data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given condition p: $|5x - 1| > a$ and condition q: $x^2 - \\frac{3}{2}x + \\frac{1}{2} > 0$, please choose an appropriate real number value for $a$, and use the given two conditions as A and B to construct the proposition: If A, then B. Make sure the constructed original propos... | math | {
"style": "rule",
"ground_truth": "a = 4"
} | {
"index": 14100,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given cos($$α+ \\frac {π}{6}$$)= $$\\frac {1}{3}$$, find the value of sin($$ \\frac {5π}{6}+2α$$)."
}
] | math | {
"style": "rule",
"ground_truth": "-$$\\frac {7}{9}$$"
} | {
"index": 14101,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given cos(π/4 - α) = 3/5 and sin(5π/4 + β) = -12/13, where α ∈ (π/4, 3π/4) and β ∈ (0, π/4), find tan(α)/tan(β)."
}
] | math | {
"style": "rule",
"ground_truth": "-17"
} | {
"index": 14102,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given data: $2$, $5$, $7$, $9$, $11$, $8$, $7$, $8$, $10$, the $80$th percentile is ______."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 14103,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given distinct natural numbers \\( k, l, m, n \\), it is known that there exist three natural numbers \\( a, b, c \\) such that each of the numbers \\( k, l, m, n \\) is a root of either the equation \\( a x^{2} - b x + c = 0 \\) or the equation \\( c x^{2} - 16 b x + 256 a = 0... | math | {
"style": "rule",
"ground_truth": "325"
} | {
"index": 14104,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given distinct prime numbers $p$ and $q$ and a natural number $n \\geq 3$, find all $a \\in \\mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1."
}
] | math | {
"style": "rule",
"ground_truth": "-1 - pq \\text{ and } 1 + pq"
} | {
"index": 14105,
"solution": "\nGiven distinct prime numbers \\( p \\) and \\( q \\) and a natural number \\( n \\geq 3 \\), we aim to find all \\( a \\in \\mathbb{Z} \\) such that the polynomial \\( f(x) = x^n + ax^{n-1} + pq \\) can be factored into two integral polynomials of degree at least 1.\n\nTo solve this... |
DeepScaleR | [
{
"role": "user",
"content": "Given eight cubes with volumes \\(1, 8, 27, 64, 125, 216, 343,\\) and \\(512\\) cubic units, and each cube having its volume decreasing upwards, determine the overall external surface area of this arrangement."
}
] | math | {
"style": "rule",
"ground_truth": "1021"
} | {
"index": 14106,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero di... | math | {
"style": "rule",
"ground_truth": "376"
} | {
"index": 14107,
"solution": "There are $\\binom{8}{5}$ ways to choose the rings, and there are $5!$ distinct arrangements to order the rings [we order them so that the first ring is the bottom-most on the first finger that actually has a ring, and so forth]. The number of ways to distribute the rings among the fi... |
DeepScaleR | [
{
"role": "user",
"content": "Given eight students, including Abby and Bridget, are randomly assigned to the 12 spots arranged in three rows of four as shown, calculate the probability that Abby and Bridget are seated directly adjacent to each other (in the same row or same column)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{66}"
} | {
"index": 14108,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given eleven books consisting of three Arabic, two English, four Spanish, and two French, calculate the number of ways to arrange the books on the shelf keeping the Arabic books together, the Spanish books together, and the English books together."
}
] | math | {
"style": "rule",
"ground_truth": "34560"
} | {
"index": 14109,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given ellipse $C_1$ and parabola $C_2$ whose foci are both on the x-axis, the center of $C_1$ and the vertex of $C_2$ are both at the origin $O$. Two points are taken from each curve, and their coordinates are recorded in the table. Calculate the distance between the left focus... | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}-1"
} | {
"index": 14110,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given equations of the form $x^2 + bx + c = 0$, determine the number of such equations that have real roots and have coefficients $b$ and $c$ selected from the set of integers $\\{1, 2, 3, 4, 5, 7\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 14111,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given events A, B, and C with respective probabilities $P(A) = 0.65$, $P(B) = 0.2$, and $P(C) = 0.1$, find the probability that the drawn product is not a first-class product."
}
] | math | {
"style": "rule",
"ground_truth": "0.35"
} | {
"index": 14112,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given every 8-digit whole number is a possible telephone number except those that begin with 0 or 1, or end in 9, determine the fraction of these telephone numbers that begin with 9 and end with 0."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{72}"
} | {
"index": 14113,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given externally tangent circles with centers at points $A$ and $B$ and radii of lengths $6$ and $4$, respectively, a line externally tangent to both circles intersects ray $AB$ at point $C$. Calculate $BC$."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 14114,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given fifty distinct natural numbers, twenty-five of which do not exceed 50, and the remaining are greater than 50 but do not exceed 100. Additionally, no two of these numbers differ by exactly 50. Find the sum of these numbers."
}
] | math | {
"style": "rule",
"ground_truth": "2525"
} | {
"index": 14115,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given five letters a, b, c, d, and e arranged in a row, find the number of arrangements where both a and b are not adjacent to c."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 14116,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given fixed points \\( A(3,0) \\), \\( B(0,4) \\), and point \\( P \\) on the incircle of triangle \\( \\triangle AOB \\) (where \\( O \\) is the origin), find the maximum value of \\( |PA|^2 + |PB|^2 + |PO|^2 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 14117,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given four circles with radii $2, 4, 6,$ and $8$ tangent to two perpendicular lines $\\ell_1$ and $\\ell_2$ intersecting at point $A$, and region $S$ consisting of all the points that lie inside exactly one of these four circles, find the maximum possible area of region $S$."
... | math | {
"style": "rule",
"ground_truth": "120\\pi"
} | {
"index": 14118,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given four distinct points P, A, B, C on a plane satisfying $\\overrightarrow{PA} + \\overrightarrow{PB} + \\overrightarrow{PC} = \\overrightarrow{0}$ and $\\overrightarrow{AB} + \\overrightarrow{AC} + m\\overrightarrow{AP} = \\overrightarrow{0}$, determine the value of the rea... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 14119,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given four distinct real numbers \\( a, b, c, d \\) such that \\(\\frac{a}{b} + \\frac{b}{c} + \\frac{c}{d} + \\frac{d}{a} = 4\\) and \\( ac = bd \\), find the maximum value of \\(\\frac{a}{c} + \\frac{b}{d} + \\frac{c}{a} + \\frac{d}{b} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"index": 14120,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given four functions: $①y=-x$, $②y=- \\frac{1}{x}$, $③y=x^{3}$, $④y=x^{ \\frac{1}{2}}$, determine the probability of the event \"the graphs of the two selected functions have exactly one common point\" when two functions are selected from these four."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 14121,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given four non-coplanar points \\(A, B, C, D\\) in space where the distances between any two points are distinct, consider a plane \\(\\alpha\\) that satisfies the following properties: The distances from three of the points \\(A, B, C, D\\) to \\(\\alpha\\) are equal, while th... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 14122,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given four numbers $101010_{(2)}$, $111_{(5)}$, $32_{(8)}$, and $54_{(6)}$, the smallest among them is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "32_{(8)}"
} | {
"index": 14123,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given four one-inch squares are placed with their bases on a line. The second square from the left is lifted out and rotated 30 degrees before reinserting it such that it just touches the adjacent square on its right. Determine the distance in inches from point B, the highest p... | math | {
"style": "rule",
"ground_truth": "\\frac{2 + \\sqrt{3}}{4}"
} | {
"index": 14124,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given four points $O,\\ A,\\ B,\\ C$ on a plane such that $OA=4,\\ OB=3,\\ OC=2,\\ \\overrightarrow{OB}\\cdot \\overrightarrow{OC}=3.$ Find the maximum area of $\\triangle{ABC}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{7} + \\frac{3\\sqrt{3}}{2}"
} | {
"index": 14125,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given four points $P, A, B, C$ on a sphere, if $PA$, $PB$, $PC$ are mutually perpendicular and $PA=PB=PC=1$, calculate the surface area of this sphere."
}
] | math | {
"style": "rule",
"ground_truth": "3\\pi"
} | {
"index": 14126,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given four points O, A, B, C on a plane satisfying OA=4, OB=3, OC=2, and $\\overrightarrow{OB} \\cdot \\overrightarrow{OC} = 3$, find the maximum area of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{7} + \\frac{3\\sqrt{3}}{2}"
} | {
"index": 14127,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given four points \\( A, B, C, \\) and \\( D \\) in space that are not coplanar, each pair of points is connected by an edge with a probability of \\( \\frac{1}{2} \\). The events of whether any two pairs of points are connected are mutually independent. What is the probability... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 14128,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given four points \\( K, L, M, N \\) that are not coplanar. A sphere touches the planes \\( K L M \\) and \\( K L N \\) at points \\( M \\) and \\( N \\) respectively. Find the surface area of the sphere, knowing that \\( M L = 1 \\), \\( K M = 2 \\), \\( \\angle M N L = 60^\\c... | math | {
"style": "rule",
"ground_truth": "\\frac{64\\pi}{11}"
} | {
"index": 14129,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given four points \\( O, A, B, C \\) on a plane, with \\( OA=4 \\), \\( OB=3 \\), \\( OC=2 \\), and \\( \\overrightarrow{OB} \\cdot \\overrightarrow{OC}=3 \\), find the maximum area of triangle \\( ABC \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{7} + \\frac{3\\sqrt{3}}{2}"
} | {
"index": 14130,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given four points \\(O, A, B, C\\) on a plane, such that \\(OA = 4\\), \\(OB = 3\\), \\(OC = 2\\), and \\(\\overrightarrow{OB} \\cdot \\overrightarrow{OC} = 3\\), find the maximum value of the area \\(S_{\\triangle ABC}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{7} + \\frac{3\\sqrt{3}}{2}"
} | {
"index": 14131,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given four points on a sphere, $A$, $B$, $C$, $D$, with the center of the sphere being point $O$, and $O$ is on $CD$. If the maximum volume of the tetrahedron $A-BCD$ is $\\frac{8}{3}$, then the surface area of sphere $O$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "16\\pi"
} | {
"index": 14132,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given four positive integers \\(a, b, c,\\) and \\(d\\) satisfying the equations \\(a^2 = c(d + 20)\\) and \\(b^2 = c(d - 18)\\). Find the value of \\(d\\)."
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"index": 14133,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given four real numbers $-9$, $a\\_1$, $a\\_2$, and $-1$ that form an arithmetic sequence, and five real numbers $-9$, $b\\_1$, $b\\_2$, $b\\_3$, and $-1$ that form a geometric sequence, find the value of $b\\_2(a\\_2-a\\_1)=$\\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"index": 14134,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given four real numbers that form an arithmetic sequence: -9, $a_1$, $a_2$, -1, and five real numbers that form a geometric sequence: -9, $b_1$, $b_2$, $b_3$, -1, find the value of $b_2(a_2-a_1)$."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"index": 14135,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given function $f(x)$ defined on $\\mathbb{R}$ satisfies: (1) The graph of $y = f(x - 1)$ is symmetric about the point $(1, 0)$; (2) For all $x \\in \\mathbb{R}$, $$f\\left( \\frac {3}{4}-x\\right) = f\\left( \\frac {3}{4}+x\\right)$$ holds; (3) When $x \\in \\left(- \\frac {3}... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 14136,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given functions $f(x)$ and $g(x)$, where the graph of $g(x)$ is symmetric about $x=1$, and $f(x) - g(x) = 1$, $f(x+1) + g(2-x) = 1$, and $g(1) = 3$, find $\\sum_{i=1}^{23}f(x)=$____."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 14137,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given functions $f(x)=-2x$ for $x<0$ and $g(x)=\\frac{x}{\\ln x}+x-2$. If $f(x_{1})=g(x_{2})$, find the minimum value of $x_{2}-2x_{1}$."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{e}-2"
} | {
"index": 14138,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given functions $f(x)=xe^x$ and $g(x)=-\\frac{lnx}{x}$, if $f(x_{1})=g(x_{2})=t\\left( \\gt 0\\right)$, find the maximum value of $\\frac{{x}_{1}}{{x}_{2}{e}^{t}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{e}"
} | {
"index": 14139,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given functions $y_1=\\frac{k_1}{x}$ and $y_{2}=k_{2}x+b$ ($k_{1}$, $k_{2}$, $b$ are constants, $k_{1}k_{2}\\neq 0$).<br/>$(1)$ If the graphs of the two functions intersect at points $A(1,4)$ and $B(a,1)$, find the expressions of functions $y_{1}$ and $y_{2}$.<br/>$(2)$ If poin... | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"index": 14140,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given functions \\(f(x)\\) and \\(g(x)\\) are defined on \\(\\mathbb{R}\\) with \\(g(x)\\neq 0\\), and \\(f(x)g'(x) > f'(x)g(x)\\). It is also given that \\(f(x)=a^{x}\\cdot g(x)\\) where \\(a > 0, a\\neq 1\\), and \\(\\frac{f(1)}{g(1)} + \\frac{f(-1)}{g(-1)} = \\frac{5}{2}\\).... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 14141,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given in $\\bigtriangleup ABC$, $AB = 75$, and $AC = 120$. A circle with center $A$ and radius $AB$ intersects $\\overline{BC}$ at points $B$ and $X$. Moreover, $\\overline{BX}$ and $\\overline{CX}$ have integer lengths. Find the length of $BC$."
}
] | math | {
"style": "rule",
"ground_truth": "117"
} | {
"index": 14142,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given in $\\triangle ABC$, $AB= \\sqrt {3}$, $BC=1$, and $\\sin C= \\sqrt {3}\\cos C$, the area of $\\triangle ABC$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {3}}{2}"
} | {
"index": 14143,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given in $\\triangle ABC$, $AC=2$, $BC=1$, $\\cos C=\\frac{3}{4}$,\n\n$(1)$ Find the value of $AB$;\n\n$(2)$ Find the value of $\\sin (A+C)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{14}}{4}"
} | {
"index": 14144,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given in $\\triangle ABC$, $\\tan A$ and $\\tan B$ are the two real roots of the equation $x^2 + ax + 4 = 0$:\n(1) If $a = -8$, find the value of $\\tan C$;\n(2) Find the minimum value of $\\tan C$, and specify the corresponding values of $\\tan A$ and $\\tan B$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 14145,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given in $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is known that $a=b\\cos C-c\\sin B$.\n\n(Ⅰ) Find the size of angle $B$;\n\n(Ⅱ) If $b=5$ and $a=3\\sqrt{2}$, find the area $S$ of $\\triangle ABC$."
... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 14146,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given in $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$, and $\\sin A=\\sqrt{3}(1-\\cos A)$.\n\n$(1)$ Find $A$;\n\n$(2)$ If $a=7$ and $\\sin B+\\sin C=\\frac{13\\sqrt{3}}{14}$, find the area of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "10 \\sqrt{3}"
} | {
"index": 14147,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given in $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$, and it is known that $2\\cos C(a\\cos C+c\\cos A)+b=0$.\n\n$(1)$ Find the magnitude of angle $C$;\n\n$(2)$ If $b=2$ and $c=2\\sqrt{3}$, find the area of $\\triangle ABC... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 14148,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given in a cube ABCD-A1B1C1D1 with edge length 1, P is a moving point inside the cube (including the surface), \nif $x + y + z = s$, and $0 \\leq x \\leq y \\leq z \\leq 1$, then the volume of the geometric body formed by all possible positions of point P is $\\_\\_\\_\\_\\_\\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 14149,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given in parallelogram $ABCD$, point $E$ is the midpoint of side $BC$. A point $F$ is arbitrarily chosen on side $AB$. The probability that the area ratio of $\\triangle ADF$ to $\\triangle BFE$ is not less than $1$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 14150,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given in the Cartesian coordinate system $xOy$, a line $l$ passing through a fixed point $P$ with an inclination angle of $\\alpha$ has the parametric equation: $$\\begin{cases} x=t\\cos\\alpha \\\\ y=-2+t\\sin\\alpha \\end{cases}$$ (where $t$ is the parameter). In the polar co... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 14151,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given in the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\\begin{cases} x=3+5\\cos \\alpha \\\\ y=4+5\\sin \\alpha \\end{cases}$, ($\\alpha$ is the parameter), points $A$ and $B$ are on curve $C$. With the origin $O$ as the pole and the positive ... | math | {
"style": "rule",
"ground_truth": "\\frac{25\\sqrt{3}}{4}"
} | {
"index": 14152,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given in the polar coordinate system, circle $C$: $p=2\\cos (\\theta+ \\frac {\\pi}{2})$ and line $l$: $\\rho\\sin (\\theta+ \\frac {\\pi}{4})= \\sqrt {2}$, point $M$ is a moving point on circle $C$. Find the maximum distance from point $M$ to line $l$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3 \\sqrt {2}}{2}+1"
} | {
"index": 14153,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given in the polar coordinate system, point P moves on the curve $\\rho^2\\cos\\theta-2\\rho=0$, the minimum distance from point P to point $Q(1, \\frac{\\pi}{3})$ is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 14154,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given in the polar coordinate system, the equation of curve Ω is $\\rho=6\\cos\\theta$. Taking the pole as the origin of the Cartesian coordinate system, with the polar axis as the positive half-axis of the x-axis, and using the same unit of length in both coordinate systems, e... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 14155,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given in the tetrahedron P-ABC, PA is perpendicular to the plane ABC, AB=AC=PA=2, and in triangle ABC, ∠BAC=120°, then the volume of the circumscribed sphere of the tetrahedron P-ABC is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{20\\sqrt{5}\\pi}{3}"
} | {
"index": 14156,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given in triangle $\\triangle ABC$ the internal angles are $A$, $B$, and $C$, and the centroid is $G$. If $2\\sin A \\cdot \\overrightarrow{GA} + \\sqrt{3}\\sin B \\cdot \\overrightarrow{GB} + 3\\sin C \\cdot \\overrightarrow{GC} = \\overrightarrow{0}$, then $\\cos B = \\_\\_\\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"index": 14157,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given integer $n\\geq 2$. Find the minimum value of $\\lambda {}$, satisfy that for any real numbers $a_1$, $a_2$, $\\cdots$, ${a_n}$ and ${b}$,\n$$\\lambda\\sum\\limits_{i=1}^n\\sqrt{|a_i-b|}+\\sqrt{n\\left|\\sum\\limits_{i=1}^na_i\\right|}\\geqslant\\sum\\limits_{i=1}^n\\sqrt... | math | {
"style": "rule",
"ground_truth": "\\frac{n-1 + \\sqrt{n-1}}{\\sqrt{n}}"
} | {
"index": 14158,
"solution": "\nGiven an integer \\( n \\geq 2 \\), we aim to find the minimum value of \\( \\lambda \\) such that for any real numbers \\( a_1, a_2, \\ldots, a_n \\) and \\( b \\), the following inequality holds:\n\\[\n\\lambda \\sum_{i=1}^n \\sqrt{|a_i - b|} + \\sqrt{n \\left| \\sum_{i=1}^n a_i \... |
DeepScaleR | [
{
"role": "user",
"content": "Given integers $a$ and $b$ satisfy: $a-b$ is a prime number, and $ab$ is a perfect square. When $a \\geq 2012$, find the minimum value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "2025"
} | {
"index": 14159,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given integers $a$ and $b$ whose sum is $20$, determine the number of distinct integers that can be written as the sum of two, not necessarily different, fractions $\\frac{a}{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 14160,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given integers $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $\\ldots $ satisfy the following conditions: $a_{1}=0$, $a_{2}=-|a+1|$, $a_{3}=-|a_{2}+2|$, $a_{4}=-|a_{3}+3|$, and so on, then the value of $a_{2022}$ is ____."
}
] | math | {
"style": "rule",
"ground_truth": "-1011"
} | {
"index": 14161,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given integers $x$ and $y$ satisfy the equation $2xy + x + y = 83$, find the values of $x + y$."
}
] | math | {
"style": "rule",
"ground_truth": "-85"
} | {
"index": 14162,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given is a isosceles triangle ABC so that AB=BC. Point K is in ABC, so that CK=AB=BC and <KAC=30°.Find <AKB=?"
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 14163,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given is a regular tetrahedron of volume 1. We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 14164,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given line segments $OA$, $OB$, $OC$ are pairwise perpendicular, with $OA=1$, $OB=1$, $OC=2$. If the projections of line segments $OA$, $OB$, $OC$ on line $OP$ have equal lengths, then the length of these projections is $\\_\\_\\_\\_\\_\\_.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 14165,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given lines $l_{1}$: $(3+a)x+4y=5-3a$ and $l_{2}$: $2x+(5+a)y=8$, find the value of $a$ such that the lines are parallel."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 14166,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given lines $l_{1}$: $\\rho\\sin(\\theta-\\frac{\\pi}{3})=\\sqrt{3}$ and $l_{2}$: $\\begin{cases} x=-t \\\\ y=\\sqrt{3}t \\end{cases}$ (where $t$ is a parameter), find the polar coordinates of the intersection point $P$ of $l_{1}$ and $l_{2}$. Additionally, three points $A$, $B... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{8}"
} | {
"index": 14167,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given natural numbers \\( m \\) and \\( n \\) where \\( n > m > 1 \\), the last three digits of the decimal representation of \\( 1978^m \\) and \\( 1978^n \\) are the same. Find \\( m \\) and \\( n \\) such that \\( m+n \\) is minimized."
}
] | math | {
"style": "rule",
"ground_truth": "106"
} | {
"index": 14168,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given non-negative numbers $x$, $y$, and $z$ such that $x+y+z=2$, determine the minimum value of $\\frac{1}{3}x^3+y^2+z$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{12}"
} | {
"index": 14169,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given non-negative real numbers $a$, $b$, $c$ satisfy $\\frac{a-1}{2}=\\frac{b-2}{3}=\\frac{3-c}{4}$, let the maximum value of $S=a+2b+c$ be $m$, and the minimum value be $n$. Then the value of $\\frac{n}{m}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{11}"
} | {
"index": 14170,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given non-negative real numbers $a_{1}, a_{2}, \\cdots, a_{2008}$ whose sum equals 1, determine the maximum value of $a_{1} a_{2} + a_{2} a_{3} + \\cdots + a_{2007} a_{2008} + a_{2008} a_{1}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 14171,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given non-negative real numbers $x_{1}, x_{2}, \\cdots, x_{n}$ satisfying the inequality $x_{1} + x_{2} + \\cdots + x_{n} \\leqslant \\frac{1}{2}$, find the minimum value of $f\\left(x_{1}, x_{2}, \\cdots, x_{n}\\right) = \\prod_{i=1}^{n}\\left(1 - x_{i}\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 14172,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given non-negative real numbers \\( a, b, c, x, y, z \\) that satisfy \\( a + b + c = x + y + z = 1 \\), find the minimum value of \\( \\left(a - x^{2}\\right)\\left(b - y^{2}\\right)\\left(c - z^{2}\\right) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-1/4"
} | {
"index": 14173,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given non-zero complex numbers \\( x \\) and \\( y \\) satisfying\n\\[ y^{2}(x^{2}-xy+y^{2})+x^{3}(x-y)=0, \\]\nfind the value of\n\\[ \\sum_{m=0}^{29} \\sum_{n=0}^{29} x^{18mn} y^{-18mn}. \\]"
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"index": 14174,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given non-zero plane vectors $\\overrightarrow{a}$, $\\overrightarrow{b}$, $\\overrightarrow{c}$ satisfy $|\\overrightarrow{a}|=2$, $|\\overrightarrow{b}-\\overrightarrow{c}|=1$. If the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is $\\frac{π}{3}$, calculate t... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} - 1"
} | {
"index": 14175,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given non-zero vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=2|\\overrightarrow{b}|$, and $(\\overrightarrow{a}-\\overrightarrow{b})\\bot \\overrightarrow{b}$, calculate the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$."... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 14176,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given non-zero vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=2|\\overrightarrow{b}|$, and $(\\overrightarrow{a}-\\overrightarrow{b})\\bot \\overrightarrow{b}$, find the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$."
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 14177,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given non-zero vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=2|\\overrightarrow{b}|$, and $(\\overrightarrow{a}-\\overrightarrow{b})\\bot \\overrightarrow{b}$, then the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is ___... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 14178,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given non-zero vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfying $\\overrightarrow{a}^{2}=(5 \\overrightarrow{a}-4 \\overrightarrow{b})\\cdot \\overrightarrow{b}$, find the minimum value of $\\cos < \\overrightarrow{a}, \\overrightarrow{b} >$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {4}{5}"
} | {
"index": 14179,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given non-zero vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfying $|\\overrightarrow{a}| = |\\overrightarrow{b}| = |\\overrightarrow{a} + \\overrightarrow{b}|$, the cosine of the angle between $\\overrightarrow{a}$ and $2\\overrightarrow{a} - \\overrightarrow{b}... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{7}}{14}"
} | {
"index": 14180,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given non-zero vectors $\\overrightarrow{a}, \\overrightarrow{b}$, if $(\\overrightarrow{a} - 2\\overrightarrow{b}) \\perp \\overrightarrow{a}$ and $(\\overrightarrow{b} - 2\\overrightarrow{a}) \\perp \\overrightarrow{b}$, find the angle between $\\overrightarrow{a}$ and $\\ove... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 14181,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given non-zero vectors $a=(-x, x)$ and $b=(2x+3, 1)$, where $x \\in \\mathbb{R}$.\n\n$(1)$ If $a \\perp b$, find the value of $x$;\n\n$(2)$ If $a \\nparallel b$, find $|a - b|$."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"index": 14182,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given non-zero vectors \\\\(a\\\\) and \\\\(b\\\\) satisfying \\\\(|b|=2|a|\\\\) and \\\\(a \\perp (\\sqrt{3}a+b)\\\\), find the angle between \\\\(a\\\\) and \\\\(b\\\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{5\\pi}{6}"
} | {
"index": 14183,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given numbers $5, 6, 7, 8, 9, 10, 11, 12, 13$ are written in a $3\\times3$ array, with the condition that two consecutive numbers must share an edge. If the sum of the numbers in the four corners is $32$, calculate the number in the center of the array."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 14184,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given numbers \\( x_{1}, \\cdots, x_{1991} \\) satisfy the condition\n$$\n\\left|x_{1}-x_{2}\\right|+\\cdots+\\left|x_{1990}-x_{1991}\\right|=1991 ,\n$$\n\nwhere \\( y_{k}=\\frac{1}{k}\\left(x_{1}+\\cdots+x_{k}\\right) \\) for \\( k = 1, \\cdots, 1991 \\). Find the maximum poss... | math | {
"style": "rule",
"ground_truth": "1990"
} | {
"index": 14185,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given numbers \\( x_{1}, \\ldots, x_{n} \\in (0,1) \\), find the maximum value of the expression\n\n$$\nA = \\frac{\\sqrt[4]{1-x_{1}} + \\ldots + \\sqrt[4]{1-x_{n}}}{\\frac{1}{\\sqrt[4]{x_{1}}} + \\ldots + \\frac{1}{\\sqrt[4]{x_{n}}}}\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 14186,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given numbers \\(a, b, c\\) satisfy \\(a b c+a+c-b\\). Then the maximum value of the algebraic expression \\(\\frac{1}{1+a^{2}}-\\frac{1}{1+b^{2}}+\\frac{1}{1+c^{2}}\\) is"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"index": 14187,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given one hundred numbers: \\(1, \\frac{1}{2}, \\frac{1}{3}, \\frac{1}{4}, \\ldots, \\frac{1}{100}\\). We compute 98 differences: \\(a_{1} = 1 - \\frac{1}{3}, a_{2} = \\frac{1}{2} - \\frac{1}{4}, \\ldots, a_{98} = \\frac{1}{98} - \\frac{1}{100}\\). What is the sum of all these ... | math | {
"style": "rule",
"ground_truth": "\\frac{14651}{9900}"
} | {
"index": 14188,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given parallelogram $ABCD$ with $E$ the midpoint of diagonal $BD$. Point $E$ is connected to a point $F$ in $DA$ so that $DF=\\frac{1}{3}DA$. What is the ratio of the area of $\\triangle DFE$ to the area of quadrilateral $ABEF$?"
}
] | math | {
"style": "rule",
"ground_truth": "1/5"
} | {
"index": 14189,
"solution": "1. **Assumption of Coordinates**: Assume $ABCD$ is a unit square for simplicity, with coordinates $A(0, 0)$, $B(0, 1)$, $C(1, 1)$, and $D(1, 0)$. \n\n2. **Locating Point $E$**: Since $E$ is the midpoint of diagonal $BD$, we find $E$ by averaging the coordinates of $B$ and $D$. Thus, $... |
DeepScaleR | [
{
"role": "user",
"content": "Given plane vectors $\\overrightarrow{a}$, $\\overrightarrow{b}$, $\\overrightarrow{c}$ satisfy $\\langle \\overrightarrow{a}, \\overrightarrow{b} \\rangle = 60^\\circ$, and $\\{|\\overrightarrow{a}|, |\\overrightarrow{b}|, |\\overrightarrow{c}|\\} = \\{1, 2, 3\\}$, calculate t... | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}+3"
} | {
"index": 14190,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given plane vectors $\\overrightarrow{a}=(-1,2)$ and $\\overrightarrow{b}=(1,-4)$. \n$(1)$ If $4\\overrightarrow{a}+\\overrightarrow{b}$ is perpendicular to $k\\overrightarrow{a}-\\overrightarrow{b}$, find the value of the real number $k$. \n$(2)$ If $\\theta$ is the angle be... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{4}"
} | {
"index": 14191,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given plane vectors $\\vec{a}, \\vec{b}, \\vec{c}$ satisfying $|\\vec{a}| = |\\vec{b}|$, $\\vec{c} = \\lambda \\vec{a} + \\mu \\vec{b}$, $|\\vec{c}| = 1 + \\vec{a} \\cdot \\vec{b}$, and $(\\vec{a} + \\vec{b}) \\cdot \\vec{c} = 1$, find the minimum possible value of $\\frac{|\\v... | math | {
"style": "rule",
"ground_truth": "2 - \\sqrt{2}"
} | {
"index": 14192,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given plane vectors $\\vec{a}, \\vec{b}, \\vec{c}$ that satisfy the following conditions: $|\\vec{a}| = |\\vec{b}| \\neq 0$, $\\vec{a} \\perp \\vec{b}$, $|\\vec{c}| = 2 \\sqrt{2}$, and $|\\vec{c} - \\vec{a}| = 1$, determine the maximum possible value of $|\\vec{a} + \\vec{b} - ... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"index": 14193,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point $A$ is on line segment $BC$ (excluding endpoints), and $O$ is a point outside line $BC$, with $\\overrightarrow{OA} - 2a \\overrightarrow{OB} - b \\overrightarrow{OC} = \\overrightarrow{0}$, then the minimum value of $\\frac{a}{a+2b} + \\frac{2b}{1+b}$ is \\_\\_\\_\... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{2} - 2"
} | {
"index": 14194,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point $A(0,2)$, and $P$ is any point on the ellipse $\\frac{x^2}{4}+y^2=1$, then the maximum value of $|PA|$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{21}}{3}"
} | {
"index": 14195,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point $A(1,2)$ and circle $C: x^{2}+y^{2}+2mx+2y+2=0$. \n$(1)$ If there are two tangents passing through point $A$, find the range of $m$. \n$(2)$ When $m=-2$, a point $P$ on the line $2x-y+3=0$ is chosen to form two tangents $PM$ and $PN$ to the circle. Find the minimu... | math | {
"style": "rule",
"ground_truth": "\\frac{7\\sqrt{15}}{5}"
} | {
"index": 14196,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point $A(3,0)$, $\\overrightarrow{EA}=(2,1)$, $\\overrightarrow{EF}=(1,2)$, and point $P(2,0)$ satisfies $\\overrightarrow{EP}=λ \\overrightarrow{EA}+μ \\overrightarrow{EF}$, find the value of $λ+μ$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 14197,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point $F(0,1)$, moving point $M$ lies on the line $l:y=-1$. The line passing through point $M$ and perpendicular to the $x$-axis intersects the perpendicular bisector of segment $MF$ at point $P$. Let the locus of point $P$ be curve $C$. \n$(1)$ Find the equation of curv... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 14198,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given point $M(2,0)$, draw two tangent lines $MA$ and $MB$ from $M$ to the circle $x^{2}+y^{2}=1$, where $A$ and $B$ are the tangent points. Calculate the dot product of vectors $\\overrightarrow{MA}$ and $\\overrightarrow{MB}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{2}"
} | {
"index": 14199,
"solution": ""
} |
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