data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given that both $α$ and $β$ are acute angles, and $\\cos(α + β) = \\frac{\\sin α}{\\sin β}$, find the maximum value of $\\tan α$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{4}"
} | {
"index": 15400,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that both $α$ and $β$ are acute angles, and $\\cos(α+β)= \\frac{\\sin α}{\\sin β}$, find the maximum value of $\\tan α$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{ \\sqrt {2}}{4}"
} | {
"index": 15401,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that bricklayer Alice can build a wall alone in 8 hours, and bricklayer Bob can build it alone in 12 hours, and they complete the wall in 6 hours when working together with a 15-brick-per-hour decrease in productivity, determine the number of bricks in the wall."
}
] | math | {
"style": "rule",
"ground_truth": "360"
} | {
"index": 15402,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that children enter at a discounted rate, half that of an adult ticket, and the total cost for $6$ adult tickets and $5$ child tickets amounts to $32.50$, calculate the total cost for $10$ adult tickets and $8$ child tickets."
}
] | math | {
"style": "rule",
"ground_truth": "53.50"
} | {
"index": 15403,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that circle $A$ has radius $150$, and circle $B$, with an integer radius $r$, is externally tangent to circle $A$ and rolls once around the circumference of circle $A$, determine the number of possible integer values of $r$."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 15404,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that circle $C$ passes through points $P(0,-4)$, $Q(2,0)$, and $R(3,-1)$. \n$(1)$ Find the equation of circle $C$. \n$(2)$ If the line $l: mx+y-1=0$ intersects circle $C$ at points $A$ and $B$, and $|AB|=4$, find the value of $m$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 15405,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that circle $C$ passes through the point $(0,2)$ with a radius of $2$, if there exist two points on circle $C$ that are symmetric with respect to the line $2x-ky-k=0$, find the maximum value of $k$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{5}}{5}"
} | {
"index": 15406,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that circle $\\odot M$ passes through the point $(1,0)$ and is tangent to the line $x=-1$, $S$ is a moving point on the trajectory of the center $M$ of the circle, and $T$ is a moving point on the line $x+y+4=0$. Find the minimum value of $|ST|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{2}"
} | {
"index": 15407,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that complex numbers $a,$ $b,$ and $c$ are zeros of a polynomial $P(z) = z^3 + qz + r,$ and $|a|^2 + |b|^2 + |c|^2 = 300$. The points corresponding to $a,$ $b,$ and $c$ on the complex plane are the vertices of a right triangle. Find the square of the length of the hypoten... | math | {
"style": "rule",
"ground_truth": "450"
} | {
"index": 15408,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that connecting all the vertices of a polygon from a point on one of the edges results in 2022 triangles, determine the number of sides of this polygon."
}
] | math | {
"style": "rule",
"ground_truth": "2023"
} | {
"index": 15409,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that construction teams A and B each have a certain number of workers. If team A lends 90 workers to team B, then team B's total number of workers will be twice that of team A. If team B lends a certain number of workers to team A, then team A's total number of workers wi... | math | {
"style": "rule",
"ground_truth": "153"
} | {
"index": 15410,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that construction teams A and B each have a certain number of workers. If team A transfers 90 workers to team B, the total number of workers in team B will be twice that of team A. If team B transfers a certain number of workers to team A, then the total number of workers... | math | {
"style": "rule",
"ground_truth": "153"
} | {
"index": 15411,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that cos(15°+α) = $\\frac{3}{5}$, where α is an acute angle, find: $$\\frac{tan(435° -α)+sin(α-165° )}{cos(195 ° +α )\\times sin(105 ° +α )}$$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{36}"
} | {
"index": 15412,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that cosα + 2cos(α + $$\\frac{π}{3}$$) = 0, find tan(α + $$\\frac{π}{6}$$)."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"index": 15413,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that each of the smaller rectangles has a shorter side of 7 feet and a longer side with a length three times that of the shorter side, and that two smaller rectangles are placed adjacent to each other by their longer sides and the third rectangle is placed adjacent by its... | math | {
"style": "rule",
"ground_truth": "588"
} | {
"index": 15414,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that each side of a large square is divided into four equal parts, a smaller square is inscribed in such a way that its corners are at the division points one-fourth and three-fourths along each side of the large square, calculate the ratio of the area of this inscribed s... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 15415,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that each side of a triangle is an integer and none exceeds 4, determine the number of such distinct triangles."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 15416,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that events $A$ and $B$ are independent, and $P(A)=\\frac{1}{2}$, $P(B)=\\frac{2}{3}$, find $P(\\overline{AB})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 15417,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that events A and B are independent, and both are mutually exclusive with event C. It is known that $P(A) = 0.2$, $P(B) = 0.6$, and $P(C) = 0.14$. Find the probability that at least one of A, B, or C occurs, denoted as $P(A+B+C)$."
}
] | math | {
"style": "rule",
"ground_truth": "0.82"
} | {
"index": 15418,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that every high school in the town of Pythagoras sent a team of 3 students to a math contest, and Andrea's score was the median among all students, and hers was the highest score on her team, and Andrea's teammates Beth and Carla placed 40th and 75th, respectively, calcul... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 15419,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that f(x) = |log₃x|, if a and b satisfy f(a - 1) = f(2b - 1), and a ≠ 2b, then the minimum value of a + b is ___."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2} + \\sqrt{2}"
} | {
"index": 15420,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that five boys, A, B, C, D, and E, are randomly assigned to stay in 3 standard rooms (with at most two people per room), calculate the probability that A and B stay in the same standard room."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 15421,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that five students from Maplewood school worked for 6 days, six students from Oakdale school worked for 4 days, and eight students from Pinecrest school worked for 7 days, and the total amount paid for the students' work was 1240 dollars, determine the total amount earned... | math | {
"style": "rule",
"ground_truth": "270.55"
} | {
"index": 15422,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that five volunteers are randomly assigned to conduct promotional activities in three communities, A, B, and C, at least 2 volunteers are assigned to community A, and at least 1 volunteer is assigned to each of communities B and C, calculate the number of different arrang... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 15423,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that for any positive integer \\( n \\), \\( 9^{2n} - 8^{2n} - 17 \\) is always divisible by \\( m \\), find the largest positive integer \\( m \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2448"
} | {
"index": 15424,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that for reals $a_1,\\cdots, a_{2004},$ equation $x^{2006}-2006x^{2005}+a_{2004}x^{2004}+\\cdots +a_2x^2+a_1x+1=0$ has $2006$ positive real solution, find the maximum possible value of $a_1.$"
}
] | math | {
"style": "rule",
"ground_truth": "-2006"
} | {
"index": 15425,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that for triangle $ABC$, the internal angles $A$ and $B$ satisfy $$\\frac {\\sin B}{\\sin A} = \\cos(A + B),$$ find the maximum value of $\\tan B$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{4}"
} | {
"index": 15426,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that four A's, four B's, four C's, and four D's are to be placed in a 4 × 4 grid so that each row and column contains one of each letter, and A is placed in the upper right corner, calculate the number of possible arrangements."
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"index": 15427,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that four integers \\( a, b, c, d \\) are all even numbers, and \\( 0 < a < b < c < d \\), with \\( d - a = 90 \\). If \\( a, b, c \\) form an arithmetic sequence and \\( b, c, d \\) form a geometric sequence, then find the value of \\( a + b + c + d \\)."
}
] | math | {
"style": "rule",
"ground_truth": "194"
} | {
"index": 15428,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that four people A, B, C, D are randomly selected for a volunteer activity, find the probability that A is selected and B is not."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 15429,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in $\\triangle ABC$, $AB=4$, $AC=6$, $BC= \\sqrt{7}$, and the center of its circumcircle is $O$, find $\\overset{⇀}{AO}· \\overset{⇀}{BC} =$ ___."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 15430,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in $\\triangle ABC$, $B= \\frac{\\pi}{4}$ and the height to side $BC$ is equal to $\\frac{1}{3}BC$, calculate the value of $\\sin A$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{10}}{10}"
} | {
"index": 15431,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in $\\triangle ABC$, $BD:DC = 3:2$ and $AE:EC = 3:4$, and the area of $\\triangle ABC$ is 1, find the area of $\\triangle BMD$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{15}"
} | {
"index": 15432,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in $\\triangle ABC$, $C = 2A$, $\\cos A = \\frac{3}{4}$, and $2 \\overrightarrow{BA} \\cdot \\overrightarrow{CB} = -27$.\n\n(I) Find the value of $\\cos B$;\n\n(II) Find the perimeter of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 15433,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in $\\triangle ABC$, $\\sin A + 2 \\sin B \\cos C = 0$, find the maximum value of $\\tan A$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 15434,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b\\sin A+a\\cos B=0$.\n(1) Find the measure of angle $B$;\n(2) If $b=2$, find the maximum area of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}-1"
} | {
"index": 15435,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in a class of 36 students, more than half purchased notebooks from a store where each notebook had the same price in cents greater than the number of notebooks bought by each student, and the total cost for the notebooks was 990 cents, calculate the price of each not... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 15436,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in a class test, $15\\%$ of the students scored $60$ points, $50\\%$ scored $75$ points, $20\\%$ scored $85$ points, and the rest scored $95$ points, calculate the difference between the mean and median score of the students' scores on this test."
}
] | math | {
"style": "rule",
"ground_truth": "2.75"
} | {
"index": 15437,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in a mathematics test, $20\\%$ of the students scored $60$ points, $25\\%$ scored $75$ points, $20\\%$ scored $85$ points, $25\\%$ scored $95$ points, and the rest scored $100$ points, calculate the difference between the mean and the median score of the students' sc... | math | {
"style": "rule",
"ground_truth": "6.5"
} | {
"index": 15438,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in acute triangle ABC, the sides opposite to angles A, B, C are a, b, c respectively, and bsinA = acos(B - $ \\frac{\\pi}{6}$).\n(1) Find the value of angle B.\n(2) If b = $\\sqrt{13}$, a = 4, and D is a point on AC such that S<sub>△ABD</sub> = 2$\\sqrt{3}$, find the... | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{13}}{3}"
} | {
"index": 15439,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in quadrilateral $ABCD$, $m\\angle B = m \\angle C = 120^{\\circ}, AB=3, BC=4,$ and $CD=5$, calculate the area of $ABCD$."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{3}"
} | {
"index": 15440,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in quadrilateral ABCD, $\\angle A : \\angle B : \\angle C : \\angle D = 1 : 3 : 5 : 6$, express the degrees of $\\angle A$ and $\\angle D$ in terms of a common variable."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 15441,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in rhombus $ABCD$, the diagonal $AC$ is twice as long as diagonal $BD$, and point $E$ is on segment $AC$ such that line segment $BE$ bisects $\\angle ABC$, determine the ratio of the area of triangle $ABE$ to the area of rhombus $ABCD$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 15442,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in square ABCD, AE = 3EC and BF = 2FB, and G is the midpoint of CD, find the ratio of the area of triangle EFG to the area of square ABCD."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{24}"
} | {
"index": 15443,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in the decimal system 2004 equates to $4 \\times 10^0 + 0 \\times 10^1 + 0 \\times 10^2 + 2 \\times 10^3$, by analogy, calculate the equivalent value of $2004$ in base 5 in decimal form."
}
] | math | {
"style": "rule",
"ground_truth": "254"
} | {
"index": 15444,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in the expansion of $(1-2x)^{n} (n \\in \\mathbb{N^*})$, the coefficient of $x^{3}$ is $-80$, find the sum of all the binomial coefficients in the expansion."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 15445,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in the expansion of $\\left(1+x\\right)^{n}$, the coefficient of $x^{3}$ is the largest, then the sum of the coefficients of $\\left(1+x\\right)^{n}$ is ____."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 15446,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in the expansion of \\\\((1+2x)^{n}\\\\), only the coefficient of the fourth term is the maximum, then the constant term in the expansion of \\\\((1+ \\dfrac {1}{x^{2}})(1+2x)^{n}\\\\) is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "61"
} | {
"index": 15447,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in the geometric sequence $\\{a_n\\}$ where all terms are positive, $a_1a_3=16$ and $a_3+a_4=24$, find the value of $a_5$."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 15448,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in the polar coordinate system, the equation of the curve $\\Omega$ is $\\rho=6\\cos \\theta$. With the pole as the origin of the rectangular coordinate system, the polar axis as the positive semi-axis of the $x$-axis, and the same length unit in both coordinate syst... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 15449,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in the rectangular coordinate system $(xOy)$, the origin is the pole and the positive semi-axis of $x$ is the polar axis to establish a polar coordinate system, the polar coordinate equation of the conic section $(C)$ is $p^{2}= \\frac {12}{3+\\sin ^{2}\\theta }$, th... | math | {
"style": "rule",
"ground_truth": "\\frac {12}{5}"
} | {
"index": 15450,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in the rectangular coordinate system $(xOy)$, the parametric equations of the curve $C$ are $ \\begin{cases} x=2+2\\cos θ \\\ny=2\\sin θ\\end{cases} $ for the parameter $(θ)$, and in the polar coordinate system $(rOθ)$ (with the same unit length as the rectangular co... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {2}"
} | {
"index": 15451,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in triangle $ABC$, $\\overrightarrow{A B} \\cdot \\overrightarrow{B C} = 3 \\overrightarrow{C A} \\cdot \\overrightarrow{A B}$, find the maximum value of $\\frac{|\\overrightarrow{A C}| + |\\overrightarrow{A B}|}{|\\overrightarrow{B C}|}$:"
}
] | math | {
"style": "rule",
"ground_truth": "$\\sqrt{3}$"
} | {
"index": 15452,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in triangle $\\triangle ABC$, $a=2$, $\\angle A=\\frac{π}{6}$, $b=2\\sqrt{3}$, find the measure of angle $\\angle C$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"index": 15453,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in triangle $\\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is known that $\\frac{{c \\sin C}}{a} - \\sin C = \\frac{{b \\sin B}}{a} - \\sin A$, $b = 4$. Find:\n\n$(1)$ The measure of angle $B$;\n\n$(2)$ If ... | math | {
"style": "rule",
"ground_truth": "4 + \\frac{{4\\sqrt{3}}}{3}"
} | {
"index": 15454,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $2\\sin A + \\sin B = 2\\sin C\\cos B$ and the area of $\\triangle ABC$ is $S = \\frac{\\sqrt{3}}{2}c$, find the minimum value of $ab$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 15455,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\\angle BAC = 60^{\\circ}$, $D$ is a point on side $BC$ such that $AD = \\sqrt{7}$, and $BD:DC = 2c:b$, then the minimum value of the area of $\\triangle... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 15456,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\\sqrt{3}a\\cos C=c\\sin A$. \n$(1)$ Find the measure of angle $C$. \n$(2)$ If $a > 2$ and $b-c=1$, find the minimum perimeter of triangle $\\trian... | math | {
"style": "rule",
"ground_truth": "9 + 6\\sqrt{2}"
} | {
"index": 15457,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and $\\left(a+b\\right)\\left(\\sin A-\\sin B\\right)=\\left(c-b\\right)\\sin C$ with $a=2$, find the maximum area of triangle $\\triangle ABC$... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 15458,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in triangle $\\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Angle $B$ is obtuse. Let the area of $\\triangle ABC$ be $S$. If $4bS=a(b^{2}+c^{2}-a^{2})$, then the maximum value of $\\sin A + \\sin C$ is... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{8}"
} | {
"index": 15459,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in triangle ABC, the lengths of the sides opposite to angles A, B, and C are a, b, and c respectively, and b = 3, c = 1, and A = 2B.\n(1) Find the value of a;\n(2) Find the value of sin(A + $\\frac{\\pi}{4}$)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4 - \\sqrt{2}}{6}"
} | {
"index": 15460,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in triangle PQR, side PR = 6 cm and side PQ = 10 cm, point S is the midpoint of QR, and the length of the altitude from P to QR is 4 cm, calculate the length of QR."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{5}"
} | {
"index": 15461,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that in △ABC, the sides opposite to the internal angles A, B, and C are a, b, and c respectively, and $b^{2}=c^{2}+a^{2}- \\sqrt {2}ac$.\n(I) Find the value of angle B;\n(II) If $a= \\sqrt {2}$ and $cosA= \\frac {4}{5}$, find the area of △ABC."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {7}{6}"
} | {
"index": 15462,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that it is currently between 4:00 and 5:00 o'clock, and eight minutes from now, the minute hand of a clock will be exactly opposite to the position where the hour hand was six minutes ago, determine the exact time now."
}
] | math | {
"style": "rule",
"ground_truth": "4:45\\frac{3}{11}"
} | {
"index": 15463,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that line $MN$ passes through the left focus $F$ of the ellipse $\\frac{x^{2}}{2}+y^{2}=1$ and intersects the ellipse at points $M$ and $N$. Line $PQ$ passes through the origin $O$ and is parallel to $MN$, intersecting the ellipse at points $P$ and $Q$. Find the value of ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 15464,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that line $l$ intersects circle $C$: $x^2+y^2+2x-4y+a=0$ at points $A$ and $B$, and the midpoint of chord $AB$ is $P(0,1)$.\n(I) If the radius of circle $C$ is $\\sqrt{3}$, find the value of the real number $a$;\n(II) If the length of chord $AB$ is $6$, find the value of ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{11}"
} | {
"index": 15465,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that line $l$ intersects the hyperbola $\\frac{x^2}{9} - \\frac{y^2}{4} = 1$ at points $A$ and $B$, and the midpoint of segment $AB$ has coordinates $(6, 2)$, determine the slope of line $l$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 15466,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that line $l$ is perpendicular to plane $\\alpha$, and line $m$ is contained in plane $\\beta$. Consider the following propositions: \n(1) If $\\alpha \\parallel \\beta$, then $l \\perp m$. \n(2) If $\\alpha \\perp \\beta$, then $l \\parallel m$. \n(3) If $l \\parallel m$... | math | {
"style": "rule",
"ground_truth": "(3)"
} | {
"index": 15467,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that line $l$ passes through point $P(-1,2)$ with a slope angle of $\\frac{2\\pi}{3}$, and the circle's equation is $\\rho=2\\cos (\\theta+\\frac{\\pi}{3})$:\n(1) Find the parametric equation of line $l$;\n(2) Let line $l$ intersect the circle at points $M$ and $N$, find ... | math | {
"style": "rule",
"ground_truth": "6+2\\sqrt{3}"
} | {
"index": 15468,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that line $l\\_1$ passes through points $A(m,1)$ and $B(-3,4)$, and line $l\\_2$ passes through points $C(1,m)$ and $D(-1,m+1)$, find the values of the real number $m$ when $l\\_1$ is parallel to $l\\_2$ or $l\\_1$ is perpendicular to $l\\_2$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{9}{2}"
} | {
"index": 15469,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that line $l_{1}$ passes through points $A(m,1)$ and $B(-3,4)$, and line $l_{2}$ passes through points $C(1,m)$ and $D(-1,m+1)$, for what value of $m$ do the lines $l_{1}$ and $l_{2}$\n\n$(1)$ intersect at a right angle;\n\n$(2)$ are parallel;\n\n$(3)$ the angle of inclin... | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"index": 15470,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that line l: x - y + 1 = 0 is tangent to the parabola C with focus F and equation y² = 2px (p > 0).\n(I) Find the equation of the parabola C;\n(II) The line m passing through point F intersects parabola C at points A and B. Find the minimum value of the sum of the distanc... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{2}"
} | {
"index": 15471,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that m > 0, p: 0 < x < m, q: x(x - 1) < 0, if p is a sufficient but not necessary condition for q, then the value of m can be _______. (Only one value of m that satisfies the condition is needed)"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 15472,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that non-negative real numbers \\( x \\) and \\( y \\) satisfy\n\\[ \nx^{2}+4y^{2}+4xy+4x^{2}y^{2}=32,\n\\]\nfind the maximum value of \\( \\sqrt{7}(x+2y)+2xy \\)."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 15473,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that odd prime numbers \\( x, y, z \\) satisfy\n\\[ x \\mid (y^5 + 1), \\quad y \\mid (z^5 + 1), \\quad z \\mid (x^5 + 1). \\]\n\nFind the minimum value of the product \\( xyz \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2013"
} | {
"index": 15474,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that one air conditioner sells for a 10% profit and the other for a 10% loss, and the two air conditioners have the same selling price, determine the percentage change in the shopping mall's overall revenue."
}
] | math | {
"style": "rule",
"ground_truth": "1\\%"
} | {
"index": 15475,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that one of the children is a boy and the probability of having a boy or a girl is equal, calculate the probability that the other child is a girl."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 15476,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that one of the roots of the function $f(x)=ax+b$ is $2$, find the roots of the function $g(x)=bx^{2}-ax$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 15477,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that one root of the equation $x^{2}+mx+3=0$ is $1$, find the other root and the value of $m$."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 15478,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that one third of the students go home on the school bus, one fifth go home by automobile, one eighth go home on their bicycles, and one tenth go home on scooters, what fractional part of the students walk home?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{29}{120}"
} | {
"index": 15479,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that our number system has a base of eight, determine the fifteenth number in the sequence."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 15480,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that out of 6 products, 2 are defective and the rest are qualified, calculate the probability of selecting exactly one defective product from these 6 products."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{15}"
} | {
"index": 15481,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that out of 8 teams, there are 3 weak teams, these 8 teams are divided into two groups $A$ and $B$ with 4 teams in each group by drawing lots.\n\n1. The probability that one of the groups $A$ or $B$ has exactly two weak teams.\n2. The probability that group $A$ has at lea... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 15482,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that p, q, r, and s are integers in the set {0, 1, 2, 3, 4}, calculate the number of ordered quadruples (p, q, r, s) such that p·s + q·r is odd."
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"index": 15483,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that plane vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are non-zero vectors, $|\\overrightarrow{a}|=2$, and $\\overrightarrow{a} \\bot (\\overrightarrow{a}+2\\overrightarrow{b})$, calculate the projection of vector $\\overrightarrow{b}$ in the direction of vec... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 15484,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that player A needs to win 2 more games and player B needs to win 3 more games, and the probability of winning each game for both players is $\\dfrac{1}{2}$, calculate the probability of player A ultimately winning."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{11}{16}"
} | {
"index": 15485,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $(a, b)$ moves on the line $x + 2y + 3 = 0$, find the maximum or minimum value of $2^a + 4^b$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 15486,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $(x, y)$ moves on the circle $x^{2}+(y-1)^{2}=1$.\n(1) Find the maximum and minimum values of $\\frac{y-1}{x-2}$;\n(2) Find the maximum and minimum values of $2x+y$."
}
] | math | {
"style": "rule",
"ground_truth": "1 - \\sqrt{5}"
} | {
"index": 15487,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $A(-2,3)$ lies on the axis of parabola $C$: $y^{2}=2px$, and the line passing through point $A$ is tangent to $C$ at point $B$ in the first quadrant. Let $F$ be the focus of $C$. Then, $|BF|=$ _____ ."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 15488,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $A(-2,α)$ lies on the terminal side of angle $α$ and $\\sin α=- \\dfrac{ \\sqrt{5}}{5}$.\n\n(1) Find the values of $α$, $\\cos α$, and $\\tan α$.\n\n(2) Find the value of $\\dfrac{\\cos ( \\dfrac{π}{2}+α)\\sin (-π-α)}{\\cos ( \\dfrac{11π}{2}-α)\\sin ( \\dfrac{9... | math | {
"style": "rule",
"ground_truth": "\\dfrac {1}{2}"
} | {
"index": 15489,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $A(1,1)$ is a point on the ellipse $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$, and $F\\_1$, $F\\_2$ are the two foci of the ellipse such that $|AF\\_1|+|AF\\_2|=4$.\n(I) Find the standard equation of the ellipse;\n(II) Find the equation of the t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 15490,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $A(1,\\sqrt{5})$ lies on the parabola $C:y^{2}=2px$, the distance from $A$ to the focus of $C$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"index": 15491,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $C$ is the midpoint between points $A$ and $B$. At 7:00 AM, Car 1 departs from $A$ towards $B$, Car 2 departs from $B$ towards $A$, and Car 3 departs from $C$ towards $A$. When Car 1 and Car 3 meet, Car 2 has traveled exactly $\\frac{3}{8}$ of the total distanc... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"index": 15492,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $M$ is the midpoint of line segment $AB$ in plane $\\alpha$, and point $P$ is a point outside plane $\\alpha$. If $AB = 2$ and the angles between lines $PA$, $PM$, $PB$ and plane $\\alpha$ are $30^{\\circ}$, $45^{\\circ}$, and $60^{\\circ}$ respectively, find t... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"index": 15493,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $M$ lies on the circle $C:x^{2}+y^{2}-4x-14y+45=0$, and point $Q(-2,3)$.\n\n(1) If $P(a,a+1)$ is on circle $C$, find the length of segment $PQ$ and the slope of line $PQ$;\n\n(2) Find the maximum and minimum values of $|MQ|$;\n\n(3) If $M(m,n)$, find the maximu... | math | {
"style": "rule",
"ground_truth": "2- \\sqrt {3}"
} | {
"index": 15494,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $M$ represents the number $9$ on the number line.<br/>$(1)$ If point $N$ is first moved $4$ units to the left and then $6$ units to the right to reach point $M$, then the number represented by point $N$ is ______.<br/>$(2)$ If point $M$ is moved $4$ units on th... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 15495,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $M(x\\_0, y\\_0)$ moves on the circle $x^{2}+y^{2}=4$, $N(4,0)$, and point $P(x,y)$ is the midpoint of segment $MN$.\n\n(1) Find the trajectory equation of point $P(x,y)$;\n\n(2) Find the maximum and minimum distances from point $P(x,y)$ to the line $3x+4y-86=0... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 15496,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $O$ is the origin of coordinates, point $A$ in the first quadrant lies on the graph of the inverse proportional function $y=\\frac{1}{x}$ for $x>0$, and point $B$ in the second quadrant lies on the graph of the inverse proportional function $y=-\\frac{4}{x}$ fo... | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{2}$"
} | {
"index": 15497,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ is a moving point on circle $C$: $x^{2}+y^{2}-2x-4y+1=0$, the maximum distance from point $P$ to a certain line $l$ is $6$. If a point $A$ is taken arbitrarily on line $l$ to form a tangent line $AB$ to circle $C$, with $B$ being the point of tangency, then... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 15498,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that point $P$ is a moving point on circle $C_{1}$: $\\left(x-1\\right)^{2}+y^{2}=1$, point $Q$ is a moving point on circle $C_{2}$: $\\left(x-4\\right)^{2}+\\left(y-1\\right)^{2}=4$, and point $R$ moves on the line $l: x-y+1=0$, find the minimum value of $|PR|+|QR|$."
... | math | {
"style": "rule",
"ground_truth": "\\sqrt{26}-3"
} | {
"index": 15499,
"solution": ""
} |
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