data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with a common ratio $q$, and the product of its first $n$ terms is $T_n$, where the first term $a_1 > 1$, and $a_{2014}a_{2015} - 1 > 0$, $\\frac{a_{2014} - 1}{a_{2015} - 1} < 0$, find the largest natural number $n$ such that $T_n > 1$."
... | math | {
"style": "rule",
"ground_truth": "4028"
} | {
"index": 13300,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with a common ratio $q=-5$, and $S_n$ denotes the sum of the first $n$ terms of the sequence, the value of $\\frac{S_{n+1}}{S_n}=\\boxed{?}$."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 13301,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with a common ratio of $2$ and the sum of the first $n$ terms denoted by $S_n$. If $a_2= \\frac{1}{2}$, find the expression for $a_n$ and the value of $S_5$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{31}{16}"
} | {
"index": 13302,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with a sum of the first $n$ terms as $S_n$, and $S_{10}:S_5 = 1:2$, find the value of $\\frac{S_5 + S_{10} + S_{15}}{S_{10} - S_5}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{9}{2}"
} | {
"index": 13303,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with all positive terms and $\\lg=6$, calculate the value of $a_1 \\cdot a_{15}$."
}
] | math | {
"style": "rule",
"ground_truth": "10^4"
} | {
"index": 13304,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with positive terms, the sum of the first $n$ terms is $S_n$. If $-3$, $S_5$, and $S_{10}$ form an arithmetic sequence, calculate the minimum value of $S_{15} - S_{10}$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 13305,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with the first term $\\frac{3}{2}$ and common ratio $- \\frac{1}{2}$, calculate the maximum value of the sum of the first $n$ terms, $S_n$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 13306,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with the first term $\\frac{3}{2}$ and common ratio $- \\frac{1}{2}$, the sum of the first $n$ terms is $S_n$. If for any $n \\in N^*$, it holds that $S_n - \\frac{1}{S_n} \\in [s, t]$, then the minimum value of $t-s$ is \\_\\_\\_\\_\\_\\_... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{12}"
} | {
"index": 13307,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with the first term $\\frac{3}{2}$ and common ratio $-\\frac{1}{2}$, and the sum of the first $n$ terms is $S_n$, then when $n\\in N^*$, the sum of the maximum and minimum values of $S_n - \\frac{1}{S_n}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 13308,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with the sum of its first n terms denoted as $S_n$, if $S_5$, $S_4$, and $S_6$ form an arithmetic sequence, determine the common ratio $q$ of the sequence $\\{a_n\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 13309,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$, the sum of the first $n$ terms is $S_n$. Given that $a_1 + a_2 + a_3 = 3$ and $a_4 + a_5 + a_6 = 6$, calculate the value of $S_{12}$."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 13310,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$, where $a_1+a_2+a_3=4$ and $a_2+a_3+a_4=-2$, find the value of $a_3+a_4+a_5+a_6+a_7+a_8$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{8}"
} | {
"index": 13311,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$, where $a_3$ and $a_7$ are the two roots of the quadratic equation $x^2+7x+9=0$, calculate the value of $a_5$."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 13312,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$, where the sum of the first $n$ terms is denoted as $S_n$, and $S_n = a\\left(\\frac{1}{4}\\right)^{n-1} + 6$, find the value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"index": 13313,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_{n}\\}$ with the sum of the first $n$ terms denoted as $S_{n}$, satisfying $S_{n} = 2^{n} + r$ (where $r$ is a constant). Define $b_{n} = 2\\left(1 + \\log_{2} a_{n}\\right)$ for $n \\in \\mathbf{N}^{*}$.\n\n(1) Find the sum of the first $n$ ter... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{2}}{4}"
} | {
"index": 13314,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_{n}\\}$, the sum of the first n terms is $S_{n}$, $S_{2}=7$ and $S_{6}=91$. Calculate $S_{4}$."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 13315,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence $a$, $a(a-1)$, $a(a-1)^{2}$, $...$, where $S_{n}$ represents the sum of its first $n$ terms.\n1) Find the range of real values for $a$ and the expression for $S_{n}$;\n2) Does there exist a real value of $a$ such that $S_{1}$, $S_{3}$, $S_{2}$ form an... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13316,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence \\(\\{a_n\\}\\) with the sum of the first \\(n\\) terms \\(S_n\\) such that \\(S_n = 2^n + r\\) (where \\(r\\) is a constant), let \\(b_n = 2(1 + \\log_2 a_n)\\) for \\(n \\in \\mathbb{N}^*\\).\n\n1. Find the sum of the first \\(n\\) terms of the sequ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4} \\sqrt{2}"
} | {
"index": 13317,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence with positive terms $\\{a_n\\}$ and a common ratio of $2$, if $a_ma_n=4a_2^2$, then the minimum value of $\\frac{2}{m}+ \\frac{1}{2n}$ equals \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 13318,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence with positive terms $\\{a_n\\}$, let $S_n$ be the sum of the first $n$ terms. If $S_3 + a_2 = 9a_3$, determine the common ratio."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13319,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric sequence {a_n} satisfies a_1 = 3, and a_1 + a_3 + a_5 = 21, find the value of a_3 + a_5 + a_7."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 13320,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a geometric series \\(\\left\\{a_{n}\\right\\}\\) with the sum of its first \\(n\\) terms denoted by \\(S_{n}\\), and satisfying the equation \\(S_{n}=\\frac{\\left(a_{n}+1\\right)^{2}}{4}\\), find the value of \\(S_{20}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "400"
} | {
"index": 13321,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a grid, identify the rectangles and squares, and describe their properties and characteristics."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 13322,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a group with the numbers $-3, 0, 5, 8, 11, 13$, and the following rules: the largest isn't first, and it must be within the first four places, the smallest isn't last, and it must be within the last four places, and the median isn't in the first or last position, determin... | math | {
"style": "rule",
"ground_truth": "5.5"
} | {
"index": 13323,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a harmonic progression with the first three terms 3, 4, 6, find the value of $S_4$."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 13324,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a hexagon \\( A B C D E F \\) with an area of 60 that is inscribed in a circle \\( \\odot O \\), where \\( AB = BC, CD = DE, \\) and \\( EF = AF \\). What is the area of \\( \\triangle B D F \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 13325,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a hyperbola $C$ that shares the same foci with the ellipse $\\frac{x^{2}}{27}+ \\frac{y^{2}}{36}=1$ and passes through the point $(\\sqrt{15},4)$.\n\n(I) Find the equation of the hyperbola $C$.\n\n(II) If $F\\_1$ and $F\\_2$ are the two foci of the hyperbola $C$, and poin... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{3}}{3}"
} | {
"index": 13326,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a hyperbola $C$ with an eccentricity of $\\sqrt {3}$, foci $F\\_1$ and $F\\_2$, and a point $A$ on the curve $C$. If $|F\\_1A|=3|F\\_2A|$, then $\\cos \\angle AF\\_2F\\_1=$ \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 13327,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a hyperbola $C$ with one of its foci on the line $l: 4x-3y+20=0$, and one of its asymptotes is parallel to $l$, and the foci of the hyperbola $C$ are on the $x$-axis, then the standard equation of the hyperbola $C$ is \\_\\_\\_\\_\\_\\_; the eccentricity is \\_\\_\\_\\_\\... | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{3}"
} | {
"index": 13328,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a hyperbola $C_{1}$ defined by $2x^{2}-y^{2}=1$, find the area of the triangle formed by a line parallel to one of the asymptotes of $C_{1}$, the other asymptote, and the x-axis."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{8}"
} | {
"index": 13329,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a hyperbola $F$ with the equation $\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, let $F$ be its left focus. Draw a line perpendicular to one asymptote of the hyperbola passing through $F$, and denote the foot of the perpendicular as $A$ and the intersec... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 13330,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ ($a>0, b>0$) with a point C on it, a line passing through the center of the hyperbola intersects the hyperbola at points A and B. Let the slopes of the lines AC and BC be $k_1$ and $k_2$ respectively. Find the eccentri... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 13331,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a hyperbola $\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ with left and right foci $F\\_1$ and $F\\_2$, respectively. One of its asymptotes is $x+\\sqrt{2}y=0$. Point $M$ lies on the hyperbola, and $MF\\_1 \\perp x$-axis. If $F\\_2$ is also a focus of the p... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{5}"
} | {
"index": 13332,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a hyperbola $x^{2}- \\frac {y^{2}}{3}=1$ and two points $M$, $N$ on it are symmetric about the line $y=x+m$, and the midpoint of $MN$ lies on the parabola $y^{2}=18x$. Find the value of the real number $m$."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"index": 13333,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a hyperbola \\( H: x^{2}-y^{2}=1 \\) with a point \\( M \\) in the first quadrant, and a line \\( l \\) tangent to the hyperbola \\( H \\) at point \\( M \\), intersecting the asymptotes of \\( H \\) at points \\( P \\) and \\( Q \\) (where \\( P \\) is in the first quadr... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 13334,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a hyperbola \\( x^{2} - y^{2} = t \\) (where \\( t > 0 \\)), the right focus is \\( F \\). Any line passing through \\( F \\) intersects the right branch of the hyperbola at points \\( M \\) and \\( N \\). The perpendicular bisector of \\( M N \\) intersects the \\( x \\)... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 13335,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a hyperbola with asymptotes $2x \\pm y=0$, that passes through the intersection of the lines $x+y-3=0$ and $2x-y+3t=0$, where $-2 \\leq t \\leq 5$. Find the maximum possible length of the real axis of the hyperbola."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"index": 13336,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a hyperbola with eccentricity $2$ and equation $\\dfrac{x^2}{a^2} - \\dfrac{y^2}{b^2} = 1$ $(a > 0, b > 0)$, the right focus $F_2$ of the hyperbola is the focus of the parabola $y^2 = 8x$. A line $l$ passing through point $F_2$ intersects the right branch of the hyperbola... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3\\sqrt{7}}{7}"
} | {
"index": 13337,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a hyperbola with eccentricity $e$ and an ellipse with eccentricity $\\frac{\\sqrt{2}}{2}$ share the same foci $F_{1}$ and $F_{2}$. If $P$ is a common point of the two curves and $\\angle F_{1}PF_{2}=60^{\\circ}$, then $e=$ ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"index": 13338,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a hyperbola with its left and right foci being $F_1$ and $F_2$ respectively, and the length of chord $AB$ on the left branch passing through $F_1$ is 5. If $2a=8$, calculate the perimeter of $\\triangle ABF_2$."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 13339,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a hyperbola with left and right foci at $F_1$ and $F_2$ respectively, a chord $AB$ on the left branch passing through $F_1$ with a length of 5. If $2a=8$, determine the perimeter of $\\triangle ABF_2$."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 13340,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a hyperbola with the equation $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$, its asymptotes intersect the parabola $y^2 = 4x$ at two points A and B, distinct from the origin O. Let F be the focus of the parabola $y^2 = 4x$. If $\\angle AFB = \\frac{2\\pi}{3}$, then the eccent... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{21}}{3}"
} | {
"index": 13341,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a hyperbola with the equation $\\frac{y^2}{4} - \\frac{x^2}{a} = 1$ and asymptote equations $y = \\pm\\frac{2\\sqrt{3}}{3}x$, find the eccentricity of this hyperbola."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{2}"
} | {
"index": 13342,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a large circle with a radius of 11 and small circles with a radius of 1, determine the maximum number of small circles that can be placed inside the large circle, such that each small circle is internally tangent to the large circle and the small circles do not overlap."
... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 13343,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a large circle with a radius of 11 and small circles with a radius of 1, find the maximum number of small circles that can be tangentially inscribed in the large circle without overlapping."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 13344,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a lawn with a rectangular shape of $120$ feet by $200$ feet, a mower with a $32$-inch swath width, and a $6$-inch overlap between each cut, and a walking speed of $4000$ feet per hour, calculate the time it will take John to mow the entire lawn."
}
] | math | {
"style": "rule",
"ground_truth": "2.8"
} | {
"index": 13345,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a line $l$ intersects the hyperbola $x^2 - \\frac{y^2}{2} = 1$ at two distinct points $A$ and $B$. If point $M(1, 2)$ is the midpoint of segment $AB$, find the equation of line $l$ and the length of segment $AB$."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"index": 13346,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a line $l$ passes through the foci of the ellipse $\\frac {y^{2}}{2}+x^{2}=1$ and intersects the ellipse at points P and Q. The perpendicular bisector of segment PQ intersects the x-axis at point M. The maximum area of $\\triangle MPQ$ is __________."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3 \\sqrt {6}}{8}"
} | {
"index": 13347,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a line $l$ passing through point $A(1,1)$ with a slope of $-m$ ($m>0$) intersects the x-axis and y-axis at points $P$ and $Q$, respectively. Perpendicular lines are drawn from $P$ and $Q$ to the line $2x+y=0$, and the feet of the perpendiculars are $R$ and $S$. Find the m... | math | {
"style": "rule",
"ground_truth": "3.6"
} | {
"index": 13348,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a line $l$ whose inclination angle $\\alpha$ satisfies the condition $\\sin \\alpha +\\cos \\alpha = \\frac{1}{5}$, determine the slope of $l$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{3}"
} | {
"index": 13349,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a line $l$ with an inclination angle of $\\theta$, if $\\cos\\theta= \\frac {4}{5}$, calculate the slope of this line."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 13350,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a line $y = \\frac{\\sqrt{3}}{3}x$ and a circle $C$ with its center on the positive x-axis and a radius of 2 intersects the line at points $A$ and $B$ such that $|AB|=2\\sqrt{3}$.\n\n(1) Given a point $P(-1, \\sqrt{7})$, and $Q$ is any point on circle $C$, find the maximu... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 13351,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a line $y=-x+1$ and an ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$), they intersect at points A and B.\n(1) If the eccentricity of the ellipse is $\\frac{\\sqrt{3}}{3}$ and the focal distance is 2, find the length of the segment AB.\n(2) (For Liberal Ar... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"index": 13352,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a line and a point at a distance of $1 \\mathrm{~cm}$ from the line. What is the volume of a cube where the point is one of the vertices and the line is one of the body diagonals?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{6}}{4}"
} | {
"index": 13353,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a line segment $\\overline{AB}=10$ cm, a point $C$ is placed on $\\overline{AB}$ such that $\\overline{AC} = 6$ cm and $\\overline{CB} = 4$ cm. Three semi-circles are drawn with diameters $\\overline{AB}$, $\\overline{AC}$, and $\\overline{CB}$, external to the segment. I... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 13354,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a line segment of length $6$ is divided into three line segments of lengths that are positive integers, calculate the probability that these three line segments can form a triangle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{10}"
} | {
"index": 13355,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a linear function \\( f(x) \\), it is known that the distance between the intersection points of the graphs \\( y=x^{2}+2 \\) and \\( y=f(x) \\) is \\(\\sqrt{10}\\), and the distance between the intersection points of the graphs \\( y=x^{2}-1 \\) and \\( y=f(x)+1 \\) is \... | math | {
"style": "rule",
"ground_truth": "\\sqrt{34}"
} | {
"index": 13356,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a linear function \\( f(x) \\). It is known that the distance between the points of intersection of the graphs \\( y = x^2 + 1 \\) and \\( y = f(x) \\) is \\( 3\\sqrt{2} \\), and the distance between the points of intersection of the graphs \\( y = x^2 \\) and \\( y = f(x... | math | {
"style": "rule",
"ground_truth": "\\sqrt{26}"
} | {
"index": 13357,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a linear function \\( f(x) \\). It is known that the distance between the points of intersection of the graphs \\( y = x^2 - 1 \\) and \\( y = f(x) + 1 \\) is \\( 3\\sqrt{10} \\), and the distance between the points of intersection of the graphs \\( y = x^2 \\) and \\( y ... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"index": 13358,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a list of $3000$ positive integers with a unique mode occurring exactly $12$ times, calculate the least number of distinct values that can occur in the list."
}
] | math | {
"style": "rule",
"ground_truth": "273"
} | {
"index": 13359,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a list of the first 12 positive integers such that for each $2\\le i\\le 12$, either $a_i + 1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list, calculate the number of such lists."
}
] | math | {
"style": "rule",
"ground_truth": "2048"
} | {
"index": 13360,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a machine that transforms a positive integer \\( N \\) based on the rule: if \\( N = 7 \\), the machine outputs \\( 3 \\times 7 + 1 = 22 \\).\n\nBy inputting the result back into the machine and repeating five times, the output sequence is:\n\\[ 7 \\rightarrow 22 \\righta... | math | {
"style": "rule",
"ground_truth": "83"
} | {
"index": 13361,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a magician's hat contains 4 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 4 of the reds are drawn or until both green chips are drawn, calculate the probability that all 4 red chips are drawn before both green chips ar... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 13362,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a mall with four categories of food: grains, vegetable oils, animal products, and fruits and vegetables, with 40, 10, 20, and 20 varieties, respectively, calculate the total sample size if 6 types of animal products are sampled."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 13363,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a matrix $\\begin{pmatrix} a_{11} & a_{12} \\\\ a_{21} & a_{22}\\end{pmatrix}$ satisfies: $a_{11}$, $a_{12}$, $a_{21}$, $a_{22} \\in \\{0,1\\}$, and $\\begin{vmatrix} a_{11} & a_{12} \\\\ a_{21} & a_{22}\\end{vmatrix} =0$, determine the total number of distinct matrices."... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 13364,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a monotonically decreasing geometric sequence $\\{a_{n}\\}$ that satisfies $a_{2}+a_{3}+a_{4}=28$, and $a_{3}+2$ is the arithmetic mean of $a_{2}$ and $a_{4}$, find the sum of the first 6 terms of the sequence $\\{a_{n}\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"index": 13365,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a monotonically increasing sequence of positive integers $\\left\\{a_{n}\\right\\}$ that satisfies the recurrence relation $a_{n+2}=3 a_{n+1}-a_{n}$, with $a_{6}=280$, find the value of $a_{7}$."
}
] | math | {
"style": "rule",
"ground_truth": "733"
} | {
"index": 13366,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a moving circle $C$ that passes through points $A(4,0)$ and $B(0,-2)$, and intersects with the line passing through point $M(1,-2)$ at points $E$ and $F$. Find the minimum value of $|EF|$ when the area of circle $C$ is at its minimum."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 13367,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a moving circle $E$ that passes through the point $F(1,0)$, and is tangent to the line $l: x=-1$.\n(Ⅰ) Find the equation of the trajectory $G$ for the center $E$ of the moving circle.\n(Ⅱ) Given the point $A(3,0)$, if a line with a slope of $1$ intersects with the line se... | math | {
"style": "rule",
"ground_truth": "\\frac{32 \\sqrt{3}}{9}"
} | {
"index": 13368,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a moving circle $M$ that passes through the fixed point $F(0,-1)$ and is tangent to the line $y=1$. The trajectory of the circle's center $M$ forms a curve $C$. Let $P$ be a point on the line $l$: $x-y+2=0$. Draw two tangent lines $PA$ and $PB$ from point $P$ to the curve... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 13369,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a moving circle P that is internally tangent to the circle M: (x+1)²+y²=8 at the fixed point N(1,0).\n(1) Find the trajectory equation of the moving circle P's center.\n(2) Suppose the trajectory of the moving circle P's center is curve C. A and B are two points on curve ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 13370,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a moving circle that passes through the fixed point $F(1,0)$ and is tangent to the fixed line $l$: $x=-1$.\n(1) Find the equation of the trajectory $C$ of the circle's center;\n(2) The midpoint of the chord $AB$ formed by the intersection of line $l$ and $C$ is $(2,1}$. $... | math | {
"style": "rule",
"ground_truth": "\\sqrt{35}"
} | {
"index": 13371,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a moving large circle $\\odot O$ tangent externally to a fixed small circle $\\odot O_{1}$ with radius 3 at point $P$, $AB$ is the common external tangent of the two circles with $A$ and $B$ as the points of tangency. A line $l$ parallel to $AB$ is tangent to $\\odot O_{1... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 13372,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a moving line $l$ that tangentially touches the circle $O: x^{2}+y^{2}=1$ and intersects the ellipse $\\frac{x^{2}}{9}+y^{2}=1$ at two distinct points $A$ and $B$, find the maximum distance from the origin to the perpendicular bisector of line segment $AB$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 13373,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a moving point $A$ on the curve $y=x^{2}$, let $m$ be the tangent line to the curve at point $A$. Let $n$ be a line passing through point $A$, perpendicular to line $m$, and intersecting the curve at another point $B$. Determine the minimum length of the line segment $AB$... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{2}"
} | {
"index": 13374,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a moving point $E$ such that the product of the slopes of the lines from $E$ to points $A(2,0)$ and $B(-2,0)$ is $- \\frac {1}{4}$, and the trajectory of point $E$ is curve $C$. \n$(1)$ Find the equation of curve $C$; \n$(2)$ Draw a line $l$ through point $D(1,0)$ that ... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{4}"
} | {
"index": 13375,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a moving point $M$ whose distance to the point $F(0,1)$ is equal to its distance to the line $y=-1$, the trajectory of point $M$ is denoted as $C$.\n\n(1) Find the equation of the trajectory $C$.\n\n(2) Let $P$ be a point on the line $l: x-y-2=0$. Construct two tangents $... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 13376,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a moving point P in the plane, the difference between the distance from point P to point F(1, 0) and the distance from point P to the y-axis is equal to 1.\n(Ⅰ) Find the equation of the trajectory C of the moving point P;\n(Ⅱ) Draw two lines $l_1$ and $l_2$ through point ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 13377,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a moving point \\( P \\) on the \\( x \\)-axis, \\( M \\) and \\( N \\) lie on the circles \\((x-1)^{2}+(y-2)^{2}=1\\) and \\((x-3)^{2}+(y-4)^{2}=3\\) respectively. Find the minimum value of \\( |PM| + |PN| \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{10} - \\sqrt{3} - 1"
} | {
"index": 13378,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a natural number \\( a \\), let \\( S(a) \\) represent the sum of its digits (for example, \\( S(123) = 1 + 2 + 3 = 6 \\) ). If a natural number \\( n \\) has all distinct digits, and \\( S(3n) = 3S(n) \\), what is the maximum value of \\( n \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "3210"
} | {
"index": 13379,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a non-empty set of numbers, the sum of its maximum and minimum elements is called the \"characteristic value\" of the set. $A\\_1$, $A\\_2$, $A\\_3$, $A\\_4$, $A\\_5$ each contain $20$ elements, and $A\\_1∪A\\_2∪A\\_3∪A\\_4∪A\\_5={x∈N^⁎|x≤slant 100}$, find the minimum val... | math | {
"style": "rule",
"ground_truth": "325"
} | {
"index": 13380,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a non-right triangle $\\triangle ABC$, where the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $c=1$, also $$C= \\frac {\\pi}{3}$$, if $\\sin C + \\sin(A-B) = 3\\sin 2B$, then the area of $\\triangle ABC$ is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3 \\sqrt {3}}{28}"
} | {
"index": 13381,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a nonnegative real number $x$, let $\\langle x\\rangle$ denote the fractional part of $x$; that is, $\\langle x\\rangle=x-\\lfloor x\\rfloor$, where $\\lfloor x\\rfloor$ denotes the greatest integer less than or equal to $x$. Suppose that $a$ is positive, $\\langle a^{-1}... | math | {
"style": "rule",
"ground_truth": "233"
} | {
"index": 13382,
"solution": "Find $a$ as shown above. Note that $a$ satisfies the equation $a^2 = a+1$ (this is the equation we solved to get it). Then, we can simplify $a^{12}$ as follows using the fibonacci numbers:\n$a^{12} = a^{11}+a^{10}= 2a^{10} + a^{9} = 3a^8+ 2a^9 = ... = 144a^1+89a^0 = 144a+89$\n So we w... |
DeepScaleR | [
{
"role": "user",
"content": "Given a number $\\overline{abcd}$ , where $a$ , $b$ , $c$ , and $d$ , represent the digits of $\\overline{abcd}$ , find the minimum value of\n\\[\\frac{\\overline{abcd}}{a+b+c+d}\\]\nwhere $a$ , $b$ , $c$ , and $d$ are distinct\n<details><summary>Answer</summary>$\\o... | math | {
"style": "rule",
"ground_truth": "60.5"
} | {
"index": 13383,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a number $x$ is randomly selected from the interval $[-1,1]$, calculate the probability that the value of $\\sin \\frac{\\pi x}{4}$ falls between $-\\frac{1}{2}$ and $\\frac{\\sqrt{2}}{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 13384,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a number \\\\(x\\\\) randomly selected from the interval \\\\(\\left[-\\frac{\\pi}{4}, \\frac{2\\pi}{3}\\right]\\\\), find the probability that the function \\\\(f(x)=3\\sin\\left(2x- \\frac{\\pi}{6}\\right)\\\\) is not less than \\\\(0\\\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{11}"
} | {
"index": 13385,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a pair of concentric circles, chords $AB,BC,CD,\\dots$ of the outer circle are drawn such that they all touch the inner circle. If $\\angle ABC = 75^{\\circ}$ , how many chords can be drawn before returning to the starting point ?\n![Image](https://i.imgur.com/Cg37vwa... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 13386,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a pair of standard $8$-sided dice is rolled once. The sum of the numbers rolled, if it is a prime number, determines the diameter of a circle. Find the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumfe... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{64}"
} | {
"index": 13387,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parabola $C$ that passes through the point $(4,4)$ and its focus lies on the $x$-axis.\n\n$(1)$ Find the standard equation of parabola $C$.\n$(2)$ Let $P$ be any point on parabola $C$. Find the minimum distance between point $P$ and the line $x - y + 4 = 0$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{2}"
} | {
"index": 13388,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parabola $C$: $y^{2}=2px (p > 0)$ with focus $F$, directrix $x=-1$, and a point $M$ on the directrix below the $x$-axis. A line $l$ passing through $M$ and $F$ intersects with $C$ at point $N$, and $F$ is the midpoint of segment $MN$.\n\n1. Find the equations of parabol... | math | {
"style": "rule",
"ground_truth": "\\frac {16}{3}"
} | {
"index": 13389,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parabola $C: y^2 = 2px (p > 0)$ that passes through the point $(1, -2)$, a line $l$ through focus $F$ intersects the parabola $C$ at points $A$ and $B$. If $Q$ is the point $(-\\frac{7}{2}, 0)$ and $BQ \\perp BF$, find the value of $|BF| - |AF|$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"index": 13390,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parabola $C: y^2 = 2px (p > 0)$, and a circle $M: (x-2)^2 + y^2 = 4$, the distance from the center $M$ of the circle to the directrix of the parabola is $3$. Point $P(x_0, y_0)(x_0 \\geqslant 5)$ is a point on the parabola in the first quadrant. Through point $P$, two t... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{2}"
} | {
"index": 13391,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parabola $C: y^2 = 3x$ with focus $F$, find the length of segment $AB$ where the line passing through $F$ at a $30^\\circ$ angle intersects the parabola at points $A$ and $B$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 13392,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parabola $C:y^2=2px (p > 0)$, the sum of the distances from any point $Q$ on the parabola to a point inside it, $P(3,1)$, and the focus $F$, has a minimum value of $4$.\n\n(I) Find the equation of the parabola;\n\n(II) Through the focus $F$, draw a line $l$ that interse... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 13393,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parabola $y = ax^2 + bx + c$ ($a \\neq 0$) with its axis of symmetry on the left side of the y-axis, where $a, b, c \\in \\{-3, -2, -1, 0, 1, 2, 3\\}$. Let the random variable $X$ represent the value of $|a-b|$. Calculate the expected value $E(X)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"index": 13394,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parabola $y=x^2+bx+c$ intersects the y-axis at point Q(0, -3), and the sum of the squares of the x-coordinates of the two intersection points with the x-axis is 15, find the equation of the function and its axis of symmetry."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 13395,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parabola $y=x^{2}-7$, find the length of the line segment $|AB|$ where $A$ and $B$ are two distinct points on it that are symmetric about the line $x+y=0$."
}
] | math | {
"style": "rule",
"ground_truth": "5 \\sqrt{2}"
} | {
"index": 13396,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parabola $y^2 = 2px$ ($p > 0$) and a hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) share a common focus $F$, and point $A$ is the intersection point of the two curves. If $AF \\perp x$-axis, find the eccentricity of the hyperbola."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + 1"
} | {
"index": 13397,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parabola $y^2=2px$ ($p>0$) with focus $F$, a circle is drawn with $F$ as the center and $p$ as the radius, intersecting the $y$-axis at points $A$ and $B$. Connect $F$ to $A$, intersecting the parabola at point $D$ (which lies on segment $FA$). Extend $FA$ to intersect ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 13398,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parabola C: $y^2=2px$ $(p>0)$ whose focus F coincides with the right focus of the ellipse $\\frac{x^2}{4}+\\frac{y^2}{3}=1$, and a line l passing through point F intersects the parabola at points A and B.\n(1) Find the equation of the parabola C.\n(2) If the line l inte... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 13399,
"solution": ""
} |
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