data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse in the Cartesian coordinate system $xoy$, its center is at the origin, the left focus is $F(-\\sqrt{3},0)$, and the right vertex is $D(2,0)$. Let point $A(1,\\frac{1}{2})$.\n\n(1) Find the standard equation of the ellipse;\n(2) If $P$ is a moving point on the e... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 14000,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse with $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$), its left focal point is $F_1(-1, 0)$, and vertex P on the ellipse satisfies $\\angle PF_1O = 45^\\circ$ (where O is the origin).\n(1) Determine the values of $a$ and $b$;\n(2) Given that line $l_1: y... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 14001,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse with a chord passing through the focus and perpendicular to the major axis of length $\\sqrt{2}$, and a distance from the focus to the corresponding directrix of $1$, determine the eccentricity of the ellipse."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 14002,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse with foci at $(1,0)$ and $(-1,0)$, it intersects with the line $y=x-2$. Determine the maximum value of the eccentricity of this ellipse."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{5}"
} | {
"index": 14003,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse with its foci on the $y$-axis, an eccentricity of $\\frac{2\\sqrt{2}}{3}$, and one focus at $(0, 2\\sqrt{2})$.\n(1) Find the standard equation of the ellipse;\n(2) A moving line $l$ passes through point $P(-1,0)$, intersecting a circle $O$ centered at the origi... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{62}}{2}"
} | {
"index": 14004,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse with its foci on the x-axis and its lower vertex at D(0, -1), the eccentricity of the ellipse is $e = \\frac{\\sqrt{6}}{3}$. A line L passes through the point P(0, 2).\n(Ⅰ) Find the standard equation of the ellipse.\n(Ⅱ) If line L is tangent to the ellipse, fin... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{4}"
} | {
"index": 14005,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse with its focus on the $y$-axis $\\frac{x^2}{m^2} + \\frac{y^2}{4} = 1$ ($m > 0$) and eccentricity $e = \\frac{1}{2}$, where $A$ is the right vertex of the ellipse and $P$ is any point on the ellipse. Find the maximum value of $|PA|$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 14006,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse with the equation $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ and an eccentricity of $\\frac{1}{2}$. $F\\_1$ and $F\\_2$ are the left and right foci of the ellipse, respectively. A line $l$ passing through $F\\_2$ intersects the ellipse at points... | math | {
"style": "rule",
"ground_truth": "m = 4"
} | {
"index": 14007,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse with the equation \\\\(\\\\dfrac{x^{2}}{a^{2}}+\\\\dfrac{y^{2}}{b^{2}}=1(a > b > 0)\\\\) and an eccentricity of \\\\(\\\\dfrac{\\\\sqrt{3}}{2}\\\\). A line $l$ is drawn through one of the foci of the ellipse, perpendicular to the $x$-axis, and intersects the el... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 14008,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an ellipse, if its two foci and the two vertices on its minor axis form a square, calculate its eccentricity."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{2}}{2}"
} | {
"index": 14009,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an equilateral triangle $ABC$ with a circle of radius $3$ tangent to line $AB$ at $B$ and to line $AC$ at $C$, find the area of the circle that passes through vertices $A$, $B$, and $C$."
}
] | math | {
"style": "rule",
"ground_truth": "36\\pi"
} | {
"index": 14010,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an equilateral triangle $DEF$ with $DE = DF = EF = 8$ units and a circle with radius $4$ units tangent to line $DE$ at $E$ and line $DF$ at $F$, calculate the area of the circle passing through vertices $D$, $E$, and $F$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{64\\pi}{3}"
} | {
"index": 14011,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an equilateral triangle $PQR$ with a side length of 8 units, a process similar to the previous one is applied, but here each time, the triangle is divided into three smaller equilateral triangles by joining the midpoints of its sides, and the middle triangle is shaded eac... | math | {
"style": "rule",
"ground_truth": "8\\sqrt{3}"
} | {
"index": 14012,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an equilateral triangle ABC, a student starts from point A and moves the chess piece using a dice-rolling method, where the direction of the movement is determined by the dice roll. Each time the dice is rolled, the chess piece is moved from one vertex of the triangle to ... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{64}"
} | {
"index": 14013,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an equilateral triangle \\( ABC \\). Point \\( K \\) is the midpoint of side \\( AB \\), and point \\( M \\) lies on side \\( BC \\) such that \\( BM : MC = 1 : 3 \\). A point \\( P \\) is chosen on side \\( AC \\) such that the perimeter of triangle \\( PKM \\) is minimi... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"index": 14014,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an equilateral triangle \\( \\triangle ABC \\) with a side length of 1,\n\\[\n\\overrightarrow{AP} = \\frac{1}{3}(\\overrightarrow{AB} + \\overrightarrow{AC}), \\quad \\overrightarrow{AQ} = \\overrightarrow{AP} + \\frac{1}{2}\\overrightarrow{BC}.\n\\]\n\nFind the area of ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{12}"
} | {
"index": 14015,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an equilateral triangle of side 10, divide each side into three equal parts, construct an equilateral triangle on the middle part, and then delete the middle part. Repeat this step for each side of the resulting polygon. Find \\( S^2 \\), where \\( S \\) is the area of th... | math | {
"style": "rule",
"ground_truth": "4800"
} | {
"index": 14016,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an equilateral triangle with one vertex at the origin and the other two vertices on the parabola $y^2 = 2\\sqrt{3}x$, find the length of the side of this equilateral triangle."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 14017,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an equilateral triangle with side length \\( n \\), divided into unit triangles as illustrated, let \\( f(n) \\) be the number of paths from the top-row triangle to the triangle in the center of the bottom row. The path must move through adjacent triangles sharing a commo... | math | {
"style": "rule",
"ground_truth": "2011!"
} | {
"index": 14018,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an equilateral triangle ∆ABC with side length 6, where all three vertices lie on the surface of sphere O with O as the center, and the angle between OA and plane ABC is 45°, find the surface area of sphere O."
}
] | math | {
"style": "rule",
"ground_truth": "96\\pi"
} | {
"index": 14019,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an even function $f(x)$ defined on $\\mathbb{R}$, for $x \\geq 0$, $f(x) = x^2 - 4x$ \n(1) Find the value of $f(-2)$; \n(2) For $x < 0$, find the expression for $f(x)$; \n(3) Let the maximum value of the function $f(x)$ on the interval $[t-1, t+1]$ (where $t > 1$) be $... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 14020,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an geometric sequence \\\\(\\{a_n\\}\\) with a common ratio less than \\\\(1\\\\), the sum of the first \\\\(n\\\\) terms is \\\\(S_n\\\\), and \\\\(a_1 = \\frac{1}{2}\\\\), \\\\(7a_2 = 2S_3\\\\).\n\n\\\\((1)\\\\) Find the general formula for the sequence \\\\(\\{a_n\\}\\... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 14021,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an increasing geometric sequence $\\{a_{n}\\}$ with a common ratio greater than $1$ such that $a_{2}+a_{4}=20$, $a_{3}=8$.<br/>$(1)$ Find the general formula for $\\{a_{n}\\}$;<br/>$(2)$ Let $b_{m}$ be the number of terms of $\\{a_{n}\\}$ in the interval $\\left(0,m\\righ... | math | {
"style": "rule",
"ground_truth": "480"
} | {
"index": 14022,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an increasing sequence $\\{a_n\\}$ that satisfies $a_{n+1}a_{n-1} = a_n^2$ (for $n \\geq 2$, $n \\in \\mathbb{N}$), the sum of the first 10 terms equals 50, and the sum of the first 15 terms is 210, calculate the sum of the first 5 terms ($S_5$)."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 14023,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an increasing sequence $\\{a_n\\}$ with $2017$ terms, and all terms are non-zero, $a_{2017}=1$. If two terms $a_i$, $a_j$ are arbitrarily chosen from $\\{a_n\\}$, when $i < j$, $a_j-a_i$ is still a term in the sequence $\\{a_n\\}$. Then, the sum of all terms in the sequen... | math | {
"style": "rule",
"ground_truth": "1009"
} | {
"index": 14024,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an increasing sequence $\\{a_{n}\\}$ where all terms are positive integers, the sum of the first $n$ terms is $S_{n}$. If $a_{1}=3$ and $S_{n}=2023$, calculate the value of $a_{n}$ when $n$ takes its maximum value."
}
] | math | {
"style": "rule",
"ground_truth": "73"
} | {
"index": 14025,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an infinite geometric sequence $\\{a_n\\}$, the product of its first $n$ terms is $T_n$, and $a_1 > 1$, $a_{2008}a_{2009} > 1$, $(a_{2008} - 1)(a_{2009} - 1) < 0$, determine the maximum positive integer $n$ for which $T_n > 1$."
}
] | math | {
"style": "rule",
"ground_truth": "4016"
} | {
"index": 14026,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an infinite grid where each cell is either red or blue, such that in any \\(2 \\times 3\\) rectangle exactly two cells are red, determine how many red cells are in a \\(9 \\times 11\\) rectangle."
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 14027,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an integer $k\\geq 2$, determine all functions $f$ from the positive integers into themselves such that $f(x_1)!+f(x_2)!+\\cdots f(x_k)!$ is divisibe by $x_1!+x_2!+\\cdots x_k!$ for all positive integers $x_1,x_2,\\cdots x_k$.\n\n$Albania$"
}
] | math | {
"style": "rule",
"ground_truth": "f(n) = n"
} | {
"index": 14028,
"solution": "\nConsider the function \\( f \\) from the positive integers into themselves. We need to determine all such functions such that for every collection of positive integers \\( x_1, x_2, \\ldots, x_k \\), the expression\n\n\\[\nf(x_1)! + f(x_2)! + \\cdots + f(x_k)!\n\\]\n\nis divisible b... |
DeepScaleR | [
{
"role": "user",
"content": "Given an integer \\( n > 4 \\), the coefficients of the terms \\( x^{n-4} \\) and \\( xy \\) in the expansion of \\( (x + 2 \\sqrt{y} - 1)^n \\) are equal. Find the value of \\( n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 14029,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an integer \\( n \\) with \\( n \\geq 2 \\), determine the smallest constant \\( c \\) such that the inequality \\(\\sum_{1 \\leq i \\leq j \\leq n} x_{i} x_{j}\\left(x_{i}^{2}+x_{j}^{2}\\right) \\leq c\\left(\\sum_{i=1}^{n} x_{i}\\right)^{4}\\) holds for all non-negative... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 14030,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an integer \\( n \\geq 2 \\), let \\( f(n) \\) be the second largest positive divisor of \\( n \\). For example, \\( f(12)=6 \\) and \\( f(13)=1 \\). Determine the largest positive integer \\( n \\) such that \\( f(n)=35 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "175"
} | {
"index": 14031,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an integer coefficient polynomial\n$$\nf(x)=x^{5}+a_{1} x^{4}+a_{2} x^{3}+a_{3} x^{2}+a_{4} x+a_{5}\n$$\n\nIf $f(\\sqrt{3}+\\sqrt{2})=0$ and $f(1)+f(3)=0$, then $f(-1)=$ $\\qquad$"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 14032,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an integer sequence \\(\\{a_i\\}\\) defined as follows:\n\\[ a_i = \\begin{cases} \ni, & \\text{if } 1 \\leq i \\leq 5; \\\\\na_1 a_2 \\cdots a_{i-1} - 1, & \\text{if } i > 5.\n\\end{cases} \\]\n\nFind the value of \\(\\sum_{i=1}^{2019} a_i^2 - a_1 a_2 \\cdots a_{2019}\\)... | math | {
"style": "rule",
"ground_truth": "1949"
} | {
"index": 14033,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an inverted cone with a base radius of $15 \\mathrm{cm}$ and a height of $15 \\mathrm{cm}$, and a cylinder with a horizontal base radius of $18 \\mathrm{cm}$, determine the height in centimeters of the water in the cylinder after $10\\%$ of the water is lost from the cone... | math | {
"style": "rule",
"ground_truth": "3.125"
} | {
"index": 14034,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an isolated ground state iron atom with atomic number Z = 26, determine the number of orbitals that contain one or more electrons."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 14035,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an isosceles right triangle \\(ABC\\) with hypotenuse \\(AB\\). Point \\(M\\) is the midpoint of side \\(BC\\). A point \\(K\\) is chosen on the smaller arc \\(AC\\) of the circumcircle of triangle \\(ABC\\). Point \\(H\\) is the foot of the perpendicular dropped from \\(... | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"index": 14036,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an isosceles right triangle ∆PQR with hypotenuse PQ = 4√2, let S be the midpoint of PR. On segment QR, point T divides it so that QT:TR = 2:1. Calculate the area of ∆PST."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"index": 14037,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an isosceles trapezoid \\(ABCD\\), where \\(AD \\parallel BC\\), \\(BC = 2AD = 4\\), \\(\\angle ABC = 60^\\circ\\), and \\(\\overrightarrow{CE} = \\frac{1}{3} \\overrightarrow{CD}\\), calculate the value of \\(\\overrightarrow{CA} \\cdot \\overrightarrow{BE}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"index": 14038,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an isosceles trapezoid \\(ABCD\\), where \\(AD \\parallel BC\\), \\(BC = 2AD = 4\\), \\(\\angle ABC = 60^\\circ\\), and \\(\\overrightarrow{CE} = \\frac{1}{3} \\overrightarrow{CD}\\), find \\(\\overrightarrow{CA} \\cdot \\overrightarrow{BE}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"index": 14039,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an isosceles trapezoid with \\(AB = 24\\) units, \\(CD = 10\\) units, and legs \\(AD\\) and \\(BC\\) each measuring \\(13\\) units. Find the length of diagonal \\(AC\\)."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 14040,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an isosceles triangle $ABC$ satisfying $AB=AC$, $\\sqrt{3}BC=2AB$, and point $D$ is on side $BC$ with $AD=BD$, then the value of $\\sin \\angle ADB$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{2}}{3}"
} | {
"index": 14041,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an isosceles triangle $ABC$ with $AB = AC = 40$ units and $BC = 24$ units, let $CX$ be the angle bisector of $\\angle BCA$. Find the ratio of the area of $\\triangle BCX$ to the area of $\\triangle ACX$. Provide your answer as a simplified fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 14042,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an isosceles triangle $PQR$ with $PQ=QR$ and the angle at the vertex $108^\\circ$. Point $O$ is located inside the triangle $PQR$ such that $\\angle ORP=30^\\circ$ and $\\angle OPR=24^\\circ$. Find the measure of the angle $\\angle QOR$."
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"index": 14043,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an isosceles triangle DEF with DE = DF = 5√3, a circle with radius 6 is tangent to DE at E and to DF at F. If the altitude from D to EF intersects the circle at its center, find the area of the circle that passes through vertices D, E, and F."
}
] | math | {
"style": "rule",
"ground_truth": "36\\pi"
} | {
"index": 14044,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an isosceles triangle \\( A B C \\) where \\( A B = A C \\) and \\( \\angle A B C = 53^\\circ \\). Point \\( K \\) is such that \\( C \\) is the midpoint of \\( A K \\). Point \\( M \\) is chosen such that:\n\n- \\( B \\) and \\( M \\) are on the same side of the line \\(... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 14045,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an isosceles triangle \\( \\triangle ABC \\) with base angles \\( \\angle ABC = \\angle ACB = 50^\\circ \\), points \\( D \\) and \\( E \\) lie on \\( BC \\) and \\( AC \\) respectively. Lines \\( AD \\) and \\( BE \\) intersect at point \\( P \\). Given \\( \\angle ABE =... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 14046,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an isosceles triangle \\(ABC\\) with \\(AB = AC\\) and \\(\\angle ABC = 53^\\circ\\), find the measure of \\(\\angle BAM\\). Point \\(K\\) is such that \\(C\\) is the midpoint of segment \\(AK\\). Point \\(M\\) is chosen such that:\n\n- \\(B\\) and \\(M\\) are on the same... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 14047,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an isosceles triangle \\(ABC\\) with \\(\\angle A = 30^\\circ\\) and \\(AB = AC\\). Point \\(D\\) is the midpoint of \\(BC\\). Point \\(P\\) is chosen on segment \\(AD\\), and point \\(Q\\) is chosen on side \\(AB\\) such that \\(PB = PQ\\). What is the measure of angle \... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 14048,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an isosceles triangle \\(XYZ\\) with \\(XY = YZ\\) and an angle at the vertex equal to \\(96^{\\circ}\\). Point \\(O\\) is located inside triangle \\(XYZ\\) such that \\(\\angle OZX = 30^{\\circ}\\) and \\(\\angle OXZ = 18^{\\circ}\\). Find the measure of angle \\(\\angle... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"index": 14049,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an isosceles triangle with a vertex angle of 36°, the ratio of the base to the leg is equal to ."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}-1}{2}"
} | {
"index": 14050,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an isosceles triangle with side lengths of $4x-2$, $x+1$, and $15-6x$, its perimeter is ____."
}
] | math | {
"style": "rule",
"ground_truth": "12.3"
} | {
"index": 14051,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an obtuse triangle \\( \\triangle ABC \\) with the following conditions:\n1. The lengths of \\( AB \\), \\( BC \\), and \\( CA \\) are positive integers.\n2. The lengths of \\( AB \\), \\( BC \\), and \\( CA \\) do not exceed 50.\n3. The lengths of \\( AB \\), \\( BC \\),... | math | {
"style": "rule",
"ground_truth": "157"
} | {
"index": 14052,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an obtuse triangle \\(ABC\\) with obtuse angle \\(C\\). Points \\(P\\) and \\(Q\\) are marked on its sides \\(AB\\) and \\(BC\\) respectively, such that \\(\\angle ACP = CPQ = 90^\\circ\\). Find the length of segment \\(PQ\\) if it is known that \\(AC = 25\\), \\(CP = 20\... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 14053,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an odd function $f(x)$ defined on $R$ satisfying $f(x+1)=f(1-x)$, when $0\\leqslant x\\leqslant 1$, $f(x)=x^{3}$, then $f(2019)=\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14054,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an odd function $f(x)$ defined on $\\mathbb{R}$ that satisfies $f(2-x) - f(x) = 0$, and $f(-1) = 1$, calculate the value of $f(1) + f(2) + f(3) + \\ldots + f(2010)$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14055,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an odd function $f(x)$ on $\\mathbb{R}$, for any $x \\in \\mathbb{R}$, $f(x+1) = -f(x)$, and when $x \\in (-1, 1)$, $f(x) = x$. Find the value of $f(3) + f(-7.5)$."
}
] | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"index": 14056,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given an odd function defined on $\\mathbb{R}$, when $x > 0$, $f(x)=x^{2}+2x-1$.\n(1) Find the value of $f(-3)$;\n(2) Find the analytic expression of the function $f(x)$."
}
] | math | {
"style": "rule",
"ground_truth": "-14"
} | {
"index": 14057,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given angles $\\alpha$ and $\\beta$ have their vertices at the origin of coordinates, and their initial sides coincide with the positive half-axis of the x-axis, $\\alpha, \\beta \\in (0, \\pi)$. The terminal side of angle $\\beta$ intersects the unit circle at a point whose x-... | math | {
"style": "rule",
"ground_truth": "\\frac{56}{65}"
} | {
"index": 14058,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given angles $α$ and $β$ satisfy $\\frac{\\tan α}{\\tan β} = \\frac{7}{13}$, and $\\sin(α+β) = \\frac{2}{3}$, find the value of $\\sin(α-β)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{5}"
} | {
"index": 14059,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given angles $α$ and $β$ whose vertices are at the origin of coordinates, and their initial sides coincide with the positive half-axis of $x$, $α$, $β$ $\\in(0,\\pi)$, the terminal side of angle $β$ intersects the unit circle at a point whose x-coordinate is $- \\dfrac{5}{13}$,... | math | {
"style": "rule",
"ground_truth": "\\dfrac{56}{65}"
} | {
"index": 14060,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given any number a from the set {1, 2, 3, ..., 99, 100} and any number b from the same set, calculate the probability that the last digit of 3^a + 7^b is 8."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"index": 14061,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given any point $P$ on the ellipse $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1\\; \\; (a > b > 0)$ with foci $F\\_{1}$ and $F\\_{2}$, if $\\angle PF\\_1F\\_2=\\alpha$, $\\angle PF\\_2F\\_1=\\beta$, $\\cos \\alpha= \\frac{ \\sqrt{5}}{5}$, and $\\sin (\\alpha+\\beta)= \\frac{3}... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{7}"
} | {
"index": 14062,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given any point $P$ on the line $l: x-y+4=0$, two tangent lines $AB$ are drawn to the circle $O: x^{2}+y^{2}=4$ with tangent points $A$ and $B$. The line $AB$ passes through a fixed point ______; let the midpoint of segment $AB$ be $Q$. The minimum distance from point $Q$ to th... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 14063,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given any positive integer, we can write the integer in base 12 and add together the digits of its base 12 representation. We perform this operation on the number $7^{6^{5^{3^{2^{1}}}}}$ repeatedly until a single base 12 digit remains. Find this digit."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 14064,
"solution": "For a positive integer $n$, let $s(n)$ be the sum of digits when $n$ is expressed in base 12. We claim that $s(n) \\equiv n(\\bmod 11)$ for all positive integers $n$. Indeed, if $n=d_{k} 12^{k}+d_{k-1} 12^{k-1}+\\cdots+d_{0}$ with each $d_{i}$ an integer between 0 and 11, inclusive, b... |
DeepScaleR | [
{
"role": "user",
"content": "Given any set $A = \\{a_1, a_2, a_3, a_4\\}$ of four distinct positive integers, we denote the sum $a_1 +a_2 +a_3 +a_4$ by $s_A$. Let $n_A$ denote the number of pairs $(i, j)$ with $1 \\leq i < j \\leq 4$ for which $a_i +a_j$ divides $s_A$. Find all sets $A$ of four distinct p... | math | {
"style": "rule",
"ground_truth": "\\{k, 5k, 7k, 11k\\} \\text{ and } \\{k, 11k, 19k, 29k\\}"
} | {
"index": 14065,
"solution": "\nLet \\( A = \\{ a_1, a_2, a_3, a_4 \\} \\) be a set of four distinct positive integers. We define \\( s_A = a_1 + a_2 + a_3 + a_4 \\) as the sum of these integers. We also define \\( n_A \\) as the number of pairs \\( (i, j) \\) with \\( 1 \\leq i < j \\leq 4 \\) such that \\( a_i +... |
DeepScaleR | [
{
"role": "user",
"content": "Given any two positive integers, a certain operation (denoted by the operator $\\oplus$) is defined as follows: when $m$ and $n$ are both positive even numbers or both positive odd numbers, $m \\oplus n = m + n$; when one of $m$ and $n$ is a positive even number and the other i... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 14066,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given any two positive real numbers $x$ and $y$, then $x \\, \\Diamond \\, y$ is a positive real number defined in terms of $x$ and $y$ by some fixed rule. Suppose the operation $x \\, \\Diamond \\, y$ satisfies the equations $(xy) \\, \\Diamond \\, y=x(y \\, \\Diamond \\, y)$... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 14067,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given any two positive real numbers $x$ and $y$, then $x \\diamond y$ is a positive real number defined in terms of $x$ and $y$ by some fixed rule. Suppose the operation $x \\diamond y$ satisfies the equations \\((x \\cdot y) \\diamond y=x(y \\diamond y)\\) and \\((x \\diamond ... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 14068,
"solution": "Note first that \\(x \\diamond 1=(x \\cdot 1) \\diamond 1=x \\cdot(1 \\diamond 1)=x \\cdot 1=x\\). Also, \\(x \\diamond x=(x \\diamond 1) \\diamond x=x \\diamond 1=x\\). Now, we have \\((x \\cdot y) \\diamond y=x \\cdot(y \\diamond y)=x \\cdot y\\). So \\(19 \\diamond 98=\\left(\\frac... |
DeepScaleR | [
{
"role": "user",
"content": "Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$ s that may occur among the $100$ numbers."
}
] | math | {
"style": "rule",
"ground_truth": "95"
} | {
"index": 14069,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given are real numbers $x, y$. For any pair of real numbers $a_{0}, a_{1}$, define a sequence by $a_{n+2}=x a_{n+1}+y a_{n}$ for $n \\geq 0$. Suppose that there exists a fixed nonnegative integer $m$ such that, for every choice of $a_{0}$ and $a_{1}$, the numbers $a_{m}, a_{m+1... | math | {
"style": "rule",
"ground_truth": "\\[ y = 0, 1, \\frac{1 + \\sqrt{5}}{2}, \\frac{1 - \\sqrt{5}}{2} \\]"
} | {
"index": 14070,
"solution": "Note that $x=1$ (or $x=0$ ), $y=0$ gives a constant sequence, so it will always have the desired property. Thus, $y=0$ is one possibility. For the rest of the proof, assume $y \\neq 0$. We will prove that $a_{m}$ and $a_{m+1}$ may take on any pair of values, for an appropriate choice ... |
DeepScaleR | [
{
"role": "user",
"content": "Given bed A has 600 plants, bed B has 500 plants, bed C has 400 plants, beds A and B share 60 plants, beds A and C share 80 plants, beds B and C share 40 plants, and beds A, B, and C share 20 plants collectively, calculate the total number of unique plants when considering just... | math | {
"style": "rule",
"ground_truth": "1340"
} | {
"index": 14071,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given circle $C$: $(x-1)^{2}+(y-2)^{2}=25$, and line $l$: $(2m+1)x+(m+1)y-7m-4=0$, if the length of the chord intercepted by line $l$ on circle $C$ is the shortest, then the value of $m$ is _____."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{4}"
} | {
"index": 14072,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given circle $C$: $(x-2)^2 + y^2 = 4$, and line $l$: $x - \\sqrt{3}y = 0$, the probability that the distance from point $A$ on circle $C$ to line $l$ is not greater than $1$ is $\\_\\_\\_\\_\\_\\_\\_.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 14073,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given circle $C$: $x^{2}+y^{2}+8x+ay-5=0$ passes through the focus of parabola $E$: $x^{2}=4y$. The length of the chord formed by the intersection of the directrix of parabola $E$ and circle $C$ is $\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{6}"
} | {
"index": 14074,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given circle $C: (x-2)^{2} + (y-2)^{2} = 8-m$, if circle $C$ has three common tangents with circle $D: (x+1)^{2} + (y+2)^{2} = 1$, then the value of $m$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"index": 14075,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given circle $C_1$: $(x-1)^2+(y-2)^2=1$ \n(1) Find the equation of the tangent line to circle $C_1$ passing through point $P(2,4)$. \n(2) If circle $C_1$ intersects with circle $C_2$: $(x+1)^2+(y-1)^2=4$ at points $A$ and $B$, find the length of segment $AB$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {4 \\sqrt {5}}{5}"
} | {
"index": 14076,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given circle $M$: $(x+1)^{2}+y^{2}=1$, and circle $N$: $(x-1)^{2}+y^{2}=9$, a moving circle $P$ is externally tangent to circle $M$ and internally tangent to circle $N$. The trajectory of the center of circle $P$ is curve $C$.\n$(1)$ Find the equation of $C$.\n$(2)$ Let $l$ be ... | math | {
"style": "rule",
"ground_truth": "\\dfrac {18}{7}"
} | {
"index": 14077,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given circle $O$ with radius $R$, the inscribed triangle $ABC$ is an acute scalene triangle, where $AB$ is the largest side. $AH_A, BH_B,CH_C$ are heights on $BC,CA,AB$. Let $D$ be the symmetric point of $H_A$ with respect to $H_BH_C$, $E$ be the symmetric point of $H_B$ with r... | math | {
"style": "rule",
"ground_truth": "R^2"
} | {
"index": 14078,
"solution": "\nGiven a circle \\( O \\) with radius \\( R \\), and an inscribed acute scalene triangle \\( ABC \\) where \\( AB \\) is the largest side, let \\( AH_A, BH_B, CH_C \\) be the altitudes from \\( A, B, C \\) to \\( BC, CA, AB \\) respectively. Let \\( D \\) be the symmetric point of \\... |
DeepScaleR | [
{
"role": "user",
"content": "Given circle $O$, points $E$ and $F$ are located such that $E$ and $F$ are on opposite sides of the diameter $\\overline{AB}$. If $\\angle AOE = 60^\\circ$ and $\\angle BOF = 30^\\circ$, find the ratio of the area of sector $EOF$ to the area of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 14079,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given circle $O$, points $E$ and $F$ are on the same side of diameter $\\overline{AB}$, $\\angle AOE = 60^\\circ$, and $\\angle FOB = 90^\\circ$. Calculate the ratio of the area of the smaller sector $EOF$ to the area of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"index": 14080,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given circle $O$: $x^{2}+y^{2}=10$, a line $l$ passing through point $P(-3,-4)$ intersects with circle $O$ at points $A$ and $B$. If the area of triangle $AOB$ is $5$, find the slope of line $l$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{2}"
} | {
"index": 14081,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given circle $O: x^2+y^2=r^2(r>0)$, $A(x_1, y_1)$, $B(x_2, y_2)$ are two points on circle $O$, satisfying $x_1+y_1=x_2+y_2=3$, $x_1x_2+y_1y_2=-\\frac{1}{2}r^2$, calculate the value of $r$."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"index": 14082,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given circle M: $(x+1)^2+y^2=1$, and circle N: $(x-1)^2+y^2=9$, a moving circle P is externally tangent to circle M and internally tangent to circle N. The trajectory of the center of circle P is curve C. \n(1) Find the equation of C: \n(2) Let $l$ be a line that is tangent t... | math | {
"style": "rule",
"ground_truth": "\\frac{18}{7}"
} | {
"index": 14083,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given circles $P$, $Q$, and $R$ each have radius 2, circle $P$ and $Q$ are tangent to each other, and circle $R$ is tangent to the midpoint of $\\overline{PQ}$. Calculate the area inside circle $R$ but outside circle $P$ and circle $Q$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"index": 14084,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given circles $P, Q,$ and $R$ where $P$ has a radius of 1 unit, $Q$ a radius of 2 units, and $R$ a radius of 1 unit. Circles $Q$ and $R$ are tangent to each other externally, and circle $R$ is tangent to circle $P$ externally. Compute the area inside circle $Q$ but outside circ... | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"index": 14085,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given circles ${C}_{1}:{x}^{2}+{y}^{2}=1$ and ${C}_{2}:(x-4)^{2}+(y-2)^{2}=1$, a moving point $M\\left(a,b\\right)$ is used to draw tangents $MA$ and $MB$ to circles $C_{1}$ and $C_{2}$ respectively, where $A$ and $B$ are the points of tangency. If $|MA|=|MB|$, calculate the mi... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{5}"
} | {
"index": 14086,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given complex number $z$, define sequence $z_{0}, z_{1}, z_{2}, \\ldots$ as $z_{0}=z$ and $z_{n+1}=2 z_{n}^{2}+2 z_{n}$ for $n \\geq 0$. Given that $z_{10}=2017$, find the minimum possible value of $|z|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt[1024]{4035}-1}{2}"
} | {
"index": 14087,
"solution": "Define $w_{n}=z_{n}+\\frac{1}{2}$, so $z_{n}=w_{n}-\\frac{1}{2}$, and the original equation becomes $$w_{n+1}-\\frac{1}{2}=2\\left(w_{n}-\\frac{1}{2}\\right)^{2}+2\\left(w_{n}-\\frac{1}{2}\\right)=2 w_{n}^{2}-\\frac{1}{2}$$ which reduces to $w_{n+1}=2 w_{n}^{2}$. it is not difficult t... |
DeepScaleR | [
{
"role": "user",
"content": "Given complex numbers $\\mathrm{z}_{\\mathrm{i}} (\\mathrm{i} = 1, 2, 3, 4, 5)$ satisfying: \n$$\\left\\{\\begin{array}{c}\n\\left|z_{1}\\right| \\leq 1, \\quad \\left|z_{2}\\right| \\leq 1 \\\\\n\\left|2 z_{3}-(\\mathrm{z}_{1}+\\mathrm{z}_{2})\\right| \\leq \\left|\\mathrm{z}_... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 14088,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given complex numbers $u$ and $v$ such that $|u+v| = 3$ and $|u^2 + v^2| = 10,$ determine the largest possible value of $|u^3 + v^3|.$"
}
] | math | {
"style": "rule",
"ground_truth": "31.5"
} | {
"index": 14089,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given complex numbers $w$ and $z$ such that $|w+z|=3$ and $|w^2+z^2|=18,$ find the smallest possible value of $|w^3+z^3|.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{81}{2}"
} | {
"index": 14090,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given complex numbers $z_1 = \\cos\\theta - i$ and $z_2 = \\sin\\theta + i$, the maximum value of the real part of $z_1 \\cdot z_2$ is \\_\\_\\_\\_\\_\\_, and the maximum value of the imaginary part is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 14091,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given complex numbers $z_{1}=m+i$ and $z_{2}=2+mi$, where $i$ is the imaginary unit and $m\\in R$. \n$(1)$ If $z_{1}\\cdot z_{2}$ is a pure imaginary number, find the value of $m$. \n$(2)$ If ${z}_{1}^{2}-2{z}_{1}+2=0$, find the imaginary part of $\\frac{{z}_{2}}{{z}_{1}}$."
... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 14092,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given complex numbers ${z_1}=2i$, ${z_2}=1-i$, where $i$ is the imaginary unit,\n\n(1) Find the conjugate of the complex number $\\frac{z_1}{z_2}$;\n\n(2) In the complex plane, let points $Z_1$, $Z_2$ correspond to ${z_1}$, ${z_2}$ respectively, and $O$ be the origin. Form a pa... | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"index": 14093,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given complex numbers \\( x \\) and \\( y \\), find the maximum value of \\(\\frac{|3x+4y|}{\\sqrt{|x|^{2} + |y|^{2} + \\left|x^{2}+y^{2}\\right|}}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{2}}{2}"
} | {
"index": 14094,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given complex numbers \\( z \\) and \\( \\omega \\) satisfying the following two conditions:\n1. \\( z + \\omega + 3 = 0 \\);\n2. \\( |z|, 2, |\\omega| \\) form an arithmetic sequence.\n\nIs there a maximum value for \\( \\cos(\\arg z - \\arg \\omega) \\)? If so, find it."
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 14095,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given complex numbers \\( z \\) and \\( \\omega \\) that satisfy the following two conditions:\n1. \\( z + \\omega + 3 = 0 \\);\n2. \\( | z |, 2, | \\omega | \\) form an arithmetic sequence.\nDoes \\( \\cos (\\arg z - \\arg \\omega) \\) have a maximum value? If it does, find th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 14096,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given complex numbers \\( z, z_{1}, z_{2} \\left( z_{1} \\neq z_{2} \\right) \\) such that \\( z_{1}^{2}=z_{2}^{2}=-2-2 \\sqrt{3} \\mathrm{i} \\), and \\(\\left|z-z_{1}\\right|=\\left|z-z_{2}\\right|=4\\), find \\(|z|=\\ \\ \\ \\ \\ .\\)"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 14097,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given complex numbers \\( z_{1}, z_{2}, z_{3} \\) such that \\( \\left|z_{1}\\right| \\leq 1 \\), \\( \\left|z_{2}\\right| \\leq 1 \\), and \\( \\left|2 z_{3}-\\left(z_{1}+z_{2}\\right)\\right| \\leq \\left|z_{1}-z_{2}\\right| \\). What is the maximum value of \\( \\left|z_{3}\... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 14098,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given complex numbers \\(z_1 = -3 + 4i\\) and \\(z_2 = 2 - i\\), find an equation in the form \\(az + b\\overline{z} = c\\) where \\(a\\), \\(b\\), and \\(c\\) are some complex numbers that represents the line joining \\(z_1\\) and \\(z_2\\), and determine the product \\(ab\\).... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 14099,
"solution": ""
} |
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