\\{0,3,6,9,12\\}$, then $A\\cap (\\complement_{\\mathbb{N}} B) = \\_\\_\\_\\_\\_\\_$; the number of proper subsets of $A\\cup B$ is $\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "255"
} | {
"index": 14372,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sets $A=\\{2, a-1\\}$, $B=\\{a^2-7, -1\\}$, and $A\\cap B=\\{2\\}$, find the real number $a$."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 14373,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sets $A=\\{2,3,4\\}$ and $B=\\{a+2,a\\}$, if $A \\cap B = B$, find $A^cB$ ___."
}
] | math | {
"style": "rule",
"ground_truth": "\\{3\\}"
} | {
"index": 14374,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sets $A=\\{x,\\frac{y}{x},1\\}$ and $B=\\{{x}^{2},x+y,0\\}$, if $A=B$, then $x^{2023}+y^{2024}=\\_\\_\\_\\_\\_\\_.$"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14375,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sets $A=\\{x|x^2-2x-3>0\\}$, $B=\\{x|x^2+ax+b\\leq0\\}$, if $A\\cup B=\\mathbb{R}$ and $A\\cap B=\\{x|3<x\\leq4\\}$, then the value of $a+b$ equals to."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 14376,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sets $A=\\{x|x^{2}+2x-3=0,x\\in R\\}$ and $B=\\{x|x^{2}-\\left(a+1\\right)x+a=0,x\\in R\\}$.<br/>$(1)$ When $a=2$, find $A\\cap C_{R}B$;<br/>$(2)$ If $A\\cup B=A$, find the set of real numbers for $a$."
}
] | math | {
"style": "rule",
"ground_truth": "\\{1\\}"
} | {
"index": 14377,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sets $M=\\{1, 2, a^2 - 3a - 1 \\}$ and $N=\\{-1, a, 3\\}$, and the intersection of $M$ and $N$ is $M \\cap N = \\{3\\}$, find the set of all possible real values for $a$."
}
] | math | {
"style": "rule",
"ground_truth": "\\{4\\}"
} | {
"index": 14378,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sets A = {-2, 1, 2} and B = {-1, 1, 3}, calculate the probability that a line represented by the equation ax - y + b = 0 will pass through the fourth quadrant."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"index": 14379,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sets A = {x | $x^2 - ax + a^2 - 19 = 0$}, B = {x | $x^2 - 5x + 6 = 0$}, C = {x | $x^2 + 2x - 8 = 0$} such that $A \\cap B \\neq \\emptyset$ and $A \\cap C = \\emptyset$. Find the value of the real number $a$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 14380,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sets \\( A = \\{ x \\mid 5x - a \\leq 0 \\} \\) and \\( B = \\{ x \\mid 6x - b > 0 \\} \\), where \\( a, b \\in \\mathbf{N} \\), and \\( A \\cap B \\cap \\mathbf{N} = \\{ 2, 3, 4 \\} \\), determine the number of integer pairs \\((a, b)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 14381,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sets \\( A = \\{ x \\mid 5x - a \\leq 0 \\} \\) and \\( B = \\{ x \\mid 6x - b > 0 \\} \\), where \\( a, b \\in \\mathbf{N} \\), and \\( A \\cap B \\cap \\mathbf{N} = \\{ 2, 3, 4 \\} \\), the number of integer pairs \\((a, b)\\) is:"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 14382,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sets \\( A = \\{1, 2, 3\\} \\) and \\( B = \\{4, 5, 6, 7\\} \\), with a mapping \\( f: A \\rightarrow B \\), if for \\( x \\in A \\), the expression \\( x + f(x) + x \\cdot f(x) \\) is odd, determine the number of such mappings \\( f \\)."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 14383,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sets \\( A = \\{1, 2, 3\\} \\) and \\( B = \\{4, 5, 6, 7\\} \\), with a mapping \\( f: A \\rightarrow B \\), if for \\( x \\in A \\), the expression \\( x + f(x) + x \\cdot f(x) \\) is odd, then the number of such mappings \\( f \\) is:"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 14384,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given seven positive integers from a list of eleven positive integers are \\(3, 5, 6, 9, 10, 4, 7\\). What is the largest possible value of the median of this list of eleven positive integers if no additional number in the list can exceed 10?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 14385,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given several numbers, one of them, $a$ , is chosen and replaced by the three numbers $\\frac{a}{3}, \\frac{a}{3}, \\frac{a}{3}$ . This process is repeated with the new set of numbers, and so on. Originally, there are $1000$ ones, and we apply the process several times. A n... | math | {
"style": "rule",
"ground_truth": "667"
} | {
"index": 14386,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given several rectangular prisms with edge lengths of $2, 3,$ and $5$, aligned in the same direction to form a cube with an edge length of $90$, how many small rectangular prisms does one diagonal of the cube intersect?"
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"index": 14387,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given sin(x + $\\frac{π}{4}$) = $\\frac{1}{3}$, find the value of sin4x - 2cos3xsinx = ___."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{9}"
} | {
"index": 14388,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given six balls numbered 1, 2, 3, 4, 5, 6 and boxes A, B, C, D, each to be filled with one ball, with the conditions that ball 2 cannot be placed in box B and ball 4 cannot be placed in box D, determine the number of different ways to place the balls into the boxes."
}
] | math | {
"style": "rule",
"ground_truth": "252"
} | {
"index": 14389,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given six cards with the digits $1, 2, 4, 5, 8$ and a comma. Using each card exactly once, various numbers are formed (the comma cannot be at the beginning or at the end of the number). What is the arithmetic mean of all such numbers?\n\n(M. V. Karlukova)"
}
] | math | {
"style": "rule",
"ground_truth": "1234.4321"
} | {
"index": 14390,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given six people are arranged in a row from left to right. Only A or B can be placed at the far left, and A cannot be placed at the far right, calculate the total number of different arrangements."
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"index": 14391,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given six people, including A, B, and C, are arranged in a row for a photo, and A and B are not on the same side as C, calculate the total number of different arrangements."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 14392,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given six senior students (including 4 boys and 2 girls) are arranged to intern at three schools, A, B, and C, with two students at each school, and the two girls cannot be at the same school or at school C, and boy A cannot go to school A, calculate the total number of differe... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 14393,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given six students – three boys and three girls – standing in a row, where girls A, B, and C are not allowed to stand next to each other and girls are not allowed to stand at the ends of the row, calculate the number of different arrangements."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 14394,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given six test scores have a mean of $85$, a median of $86$, and a mode of $88$. Determine the sum of the two lowest test scores."
}
] | math | {
"style": "rule",
"ground_truth": "162"
} | {
"index": 14395,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given six thin wooden sticks, the two longer ones are $\\sqrt{3}a$ and $\\sqrt{2}a$, while the remaining four are of length $a$. If they are used to form a triangular prism, find the cosine of the angle between the lines containing the two longer edges."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{3}"
} | {
"index": 14396,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given square $ABCD$ with side $8$ feet. A circle is drawn through vertices $A$ and $D$ and tangent to side $BC$. The radius of the circle, in feet, is:"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 14397,
"solution": "1. **Identify the Geometry and Setup**: We are given a square $ABCD$ with side length $8$ feet. A circle passes through vertices $A$ and $D$ and is tangent to side $BC$. We need to find the radius of this circle.\n\n2. **Define Points and Variables**: Let $O$ be the center of the circ... |
DeepScaleR | [
{
"role": "user",
"content": "Given square $ABCD$, points $E$ and $F$ lie on $\\overline{AB}$ so that $\\overline{ED}$ and $\\overline{FD}$ bisect $\\angle ADC$. Calculate the ratio of the area of $\\triangle DEF$ to the area of square $ABCD$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 14398,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given square $PQRS$ with side length $12$ feet, a circle is drawn through vertices $P$ and $S$, and tangent to side $QR$. If the point of tangency divides $QR$ into segments of $3$ feet and $9$ feet, calculate the radius of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{(6 - 3\\sqrt{2})^2 + 9^2}"
} | {
"index": 14399,
"solution": ""
} |
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