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 parabola \\( y^2 = 6x \\) with two variable points \\( A(x_1, y_1) \\) and \\( B(x_2, y_2) \\), where \\( x_1 \\neq x_2 \\) and \\( x_1 + x_2 = 4 \\). The perpendicular bisector of segment \\( AB \\) intersects the x-axis at point \\( C \\). Find the maximum area of tri... | math | {
"style": "rule",
"ground_truth": "\\frac{14}{3}\\sqrt{7}"
} | {
"index": 13400,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parabola \\(C\\) with the center of ellipse \\(E\\) as its focus, the parabola \\(C\\) passes through the two foci of the ellipse \\(E\\), and intersects the ellipse \\(E\\) at exactly three points. Find the eccentricity of the ellipse \\(E\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{5}}{5}"
} | {
"index": 13401,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parabola with vertex \\( V \\) and a focus \\( F \\), and points \\( B \\) and \\( C \\) on the parabola such that \\( BF=25 \\), \\( BV=24 \\), and \\( CV=20 \\), determine the sum of all possible values of the length \\( FV \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{50}{3}"
} | {
"index": 13402,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parallelepiped $A B C D A_1 B_1 C_1 D_1$, point $X$ is selected on the edge $A_1 D_1$, and point $Y$ is selected on the edge $B C$. It is known that $A_1 X = 5$, $B Y = 3$, and $B_1 C_1 = 14$. The plane $C_1 X Y$ intersects the ray $D A$ at point $Z$. Find $D Z$."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 13403,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parallelepiped $A B C D A_1 B_1 C_1 D_1$, points $M, N, K$ are midpoints of edges $A B$, $B C$, and $D D_1$ respectively. Construct the cross-sectional plane of the parallelepiped with the plane $MNK$. In what ratio does this plane divide the edge $C C_1$ and the diagon... | math | {
"style": "rule",
"ground_truth": "3:7"
} | {
"index": 13404,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the point $X$ is chosen on the edge $A_{1} D_{1}$ and the point $Y$ is chosen on the edge $B C$. It is known that $A_{1} X=5$, $B Y=3$, and $B_{1} C_{1}=14$. The plane $C_{1} X Y$ intersects the ray $D A$ at point $Z$. F... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 13405,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$. Point $X$ is chosen on edge $A_{1} D_{1}$, and point $Y$ is chosen on edge $B C$. It is known that $A_{1} X=5$, $B Y=3$, and $B_{1} C_{1}=14$. The plane $C_{1} X Y$ intersects ray $D A$ at point $Z$. Find $D Z$."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 13406,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parallelepiped \\( A B C D A_{1} B_{1} C_{1} D_{1} \\). On edge \\( A_{1} D_{1} \\), point \\( X \\) is selected, and on edge \\( B C \\), point \\( Y \\) is selected. It is known that \\( A_{1} X = 5 \\), \\( B Y = 3 \\), and \\( B_{1} C_{1} = 14 \\). The plane \\( C_{... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 13407,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parallelepiped \\(A B C D A_{1} B_{1} C_{1} D_{1}\\), a point \\(X\\) is chosen on edge \\(A_{1} D_{1}\\), and a point \\(Y\\) is chosen on edge \\(B C\\). It is known that \\(A_{1} X = 5\\), \\(B Y = 3\\), and \\(B_{1} C_{1} = 14\\). The plane \\(C_{1} X Y\\) intersect... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 13408,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parallelepiped \\(A B C D A_{1} B_{1} C_{1} D_{1}\\). Point \\(X\\) is chosen on edge \\(A_{1} D_{1}\\) and point \\(Y\\) is chosen on edge \\(B C\\). It is known that \\(A_{1} X = 5\\), \\(B Y = 3\\), and \\(B_{1} C_{1} = 14\\). The plane \\(C_{1} X Y\\) intersects the... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 13409,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parallelogram \\(A B C D\\) with \\(\\angle B = 111^\\circ\\) and \\(B C = B D\\). On the segment \\(B C\\), there is a point \\(H\\) such that \\(\\angle B H D = 90^\\circ\\). Point \\(M\\) is the midpoint of side \\(A B\\). Find the angle \\(A M H\\). Provide the answ... | math | {
"style": "rule",
"ground_truth": "132"
} | {
"index": 13410,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parallelogram \\(ABCD\\) with \\(\\angle B = 60^\\circ\\). Point \\(O\\) is the center of the circumcircle of triangle \\(ABC\\). Line \\(BO\\) intersects the bisector of the exterior angle \\(\\angle D\\) at point \\(E\\). Find the ratio \\(\\frac{BO}{OE}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 13411,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parallelogram \\(ABCD\\) with sides \\(AB=2\\) and \\(BC=3\\), find the area of this parallelogram, given that the diagonal \\(AC\\) is perpendicular to the segment \\(BE\\), where \\(E\\) is the midpoint of side \\(AD\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{35}"
} | {
"index": 13412,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parallelogram \\\\(ABCD\\\\) where \\\\(AD=2\\\\), \\\\(∠BAD=120^{\\\\circ}\\\\), and point \\\\(E\\\\) is the midpoint of \\\\(CD\\\\), if \\\\( \\overrightarrow{AE} \\cdot \\overrightarrow{BD}=1\\\\), then \\\\( \\overrightarrow{BD} \\cdot \\overrightarrow{BE}=\\\\) \... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 13413,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parallelogram with an acute angle of \\(60^{\\circ}\\). Find the ratio of the sides of the parallelogram if the ratio of the squares of the diagonals is \\(\\frac{1}{3}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "1:1"
} | {
"index": 13414,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parallelogram with area $1$ and we will construct lines where this lines connect a vertex with a midpoint of the side no adjacent to this vertex; with the $8$ lines formed we have a octagon inside of the parallelogram. Determine the area of this octagon"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13415,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a parameterized curve $ C: x\\equal{}e^t\\minus{}e^{\\minus{}t},\\ y\\equal{}e^{3t}\\plus{}e^{\\minus{}3t}$ .\r\n\r\nFind the area bounded by the curve $ C$ , the $ x$ axis and two lines $ x\\equal{}\\pm 1$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{5}}{2}"
} | {
"index": 13416,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a pedestrian signal light that alternates between red and green, with the red light lasting for 50 seconds, calculate the probability that a student needs to wait at least 20 seconds for the green light to appear."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{5}"
} | {
"index": 13417,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a periodic sequence $\\left\\{x_{n}\\right\\}$ that satisfies $x_{n}=\\left|x_{n-1}-x_{n-2}\\right|(n \\geqslant 3)$, if $x_{1}=1$ and $x_{2}=a \\geqslant 0$, calculate the sum of the first 2002 terms when the period of the sequence is minimized."
}
] | math | {
"style": "rule",
"ground_truth": "1335"
} | {
"index": 13418,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a permutation $\\pi$ of the set $\\{1,2, \\ldots, 10\\}$, define a rotated cycle as a set of three integers $i, j, k$ such that $i<j<k$ and $\\pi(j)<\\pi(k)<\\pi(i)$. What is the total number of rotated cycles over all permutations $\\pi$ of the set $\\{1,2, \\ldots, 10\\... | math | {
"style": "rule",
"ground_truth": "72576000"
} | {
"index": 13419,
"solution": "Let us consider a triple $(i, j, k)$ with $i<j<k$ and determine how many permutations rotate it. There are $\\binom{10}{3}$ choices for the values of $\\pi(i), \\pi(j), \\pi(k)$ and the choice of this set of three determines the values of $\\pi(i), \\pi(j), \\pi(k)$. The other 7 value... |
DeepScaleR | [
{
"role": "user",
"content": "Given a permutation $\\sigma$ of $\\{1,2, \\ldots, 2013\\}$, let $f(\\sigma)$ to be the number of fixed points of $\\sigma$ - that is, the number of $k \\in\\{1,2, \\ldots, 2013\\}$ such that $\\sigma(k)=k$. If $S$ is the set of all possible permutations $\\sigma$, compute $$\\... | math | {
"style": "rule",
"ground_truth": "15(2013!)"
} | {
"index": 13420,
"solution": "First, note that $$\\sum_{\\sigma \\in S} f(\\sigma)^{4}=\\sum_{\\sigma \\in S} \\sum_{1 \\leq a_{1}, a_{2}, a_{3}, a_{4} \\leq 2013} g\\left(\\sigma, a_{1}, a_{2}, a_{3}, a_{4}\\right)$$ where $g\\left(\\sigma, a_{1}, a_{2}, a_{3}, a_{4}\\right)=1$ if all $a_{i}$ are fixed points of ... |
DeepScaleR | [
{
"role": "user",
"content": "Given a plane $\\alpha$ and two non-coincident straight lines $m$ and $n$, consider the following four propositions:\n(1) If $m \\parallel \\alpha$ and $n \\subseteq \\alpha$, then $m \\parallel n$.\n(2) If $m \\parallel \\alpha$ and $n \\parallel \\alpha$, then $m \\parallel n... | math | {
"style": "rule",
"ground_truth": "(4)"
} | {
"index": 13421,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a plane intersects all 12 edges of a cube at an angle $\\alpha$, find $\\sin \\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 13422,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a point $P$ on a triangular piece of paper $ABC,\\,$ consider the creases that are formed in the paper when $A, B,\\,$ and $C\\,$ are folded onto $P.\\,$ Let us call $P$ a fold point of $\\triangle ABC\\,$ if these creases, which number three unless $P$ is one of the vert... | math | {
"style": "rule",
"ground_truth": "597"
} | {
"index": 13423,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a point $P$ on the curve $y= \\frac{1}{2}e^{x}$ and a point $Q$ on the curve $y=\\ln (2x)$, determine the minimum value of $|PQ|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}(1-\\ln 2)"
} | {
"index": 13424,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a point $P$ on the ellipse $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with foci $F_{1}$, $F_{2}$, find the eccentricity of the ellipse given that $\\overrightarrow{PF_{1}} \\cdot \\overrightarrow{PF_{2}}=0$ and $\\tan \\angle PF_{1}F_{2}= \\frac{1}{2}$."
... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{3}"
} | {
"index": 13425,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a point $P(x,y)$ moving on the ellipse $\\frac{x^{2}}{4} + \\frac{y^{2}}{3} = 1$, let $d = \\sqrt{x^{2} + y^{2} + 4y + 4} - \\frac{x}{2}$. Find the minimum value of $d$.\nA) $\\sqrt{5} - 2$\nB) $2\\sqrt{2} - 1$\nC) $\\sqrt{5} - 1$\nD) $\\sqrt{6} - 1$"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} - 1"
} | {
"index": 13426,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a point $P^{}_{}$ on a triangular piece of paper $ABC,\\,$ consider the creases that are formed in the paper when $A, B,\\,$ and $C\\,$ are folded onto $P.\\,$ Let us call $P_{}^{}$ a fold point of $\\triangle ABC\\,$ if these creases, which number three unless $P^{}_{}$ ... | math | {
"style": "rule",
"ground_truth": "597"
} | {
"index": 13427,
"solution": "Let $O_{AB}$ be the intersection of the perpendicular bisectors (in other words, the intersections of the creases) of $\\overline{PA}$ and $\\overline{PB}$, and so forth. Then $O_{AB}, O_{BC}, O_{CA}$ are, respectively, the circumcenters of $\\triangle PAB, PBC, PCA$. According to the... |
DeepScaleR | [
{
"role": "user",
"content": "Given a point $Q$ on a rectangular piece of paper $DEF$, where $D, E, F$ are folded onto $Q$. Let $Q$ be a fold point of $\\triangle DEF$ if the creases, which number three unless $Q$ is one of the vertices, do not intersect within the triangle. Suppose $DE=24, DF=48,$ and $\\a... | math | {
"style": "rule",
"ground_truth": "147"
} | {
"index": 13428,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a point $p$ and a line segment $l$, let $d(p, l)$ be the distance between them. Let $A, B$, and $C$ be points in the plane such that $A B=6, B C=8, A C=10$. What is the area of the region in the $(x, y)$-plane formed by the ordered pairs $(x, y)$ such that there exists a ... | math | {
"style": "rule",
"ground_truth": "\\frac{288}{5}"
} | {
"index": 13429,
"solution": "Place $A B C$ in the coordinate plane so that $A=(0,6), B=(0,0), C=(8,0)$. Consider a point $P=(a, b)$ inside triangle $A B C$. Clearly, $d(P, A B)=a, d(P, B C)=b$. Now, we see that the area of triangle $A B C$ is $\\frac{6 \\cdot 8}{2}=24$, but may also be computed by summing the are... |
DeepScaleR | [
{
"role": "user",
"content": "Given a point M$(x_0, y_0)$ moves on the circle $x^2+y^2=4$, and N$(4, 0)$, the point P$(x, y)$ is the midpoint of the line segment MN. \n(1) Find the trajectory equation of point P$(x, y)$. \n(2) Find the maximum and minimum distances from point P$(x, y)$ to the line $3x+4y-... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 13430,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a point P $(x, y)$ moves on the circle $x^2 + (y-1)^2 = 1$, the maximum value of $\\frac{y-1}{x-2}$ is \\_\\_\\_\\_\\_\\_, and the minimum value is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{3}"
} | {
"index": 13431,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a point P $(x, y)$ on the circle $x^2 - 4x - 4 + y^2 = 0$, find the maximum value of $x^2 + y^2$."
}
] | math | {
"style": "rule",
"ground_truth": "12 + 8\\sqrt{2}"
} | {
"index": 13432,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a point P $(x, y)$ on the ellipse $\\frac{x^2}{4} + \\frac{y^2}{9} = 1$, find the minimum distance from point P to the line $2x + y - 10 = 0$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 13433,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a point P in the plane satisfying $|PM| - |PN| = 2\\sqrt{2}$, with $M(-2,0)$, $N(2,0)$, and $O(0,0)$,\n(1) Find the locus S of point P;\n(2) A straight line passing through the point $(2,0)$ intersects with S at points A and B. Find the minimum value of the area of triang... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 13434,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a point P is 9 units away from the center of a circle with a radius of 15 units, find the number of chords passing through point P that have integer lengths."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 13435,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a point P on the curve $y = x^2 - \\ln x$, find the minimum distance from point P to the line $y = x + 2$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 13436,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a point P on the hyperbola $C_1: \\frac{x^2}{16} - \\frac{y^2}{9} = 1$, a point Q on the circle $C_2: (x - 5)^2 + y^2 = 1$, and a point R on the circle $C_3: (x + 5)^2 + y^2 = 1$, find the maximum value of $|PQ| - |PR|$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 13437,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a point P on the parabola $y^2=4x$, let the distance from point P to the directrix of this parabola be $d_1$, and the distance from point P to the line $x+2y-12=0$ be $d_2$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11 \\sqrt{5}}{5}"
} | {
"index": 13438,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a point P(3, 2) outside the circle $x^2+y^2-2x-2y+1=0$, find the cosine of the angle between the two tangents drawn from this point to the circle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 13439,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a point \\\\(P(m,-\\sqrt{3})\\\\) (\\\\(m \\neq 0\\\\)) on the terminal side of angle \\\\(\\alpha\\\\), and \\\\(\\cos \\alpha = \\frac{\\sqrt{2}m}{4}\\\\),\n\\\\((i)\\\\) find the value of \\\\(m\\\\); \n\n\\\\((ii)\\\\) calculate \\\\(\\sin \\alpha\\\\) and \\\\(\\tan... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{15}}{5}"
} | {
"index": 13440,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a point on the parabola $y^2=6x$ whose distance to the focus is twice the distance to the y-axis, find the x-coordinate of this point."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 13441,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a polygon drawn on graph paper with a perimeter of 2014 units, and whose sides follow the grid lines, what is the maximum possible area of this polygon?"
}
] | math | {
"style": "rule",
"ground_truth": "253512"
} | {
"index": 13442,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a polynomial $(a+b+c+d+e+1)^N$, when expanded and like terms are combined, the expression contains exactly 2002 terms that include all five variables $a, b, c, d, e$, each to some positive power, determine the value of $N$."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 13443,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a polynomial $f(x) = 2x^7 + x^6 + x^4 + x^2 + 1$, calculate the value of $V_2$ using the Horner's method when $x=2$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 13444,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a polynomial \\( P(x) = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \\ldots + a_1 x + a_0 \\) where each coefficient \\( a_i \\) belongs to the interval \\([100,101]\\), what is the smallest natural number \\( n \\) such that this polynomial can have a real root?"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 13445,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a polynomial \\( P(x) \\) with integer coefficients. It is known that \\( P(1) = 2013 \\), \\( P(2013) = 1 \\), and \\( P(k) = k \\), where \\( k \\) is some integer. Find \\( k \\)."
}
] | math | {
"style": "rule",
"ground_truth": "1007"
} | {
"index": 13446,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a polynomial of degree 2022 with integer coefficients and a leading coefficient of 1, what is the maximum number of roots it can have within the interval \\( (0,1) \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"index": 13447,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a polynomial with integer coefficients,\n\\[16x^5 + b_4x^4 + b_3x^3 + b_2x^2 + b_1x + 24 = 0,\\]\nfind the number of different possible rational roots of this polynomial."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 13448,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a population of $100$ individuals randomly numbered from $0$ to $99$, and a sample of size $10$ is drawn, with the units digit of the number drawn from the $k$-th group being the same as the units digit of $m + k$, where $m = 6$, find the number drawn from the 7-th group.... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"index": 13449,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive geometric sequence $\\{a_{n}\\}$, if ${a_m}{a_n}=a_3^2$, find the minimum value of $\\frac{2}{m}+\\frac{1}{{2n}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 13450,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive integer $n,$ let $s(n)$ denote the sum of the digits of $n.$ Compute the largest positive integer $n$ such that $n = s(n)^2 + 2s(n) - 2.$"
}
] | math | {
"style": "rule",
"ground_truth": "397"
} | {
"index": 13451,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive integer $ n$, for all positive integers $ a_1, a_2, \\cdots, a_n$ that satisfy $ a_1 \\equal{} 1$, $ a_{i \\plus{} 1} \\leq a_i \\plus{} 1$, find $ \\displaystyle \\sum_{i \\equal{} 1}^{n} a_1a_2 \\cdots a_i$."
}
] | math | {
"style": "rule",
"ground_truth": "(2n-1)!!"
} | {
"index": 13452,
"solution": "\nGiven a positive integer \\( n \\), for all positive integers \\( a_1, a_2, \\cdots, a_n \\) that satisfy \\( a_1 = 1 \\) and \\( a_{i+1} \\leq a_i + 1 \\), we aim to find the sum \\( \\displaystyle \\sum_{i=1}^{n} a_1 a_2 \\cdots a_i \\).\n\nTo solve this problem, we denote \\( f(m... |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive integer $k$, let \\|k\\| denote the absolute difference between $k$ and the nearest perfect square. For example, \\|13\\|=3 since the nearest perfect square to 13 is 16. Compute the smallest positive integer $n$ such that $\\frac{\\|1\\|+\\|2\\|+\\cdots+\\|n\\|... | math | {
"style": "rule",
"ground_truth": "89800"
} | {
"index": 13453,
"solution": "Note that from $n=m^{2}$ to $n=(m+1)^{2},\\|n\\|$ increases from 0 to a peak of $m$ (which is repeated twice), and then goes back down to 0. Therefore $\\sum_{n=1}^{m^{2}}\\|n\\|=\\sum_{k=1}^{m-1} 2(1+2+\\cdots+k)=\\sum_{k=1}^{m-1} 2\\binom{k+1}{2}=2\\binom{m+1}{3}=\\frac{m}{3}\\left(... |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive integer $n$, find all $n$-tuples of real number $(x_1,x_2,\\ldots,x_n)$ such that\n\\[ f(x_1,x_2,\\cdots,x_n)=\\sum_{k_1=0}^{2} \\sum_{k_2=0}^{2} \\cdots \\sum_{k_n=0}^{2} \\big| k_1x_1+k_2x_2+\\cdots+k_nx_n-1 \\big| \\]\nattains its minimum."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{1}{n+1}, \\frac{1}{n+1}, \\ldots, \\frac{1}{n+1} \\right)"
} | {
"index": 13454,
"solution": "\nGiven a positive integer \\( n \\), we aim to find all \\( n \\)-tuples of real numbers \\( (x_1, x_2, \\ldots, x_n) \\) such that\n\\[\nf(x_1, x_2, \\cdots, x_n) = \\sum_{k_1=0}^{2} \\sum_{k_2=0}^{2} \\cdots \\sum_{k_n=0}^{2} \\left| k_1 x_1 + k_2 x_2 + \\cdots + k_n x_n - 1 \\righ... |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive integer $n$, it can be shown that every complex number of the form $r+si$, where $r$ and $s$ are integers, can be uniquely expressed in the \"base\" $-n+i$ using the integers $1,2,\\ldots,n^2$ as \"digits.\" That is, the equation\\[r+si=a_m(-n+i)^m+a_{m-1}(-n+i... | math | {
"style": "rule",
"ground_truth": "490"
} | {
"index": 13455,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive integer $n$, it can be shown that every complex number of the form $r+si$, where $r$ and $s$ are integers, can be uniquely expressed in the base $-n+i$ using the integers $0,1,2,\\ldots,n^2$ as digits. That is, the equation\n$r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1... | math | {
"style": "rule",
"ground_truth": "490"
} | {
"index": 13456,
"solution": "First, we find the first three powers of $-3+i$:\n$(-3+i)^1=-3+i ; (-3+i)^2=8-6i ; (-3+i)^3=-18+26i$\n So we need to solve the diophantine equation $a_1-6a_2+26a_3=0 \\Longrightarrow a_1-6a_2=-26a_3$.\n The minimum the left hand side can go is -54, so $a_3\\leq 2$, so we try cases:\nC... |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive integer $n$, let $M(n)$ be the largest integer $m$ such that \\[ \\binom{m}{n-1} > \\binom{m-1}{n}. \\] Evaluate \\[ \\lim_{n \\to \\infty} \\frac{M(n)}{n}. \\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3+\\sqrt{5}}{2}"
} | {
"index": 13457,
"solution": "The answer is $\\frac{3+\\sqrt{5}}{2}$. Note that for $m > n+1$, both binomial coefficients are nonzero and their ratio is \\[ {m\\choose n-1}/{m-1\\choose n} = \\frac{m!n!(m-n-1)!}{(m-1)!(n-1)!(m-n+1)!} = \\frac{mn}{(m-n+1)(m-n)}. \\] Thus the condition ${m\\choose{n-1}} > {{m-1}\\ch... |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive integer $n\\,$, let $p(n)\\,$ be the product of the non-zero digits of $n\\,$. (If $n\\,$ has only one digits, then $p(n)\\,$ is equal to that digit.) Let\n$S=p(1)+p(2)+p(3)+\\cdots+p(999)$\n .\n What is the largest prime factor of $S\\,$?"
}
] | math | {
"style": "rule",
"ground_truth": "103"
} | {
"index": 13458,
"solution": "Note that $p(1)=p(11), p(2)=p(12), p(3)=p(13), \\cdots p(19)=p(9)$, and $p(37)=3p(7)$. So $p(10)+p(11)+p(12)+\\cdots +p(19)=46$, $p(10)+p(11)+\\cdots +p(99)=46*45=2070$. We add $p(1)+p(2)+p(3)+\\cdots +p(10)=45$ to get 2115. When we add a digit we multiply the sum by that digit. Thus ... |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive integer \\( A \\) whose prime factorization can be written as \\( A = 2^{\\alpha} \\times 3^{\\beta} \\times 5^{\\gamma} \\), where \\(\\alpha, \\beta, \\gamma\\) are natural numbers. If one-half of \\( A \\) is a perfect square, one-third of \\( A \\) is a per... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 13459,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive integer \\( N \\) that has exactly nine positive divisors, with three of these divisors \\( a, b, \\) and \\( c \\) satisfying\n\\[ a + b + c = 2017 \\]\nand\n\\[ ac = b^2. \\]\n\nFind the value of \\( N \\)."
}
] | math | {
"style": "rule",
"ground_truth": "82369"
} | {
"index": 13460,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive integer \\( n \\) (\\( n \\geqslant 6 \\)), let set \\( A \\) be composed of sums of any 5 consecutive positive integers not greater than \\( n \\), and let set \\( B \\) be composed of sums of any 6 consecutive positive integers not greater than \\( n \\). If ... | math | {
"style": "rule",
"ground_truth": "12106"
} | {
"index": 13461,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive integer \\( n \\) such that \\( n \\leq 2016 \\) and \\(\\left\\{\\frac{n}{2}\\right\\}+\\left\\{\\frac{n}{4}\\right\\}+\\left\\{\\frac{n}{6}\\right\\}+\\left\\{\\frac{n}{12}\\right\\}=3\\), where \\(\\{x\\} = x - \\lfloor x \\rfloor\\) and \\(\\lfloor x \\rflo... | math | {
"style": "rule",
"ground_truth": "168"
} | {
"index": 13462,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive integer \\( n \\geq 3 \\), for an \\( n \\)-element real array \\(\\left(x_{1}, x_{2}, \\cdots, x_{n}\\right)\\), if every permutation \\( y_{1}, y_{2}, \\cdots, y_{n} \\) of it satisfies \\(\\sum_{i=1}^{n-1} y_{i} y_{i+1} \\geq -1\\), then the real array \\(\\... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 13463,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive integer \\( n \\geqslant 2 \\), positive real numbers \\( a_1, a_2, \\ldots, a_n \\), and non-negative real numbers \\( b_1, b_2, \\ldots, b_n \\), which satisfy the following conditions:\n(a) \\( a_1 + a_2 + \\cdots + a_n + b_1 + b_2 + \\cdots + b_n = n \\);\n... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13464,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive integer \\(N\\) (written in base 10), define its integer substrings to be integers that are equal to strings of one or more consecutive digits from \\(N\\), including \\(N\\) itself. For example, the integer substrings of 3208 are \\(3, 2, 0, 8, 32, 20, 320, 20... | math | {
"style": "rule",
"ground_truth": "88,888,888"
} | {
"index": 13465,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive number $x$ has two square roots, which are $2a-3$ and $5-a$, find the values of $a$ and $x$."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 13466,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive number \\( r \\), let the set \\( T = \\left\\{(x, y) \\mid x, y \\in \\mathbb{R}, \\text{ and } x^{2} + (y-7)^{2} \\leq r^{2} \\right\\} \\). This set \\( T \\) is a subset of the set \\( S = \\{(x, y) \\mid x, y \\in \\mathbb{R}, \\text{ and for any } \\theta... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"index": 13467,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive number \\(r\\) such that the set \\(T=\\left\\{(x, y) \\mid x, y \\in \\mathbf{R}\\right.\\) and \\(\\left.x^{2}+(y-7)^{2} \\leqslant r^{2}\\right\\}\\) is a subset of the set \\(S=\\{(x, y) \\mid x, y \\in \\mathbf{R}\\right.\\) and for any \\(\\theta \\in \\m... | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{2}"
} | {
"index": 13468,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive real number \\(\\alpha\\), determine the greatest real number \\(C\\) such that the inequality\n\n$$\n\\left(1+\\frac{\\alpha}{x^{2}}\\right)\\left(1+\\frac{\\alpha}{y^{2}}\\right)\\left(1+\\frac{\\alpha}{z^{2}}\\right) \\geq C \\cdot\\left(\\frac{x}{z}+\\frac{... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 13469,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive sequence $\\{a_n\\}$ whose sum of the first $n$ terms is $S_n$, if both $\\{a_n\\}$ and $\\{\\sqrt{S_n}\\}$ are arithmetic sequences with the same common difference, calculate $S_{100}$."
}
] | math | {
"style": "rule",
"ground_truth": "2500"
} | {
"index": 13470,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a positive sequence $\\{a_n\\}$ with the first term being 1, it satisfies $a_{n+1}^2 + a_n^2 < \\frac{5}{2}a_{n+1}a_n$, where $n \\in \\mathbb{N}^*$, and $S_n$ is the sum of the first $n$ terms of the sequence $\\{a_n\\}$.\n1. If $a_2 = \\frac{3}{2}$, $a_3 = x$, and $a_4 ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 13471,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a power function $y = f(x)$ whose graph passes through the point $(4, 2)$, find $f\\left( \\frac{1}{2} \\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 13472,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a prime $p$ and an integer $a$, we say that $a$ is a $\\textit{primitive root} \\pmod p$ if the set $\\{a,a^2,a^3,\\ldots,a^{p-1}\\}$ contains exactly one element congruent to each of $1,2,3,\\ldots,p-1\\pmod p$.\n\nFor example, $2$ is a primitive root $\\pmod 5$ because ... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 13473,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a prism \\(ABC-A'B'C'\\) with a base that is an equilateral triangle with side length 2, the lateral edge \\(AA'\\) forms a 45-degree angle with the edges \\(AB\\) and \\(AC\\) of the base. Point \\(A'\\) is equidistant from the planes \\(ABC\\) and \\(BB'C'C\\). Find \\(... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"index": 13474,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a pyramid $P-ABC$ where $PA=PB=2PC=2$, and $\\triangle ABC$ is an equilateral triangle with side length $\\sqrt{3}$, the radius of the circumscribed sphere of the pyramid $P-ABC$ is _______."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{5}}{2}"
} | {
"index": 13475,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a pyramid A-PBC, where PA is perpendicular to plane ABC, AB is perpendicular to AC, and BA=CA=2=2PA, calculate the height from the base PBC to the apex A."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{3}"
} | {
"index": 13476,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a pyramid P-ABCD whose base ABCD is a rectangle with side lengths AB = 2 and BC = 1, the vertex P is equidistant from all the vertices A, B, C, and D, and ∠APB = 90°. Calculate the volume of the pyramid."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{3}"
} | {
"index": 13477,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a pyramid \\( S-ABCD \\) with a square base where each side measures 2, and \\( SD \\perp \\) plane \\( ABCD \\) and \\( SD = AB \\). Determine the surface area of the circumscribed sphere of the pyramid \\( S-ABCD \\)."
}
] | math | {
"style": "rule",
"ground_truth": "12\\pi"
} | {
"index": 13478,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a pyramid with a vertex and base ABCD, each vertex is painted with one color, ensuring that two vertices on the same edge are of different colors. There are 5 different colors available. Calculate the total number of distinct coloring methods. (Answer with a number)"
}
... | math | {
"style": "rule",
"ground_truth": "420"
} | {
"index": 13479,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a pyramid-like structure with a rectangular base consisting of $4$ apples by $7$ apples, each apple above the first level resting in a pocket formed by four apples below, and the stack topped off with a single row of apples, determine the total number of apples in the sta... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 13480,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a quadratic equation \\( x^{2} + bx + c = 0 \\) with roots 98 and 99, within the quadratic function \\( y = x^{2} + bx + c \\), if \\( x \\) takes on values 0, 1, 2, 3, ..., 100, how many of the values of \\( y \\) are divisible by 6?"
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 13481,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a quadratic function $f(x) = ax^2 + bx + 1$ that satisfies $f(-1) = 0$, and when $x \\in \\mathbb{R}$, the range of $f(x)$ is $[0, +\\infty)$. \n(1) Find the expression for $f(x)$. \n(2) Let $g(x) = f(x) - 2kx$, where $k \\in \\mathbb{R}$. \n (i) If $g(x)$ is monoton... | math | {
"style": "rule",
"ground_truth": "k = 6"
} | {
"index": 13482,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a quadratic function $f(x) = ax^2 + bx + c$ (where $a$, $b$, and $c$ are constants). If the solution set of the inequality $f(x) \\geq 2ax + b$ is $\\mathbb{R}$ (the set of all real numbers), then the maximum value of $\\frac{b^2}{a^2 + c^2}$ is __________."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} - 2"
} | {
"index": 13483,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a quadratic function $f(x) = ax^2 - 4bx + 1$.\n\n(1) Let set $P = \\{-1,1,2,3,4,5\\}$ and set $Q = \\{-2,-1,1,2,3,4\\}$. Randomly select a number from set $P$ as $a$ and from set $Q$ as $b$. Calculate the probability that the function $y = f(x)$ is increasing on the inter... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{3}"
} | {
"index": 13484,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a quadratic function $f(x) = x^2 + mx + n$.\n1. If $f(x)$ is an even function and its minimum value is 1, find the expression for $f(x)$.\n2. Based on (1), for the function $g(x) = \\frac{6x}{f(x)}$, solve the inequality $g(2^x) > 2^x$ for $x$.\n3. Let $h(x) = |f(x)|$ and... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13485,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a quadratic function $f(x)$ with a second-degree coefficient $a$, and the inequality $f(x) > -2x$ has the solution set $(1,3)$:\n(1) If the function $y = f(x) + 6a$ has exactly one zero, find the explicit form of $f(x)$.\n(2) Let $h(a)$ be the maximum value of $f(x)$, fin... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 13486,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a quadratic function $f\\left(x\\right)=ax^{2}+bx+c$, where $f\\left(0\\right)=1$, $f\\left(1\\right)=0$, and $f\\left(x\\right)\\geqslant 0$ for all real numbers $x$. <br/>$(1)$ Find the analytical expression of the function $f\\left(x\\right)$; <br/>$(2)$ If the maximum... | math | {
"style": "rule",
"ground_truth": "m = 2"
} | {
"index": 13487,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a quadratic function $y=-x^{2}+bx+c$ where $b$ and $c$ are constants. \n$(1)$ If $y=0$ and the corresponding values of $x$ are $-1$ and $3$, find the maximum value of the quadratic function. \n$(2)$ If $c=-5$, and the quadratic function $y=-x^{2}+bx+c$ intersects the li... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 13488,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a quadratic function $y=ax^{2}-4ax+3+b\\left(a\\neq 0\\right)$. \n$(1)$ Find the axis of symmetry of the graph of the quadratic function; \n$(2)$ If the graph of the quadratic function passes through the point $\\left(1,3\\right)$, and the integers $a$ and $b$ satisfy $... | math | {
"style": "rule",
"ground_truth": "t = \\frac{5}{2}"
} | {
"index": 13489,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a quadratic function \\( f(x) = x^2 + ax + b \\), determine the minimum value of \\( F = \\max_{|x| \\leq 1} \\mid f(x) \\), where \\( a \\) and \\( b \\) are real numbers."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13490,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a quadratic function in terms of \\\\(x\\\\), \\\\(f(x)=ax^{2}-4bx+1\\\\).\n\\\\((1)\\\\) Let set \\\\(P=\\\\{1,2,3\\\\}\\\\) and \\\\(Q=\\\\{-1,1,2,3,4\\\\}\\\\), randomly pick a number from set \\\\(P\\\\) as \\\\(a\\\\) and from set \\\\(Q\\\\) as \\\\(b\\\\), calculat... | math | {
"style": "rule",
"ground_truth": "\\dfrac{961}{1280}"
} | {
"index": 13491,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a quadratic polynomial $q(x) = x^2 - px + q$ known to be \"mischievous\" if the equation $q(q(x)) = 0$ is satisfied by exactly three different real numbers, determine the value of $q(2)$ for the unique polynomial $q(x)$ for which the product of its roots is minimized."
... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 13492,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a quadratic polynomial \\( P(x) \\). It is known that the equations \\( P(x) = x - 2 \\) and \\( P(x) = 1 - x / 2 \\) each have exactly one root. What is the discriminant of \\( P(x) \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 13493,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a quadrilateral $ABCD$ with $AB = BC =3$ cm, $CD = 4$ cm, $DA = 8$ cm and $\\angle DAB + \\angle ABC = 180^o$ . Calculate the area of the quadrilateral."
}
] | math | {
"style": "rule",
"ground_truth": "13.2"
} | {
"index": 13494,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a quadrilateral $ABCD$ inscribed in a circle with side $AB$ extended beyond $B$ to point $E$, if $\\angle BAD=92^\\circ$ and $\\angle ADC=68^\\circ$, find $\\angle EBC$."
}
] | math | {
"style": "rule",
"ground_truth": "68^\\circ"
} | {
"index": 13495,
"solution": "1. **Identify Properties of Cyclic Quadrilateral**: Since $ABCD$ is a cyclic quadrilateral, by the Inscribed Angle Theorem, opposite angles in a cyclic quadrilateral sum to $180^\\circ$. Thus, we have:\n \\[\n \\angle ABC + \\angle ADC = 180^\\circ\n \\]\n\n2. **Calculate $\\ang... |
DeepScaleR | [
{
"role": "user",
"content": "Given a quadrilateral $ABCD$ where the internal angles form an arithmetic progression, the angles are such that $\\angle ADB$ is double the $\\angle DBA$. Triangles $ABD$ and $DCB$ are similar with $\\angle DBA = \\angle DCB$ and $\\angle ADB = \\angle CBD$. Furthermore, the an... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 13496,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a quadrilateral formed by the two foci and the two endpoints of the conjugate axis of a hyperbola $C$, one of its internal angles is $60^{\\circ}$. Determine the eccentricity of the hyperbola $C$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"index": 13497,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a random variable $0.4987X \\sim N\\left( 9, \\sigma^2 \\right)$, and $P(X < 6) = 0.2$, determine the probability that $9 < X < 12$."
}
] | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"index": 13498,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a random variable $X \\sim B(10, 0.6)$, calculate the values of $E(X)$ and $D(X)$."
}
] | math | {
"style": "rule",
"ground_truth": "2.4"
} | {
"index": 13499,
"solution": ""
} |
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