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 \\(\\triangle ABC\\) has sides \\(a\\), \\(b\\), and \\(c\\) corresponding to angles \\(A\\), \\(B\\), and \\(C\\) respectively, and knowing that \\(a + b + c = 16\\), find the value of \\(b^2 \\cos^2 \\frac{C}{2} + c^2 \\cos^2 \\frac{B}{2} + 2bc \\cos \\frac{B}{2} \... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 15300,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(a - b = 2 + \\sqrt{3}\\) and \\(b - c = 2 - \\sqrt{3}\\), find the value of \\(a^2 + b^2 + c^2 - ab - bc - ca\\)."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 15301,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(a \\leq b < c\\) are the side lengths of a right triangle, find the maximum constant \\(M\\) such that\n$$\n\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c} \\geq \\frac{M}{a+b+c} .\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "5 + 3\\sqrt{2}"
} | {
"index": 15302,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(a, b, c, a+b-c, a+c-b, b+c-a, a+b+c\\) are 7 distinct prime numbers, and the sum of any two of \\(a, b, c\\) is 800. Let \\(d\\) be the difference between the largest and smallest of these 7 prime numbers. Find the maximum possible value of \\(d\\)."
}
] | math | {
"style": "rule",
"ground_truth": "1594"
} | {
"index": 15303,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(a, b, c, d\\) are within the interval \\(\\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right]\\), and \\(\\sin a + \\sin b + \\sin c + \\sin d = 1\\), and \\(\\cos 2a + \\cos 2b + \\cos 2c + \\cos 2d \\geq \\frac{10}{3}\\), what is the maximum value of \\(a\\)? \\(\\q... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"index": 15304,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(a, b, c\\) are positive integers and the quadratic equation \\(a x^{2}+b x+c=0\\) has two real roots whose absolute values are both less than \\(\\frac{1}{3}\\), find the minimum value of \\(a + b + c\\)."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 15305,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(a\\) and \\(b\\) are real numbers, and the equation \\( x^{4} + a x^{3} + b x^{2} + a x + 1 = 0 \\) has at least one real root, find the minimum value of \\(a^{2} + b^{2}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "4/5"
} | {
"index": 15306,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(a\\) and \\(b\\) are real numbers, and the polynomial \\(x^{4} + a x^{3} + b x^{2} + a x + 1 = 0\\) has at least one real root, determine the minimum value of \\(a^{2} + b^{2}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "4/5"
} | {
"index": 15307,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(a\\), \\(b\\), \\(c\\), and \\(d\\) are four positive prime numbers such that the product of these four prime numbers is equal to the sum of 55 consecutive positive integers, find the smallest possible value of \\(a + b + c + d\\). Note that the four numbers \\(a\... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 15308,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(a_{1}, a_{2}, \\cdots, a_{n}\\) are \\(n\\) people corresponding to \\(A_{1}, A_{2}, \\cdots, A_{n}\\) cards (\\(n \\geq 2\\), \\(a_{i}\\) corresponds to \\(A_{i}\\)). Now \\(a_{1}\\) picks a card from the deck randomly, and then each person in sequence picks a ca... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 15309,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(x\\) is a positive real, find the maximum possible value of \\(\\sin \\left(\\tan ^{-1}\\left(\\frac{x}{9}\\right)-\\tan ^{-1}\\left(\\frac{x}{16}\\right)\\right)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{25}"
} | {
"index": 15310,
"solution": "Consider a right triangle \\(A O C\\) with right angle at \\(O, A O=16\\) and \\(C O=x\\). Moreover, let \\(B\\) be on \\(A O\\) such that \\(B O=9\\). Then \\(\\tan ^{-1} \\frac{x}{9}=\\angle C B O\\) and \\(\\tan ^{-1} \\frac{x}{16}=\\angle C A O\\), so their difference is equal to ... |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(x\\) is a real number, find the least possible value of \\((x+2)(x+3)(x+4)(x+5)+3033\\)."
}
] | math | {
"style": "rule",
"ground_truth": "3032"
} | {
"index": 15311,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\(x\\) satisfies \\(\\log _{5x} (2x) = \\log _{625x} (8x)\\), find the value of \\(\\log _{2} x\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\ln 5}{2 \\ln 2 - 3 \\ln 5}"
} | {
"index": 15312,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\\\(AB\\\\) is a chord passing through the focus of the parabola \\\\(y^{2} = 4\\sqrt{3}x\\\\), and the midpoint \\\\(M\\\\) of \\\\(AB\\\\) has an x-coordinate of \\\\(2\\\\), calculate the length of \\\\(AB\\\\."
}
] | math | {
"style": "rule",
"ground_truth": "4 + 2\\sqrt{3}"
} | {
"index": 15313,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\\\(F_1\\\\) and \\\\(F_2\\\\) are the left and right foci of the hyperbola \\\\( \\frac {x^{2}}{a^{2}}- \\frac {y^{2}}{b^{2}}=1(a > 0,b > 0)\\\\), if there exists a point \\\\(P\\\\) on the left branch of the hyperbola that is symmetric to point \\\\(F_2\\\\) with ... | math | {
"style": "rule",
"ground_truth": "\\sqrt {5}"
} | {
"index": 15314,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\\\(\\alpha\\\\) and \\\\(\\beta\\\\) are acute angles, and \\\\(\\cos \\alpha= \\frac{\\sqrt{5}}{5}\\\\), \\\\(\\sin (\\alpha+\\beta)= \\frac{3}{5}\\\\), find the value of \\\\(\\cos \\beta\\\\."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5}}{25}"
} | {
"index": 15315,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\\\(\\alpha\\\\) is an angle in the second quadrant, and \\\\(\\sin (π+α)=- \\frac {3}{5}\\\\), find the value of \\\\(\\tan 2α\\\\)."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {24}{7}"
} | {
"index": 15316,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\\\(\\theta\\\\) is an angle in the fourth quadrant, and \\\\(\\sin (\\theta+ \\frac {\\pi}{4})= \\frac {3}{5}\\\\), then \\\\(\\tan (\\theta- \\frac {\\pi}{4})=\\\\) \\_\\_\\_\\_\\_\\_ ."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {4}{3}"
} | {
"index": 15317,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that \\\\(y=f(x)+x^{2}\\\\) is an odd function, and \\\\(f(1)=1\\\\), if \\\\(g(x)=f(x)+2\\\\), then \\\\(g(-1)=\\\\) ."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 15318,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a 1-step requires 4 toothpicks, a 2-step requires 10 toothpicks, a 3-step requires 18 toothpicks, and a 4-step requires 28 toothpicks, determine the number of additional toothpicks needed to build a 6-step staircase."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 15319,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a 4-digit positive integer has four different digits, the leading digit is not zero, the integer is a multiple of 4, and 6 is the largest digit, determine the total count of such integers."
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 15320,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a 4-digit positive integer has only even digits (0, 2, 4, 6, 8) and is divisible by 4, calculate the number of such integers."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 15321,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a bank has 320 employees with an average annual profit of 200,000 yuan per employee, and each remaining employee's profit increases by 20,000 yuan for every laid-off employee, while the bank must pay 60,000 yuan per year for each laid-off employee, and the number of ... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 15322,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a basketball player has a 40% chance of scoring with each shot, estimate the probability that the player makes exactly two out of three shots using a random simulation method. The simulation uses a calculator to generate a random integer between 0 and 9, with 1, 2, 3... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{4}"
} | {
"index": 15323,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a bin contains 10 kg of peanuts, 2 kg of peanuts are removed and 2 kg of raisins are added and thoroughly mixed in, and then 2 kg of this mixture are removed and 2 kg of raisins are added and thoroughly mixed in again, determine the ratio of the mass of peanuts to th... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{9}"
} | {
"index": 15324,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a blue ball and an orange ball are randomly and independently tossed into bins numbered with the positive integers, where for each ball the probability that it is tossed into bin k is 3^(-k) for k = 1, 2, 3, ..., determine the probability that the blue ball is tossed... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"index": 15325,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a bridge is $800$ meters long, a train passes over it and it takes $1$ minute for the train to completely pass through the bridge. The train is entirely on the bridge for $40$ seconds. Calculate the speed of the train."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 15326,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a certain basketball player has a 50% chance of making each shot, we use a random simulation method to estimate the probability that the player makes exactly two out of four shots: First, we generate a random integer between 0 and 9 using a calculator, where 0, 1, 2,... | math | {
"style": "rule",
"ground_truth": "0.35"
} | {
"index": 15327,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a certain middle school has 3500 high school students and 1500 junior high school students, if 70 students are drawn from the high school students, calculate the total sample size $n$."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 15328,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a certain product requires $6$ processing steps, where $2$ of these steps must be consecutive and another $2$ steps cannot be consecutive, calculate the number of possible processing sequences."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 15329,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a child builds towers with $2$ red cubes, $3$ blue cubes, and $4$ green cubes, determine the number of different towers with a height of $8$ cubes that can be built, with one cube left out."
}
] | math | {
"style": "rule",
"ground_truth": "1,260"
} | {
"index": 15330,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a class has 5 students participating in the duty roster from Monday to Friday, with one student arranged each day, student A can only be arranged on Monday or Tuesday, and student B cannot be arranged on Friday, calculate the number of different duty arrangements for... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 15331,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a class selects 4 athletes from 5 male and 4 female track and field athletes to participate in the competition, where the selection must include both male and female athletes, and at least one of the male athlete A or female athlete B must be selected, calculate the ... | math | {
"style": "rule",
"ground_truth": "86"
} | {
"index": 15332,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a five-digit palindromic number is equal to the product of 45 and a four-digit palindromic number (i.e., $\\overline{\\mathrm{abcba}} = 45 \\times \\overline{\\text{deed}}$), find the largest possible value of the five-digit palindromic number."
}
] | math | {
"style": "rule",
"ground_truth": "59895"
} | {
"index": 15333,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a floor is tiled in a similar pattern with a $4 \\times 4$ unit repeated pattern and each of the four corners looks like the scaled down version of the original, determine the fraction of the tiled floor made up of darker tiles, assuming symmetry and pattern are pres... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 15334,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a flower bouquet contains pink roses, red roses, pink tulips, and red tulips, and that one fourth of the pink flowers are roses, one third of the red flowers are tulips, and seven tenths of the flowers are red, calculate the percentage of the flowers that are tulips.... | math | {
"style": "rule",
"ground_truth": "46\\%"
} | {
"index": 15335,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a four-digit integer $MMMM$, with all identical digits, is multiplied by the one-digit integer $M$, the result is the five-digit integer $NPMPP$. Assuming $M$ is the largest possible single-digit integer that maintains the units digit property of $M^2$, find the grea... | math | {
"style": "rule",
"ground_truth": "89991"
} | {
"index": 15336,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a full circle is 800 clerts on Venus and is 360 degrees, calculate the number of clerts in an angle of 60 degrees."
}
] | math | {
"style": "rule",
"ground_truth": "133.\\overline{3}"
} | {
"index": 15337,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a geometric sequence $\\{a_n\\}$ consists of positive terms, and $(a_3, \\frac{1}{2}a_5,a_4)$ form an arithmetic sequence, find the value of $\\frac{a_3+a_5}{a_4+a_6}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}-1}{2}"
} | {
"index": 15338,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a group of students is sitting evenly spaced around a circular table and a bag containing 120 pieces of candy is circulated among them, determine the possible number of students if Sam picks both the first and a final piece after the bag has completed exactly two ful... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 15339,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a high school senior year has 12 classes, with exactly 8 classes to be proctored by their own homeroom teachers, find the number of different proctoring arrangements for the math exam."
}
] | math | {
"style": "rule",
"ground_truth": "4455"
} | {
"index": 15340,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a hyperbola $\\frac{{x}^{2}}{{a}^{2}}-\\frac{{y}^{2}}{{b}^{2}}=1$ has only one common point with the parabola $y=x^{2}+1$, calculate the eccentricity of the hyperbola."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 15341,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a hyperbola $mx^{2}+ny^{2}=1$ has a focus that is the same as the focus of the parabola $y=\\frac{1}{8}{x^2}$ and an eccentricity of $2$, calculate the distance from the focus of the parabola to one of the asymptotes of the hyperbola."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 15342,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a hyperbola shares common foci $F_1$ and $F_2$ with the ellipse $\\dfrac {x^{2}}{9}+ \\dfrac {y^{2}}{25}=1$, and their sum of eccentricities is $2 \\dfrac {4}{5}$. \n$(1)$ Find the standard equation of the hyperbola; \n$(2)$ Let $P$ be a point of intersection betwe... | math | {
"style": "rule",
"ground_truth": "- \\dfrac {1}{7}"
} | {
"index": 15343,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a light bulb is located $15$ centimeters below the ceiling in Bob's living room, the ceiling is $2.8$ meters above the floor, Bob is $1.65$ meters tall and can reach $55$ centimeters above his head, and Bob standing on a chair can just reach the light bulb, calculate... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 15344,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a match between two people is played with a best-of-five-games format, where the winner is the first to win three games, and that the probability of person A winning a game is $\\dfrac{2}{3}$, calculate the probability that person A wins with a score of $3:1$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{8}{27}"
} | {
"index": 15345,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a meeting is convened with 2 representatives from one company and 1 representative from each of the other 3 companies, find the probability that 3 randomly selected individuals to give speeches come from different companies."
}
] | math | {
"style": "rule",
"ground_truth": "0.6"
} | {
"index": 15346,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a multiple-choice math question has incorrect options that cannot be adjacent, calculate the total number of arrangements of $4$ different options that meet the requirements."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 15347,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a new kitchen mixer is listed in a store for $\\textdollar 129.99$ and an online advertisement offers the same mixer for four easy payments of $\\textdollar 29.99$ and a one-time shipping and handling fee of $\\textdollar 19.99$, calculate how many cents are saved by... | math | {
"style": "rule",
"ground_truth": "996"
} | {
"index": 15348,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a normal vector of line $l$ is $\\overrightarrow{n}=(\\sqrt{3}, -1)$, find the size of the slope angle of line $l$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 15349,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a normal vector of the straight line $l$ is $\\overrightarrow{n} = (1, -\\sqrt{3})$, find the size of the inclination angle of this straight line."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"index": 15350,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a person can click four times in sequence and receive one of three types of red packets each time, with the order of appearance corresponding to different prize rankings, calculate the number of different prize rankings that can be obtained if all three types of red ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 15351,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a point on the terminal side of angle \\(\\alpha\\) has coordinates \\((\\sin \\frac{2\\pi }{3},\\cos \\frac{2\\pi }{3})\\), find the smallest positive angle for \\(\\alpha\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11\\pi }{6}"
} | {
"index": 15352,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a positive integer \\( A \\) can be factorized as \\( A = 2^{\\alpha} \\times 3^{\\beta} \\times 5^{\\gamma} \\), where \\( \\alpha \\), \\( \\beta \\), and \\( \\gamma \\) are natural numbers, and given that half of \\( A \\) is a perfect square, one-third of \\( A ... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 15353,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a product originally priced at an unknown value is raised by 20% twice consecutively, and another product originally priced at an unknown value is reduced by 20% twice consecutively, ultimately selling both at 23.04 yuan each, determine the profit or loss situation w... | math | {
"style": "rule",
"ground_truth": "5.92"
} | {
"index": 15354,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a rancher bought 600 sheep at an unknown cost price, then sold 550 sheep for the cost price of 600 sheep, and the remaining 50 sheep at the same price per head as the first 550, calculate the percent gain on the entire transaction."
}
] | math | {
"style": "rule",
"ground_truth": "9.09\\%"
} | {
"index": 15355,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a recipe calls for \\( 4 \\frac{1}{2} \\) cups of flour, calculate the amount of flour needed if only half of the recipe is made."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\frac{1}{4}"
} | {
"index": 15356,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a rectangle with length $3x$ inches and width $x + 5$ inches has the property that its area and perimeter have equal values, what is $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 15357,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a rectangular room is 15 feet long and 108 inches wide, calculate the area of the new extended room after adding a 3 feet wide walkway along the entire length of one side, in square yards, where 1 yard equals 3 feet and 1 foot equals 12 inches."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 15358,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a regular tetrahedron has all edge lengths of $\\sqrt {2}$, and all four vertices are on the same spherical surface, find the surface area of this sphere."
}
] | math | {
"style": "rule",
"ground_truth": "3\\pi"
} | {
"index": 15359,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a school has 5 top students and 3 teachers, where each teacher mentors no more than 2 students, calculate the number of different mentorship arrangements possible."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 15360,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a sequence satisfies $x_0=0$ and $|x_k|=|x_{k-1}+3|$ for all integers $k\\ge 1,$ find the minimum possible value of $|x_1+x_2+\\cdots+x_{2006}|.$"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 15361,
"solution": "Playing around with a couple numbers, we see that we can generate the sequence $0, 3, -6, 3, -6, \\cdots$, and we can also generate the sequence $3, 6, 9, 12, \\cdots$ after each $-6$ value. Thus, we will apply this to try and find some bounds. We can test if the first $1000$ pairs of... |
DeepScaleR | [
{
"role": "user",
"content": "Given that a set of $n$ people participate in an online video soccer tournament, the statistics from the tournament reveal: The average number of complete teams wholly contained within randomly chosen subsets of $10$ members equals twice the average number of complete teams fou... | math | {
"style": "rule",
"ground_truth": "450"
} | {
"index": 15362,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a ship travels in one direction and Emily walks parallel to the riverbank in the opposite direction, counting 210 steps from back to front and 42 steps from front to back, determine the length of the ship in terms of Emily's equal steps."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 15363,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a shop advertises everything as \"half price in today's sale,\" and a 20% discount is applied to sale prices, and a promotional offer is available where if a customer buys two items, they get the lesser priced item for free, calculate the percentage off the total ori... | math | {
"style": "rule",
"ground_truth": "20\\%"
} | {
"index": 15364,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a smaller circle is entirely inside a larger circle, such that the larger circle has a radius $R = 2$, and the areas of the two circles form an arithmetic progression, with the largest area being that of the larger circle, find the radius of the smaller circle."
}
... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 15365,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a square $S_1$ has an area of $25$, the area of the square $S_3$ constructed by bisecting the sides of $S_2$ is formed by the points of bisection of $S_2$."
}
] | math | {
"style": "rule",
"ground_truth": "6.25"
} | {
"index": 15366,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a tetrahedron $ABCD$ is inscribed in a sphere $O$, and $AD$ is the diameter of the sphere $O$. If triangles $\\triangle ABC$ and $\\triangle BCD$ are equilateral triangles with side length 1, calculate the volume of tetrahedron $ABCD$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{12}"
} | {
"index": 15367,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a three-digit positive integer (a_1 a_2 a_3) is said to be a convex number if it satisfies (a_1 < a_2 > a_3), determine the number of three-digit convex numbers."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 15368,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a travel agency plans to arrange a trip for 900 passengers using two types of buses, A and B, with capacities 36 and 60 passengers respectively, rental costs 1600 yuan and 2400 yuan per bus respectively, and the total number of buses rented does not exceed 21, and th... | math | {
"style": "rule",
"ground_truth": "36800"
} | {
"index": 15369,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a triangle with integral sides is isosceles and has a perimeter of 12, find the area of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"index": 15370,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a triangular corner with side lengths DB=EB=1.5 is cut from an equilateral triangle ABC of side length 4.5, determine the perimeter of the remaining quadrilateral."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 15371,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a water tower stands 60 meters high and contains 150,000 liters of water, and a model of the tower holds 0.15 liters, determine the height of Liam's model tower."
}
] | math | {
"style": "rule",
"ground_truth": "0.6"
} | {
"index": 15372,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a website publishes pictures of five celebrities along with five photos of the celebrities when they were teenagers, but only three of the teen photos are correctly labeled, and two are unlabeled, determine the probability that a visitor guessing at random will match... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{20}"
} | {
"index": 15373,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that a, b, and c are the sides opposite to angles A, B, and C respectively in triangle ABC, and c = 2, sinC(cosB - $\\sqrt{3}$sinB) = sinA.\n(1) Find the measure of angle C;\n(2) If cosA = $\\frac{2\\sqrt{2}}{3}$, find the length of side b."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{2} - 2\\sqrt{3}}{3}"
} | {
"index": 15374,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that acute angles $\\alpha$ and $\\beta$ satisfy $\\alpha+2\\beta=\\frac{2\\pi}{3}$ and $\\tan\\frac{\\alpha}{2}\\tan\\beta=2-\\sqrt{3}$, find the value of $\\alpha +\\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{12}"
} | {
"index": 15375,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that acute angles $\\alpha$ and $\\beta$ satisfy $\\sin\\alpha=\\frac{4}{5}$ and $\\cos(\\alpha+\\beta)=-\\frac{12}{13}$, determine the value of $\\cos \\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{16}{65}"
} | {
"index": 15376,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that all edges of a tetrahedron have a length of $\\sqrt{2}$, and the four vertices are on the same sphere, calculate the surface area of this sphere."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\pi"
} | {
"index": 15377,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that all faces of the tetrahedron P-ABC are right triangles, and the longest edge PC equals $2\\sqrt{3}$, the surface area of the circumscribed sphere of this tetrahedron is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "12\\pi"
} | {
"index": 15378,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that all four roots of the equation $2x^4 + mx^2 + 8 = 0$ are integers, then $m = \\ $, and the polynomial $2x^4 + mx^2 + 8$ can be factored into $\\ $."
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"index": 15379,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that all terms are positive in the geometric sequence $\\{a_n\\}$, and the sum of the first $n$ terms is $S_n$, if $S_1 + 2S_5 = 3S_3$, then the common ratio of $\\{a_n\\}$ equals \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 15380,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that all three vertices of \\(\\triangle ABC\\) lie on the parabola defined by \\(y = 4x^2\\), with \\(A\\) at the origin and \\(\\overline{BC}\\) parallel to the \\(x\\)-axis, calculate the length of \\(BC\\), given that the area of the triangle is 128."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt[3]{4}"
} | {
"index": 15381,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that all vertices of the tetrahedron S-ABC are on the surface of sphere O, SC is the diameter of sphere O, and if plane SCA is perpendicular to plane SCB, with SA = AC and SB = BC, and the volume of tetrahedron S-ABC is 9, find the surface area of sphere O."
}
] | math | {
"style": "rule",
"ground_truth": "36\\pi"
} | {
"index": 15382,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that an ellipse and a hyperbola $(x^{2}-y^{2}=1)$ share the same foci and the eccentricity is $\\frac{\\sqrt{2}}{2}$.\n(I) Find the standard equation of the ellipse;\n(II) A line passing through point $P(0,1)$ intersects the ellipse at points $A$ and $B$. $O$ is the origi... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{126}}{8}"
} | {
"index": 15383,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that an ellipse and a hyperbola share common foci $F_1$, $F_2$, and $P$ is one of their intersection points, $\\angle F_1PF_2=60^\\circ$, find the minimum value of $e_1^2+e_2^2$."
}
] | math | {
"style": "rule",
"ground_truth": "1+\\frac{\\sqrt{3}}{2}"
} | {
"index": 15384,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that an ellipse has the equation $\\frac {x^{2}}{a^{2}} + \\frac {y^{2}}{b^{2}} = 1$ with $a > b > 0$ and eccentricity $e = \\frac {\\sqrt {6}}{3}$. The distance from the origin to the line that passes through points $A(0,-b)$ and $B(a,0)$ is $\\frac {\\sqrt {3}}{2}$.\n\n... | math | {
"style": "rule",
"ground_truth": "\\frac {7}{6}"
} | {
"index": 15385,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that an integer is either only even or only odd and must be divisible by 4, calculate the number of 4-digit positive integers that satisfy these conditions."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 15386,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that an isosceles trapezoid is circumscribed around a circle, find the ratio of the area of the trapezoid to the area of the circle if the distance between the points where the circle touches the non-parallel sides of the trapezoid is related to the radius of the circle a... | math | {
"style": "rule",
"ground_truth": "\\frac{8\\sqrt{3}}{3\\pi}"
} | {
"index": 15387,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that an odd function \\( f(x) \\) satisfies the condition \\( f(x+3) = f(x) \\). When \\( x \\in [0,1] \\), \\( f(x) = 3^x - 1 \\). Find the value of \\( f\\left(\\log_1 36\\right) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-1/3"
} | {
"index": 15388,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that an office at a school needs to arrange a duty roster from the 1st to the 6th day with six designated individuals participating, find the total number of different arrangements possible, given that person A and person B cannot be adjacent, and person C and person D al... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"index": 15389,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that angle $A$ is an internal angle of a triangle and $\\cos A= \\frac{3}{5}$, find $\\tan A=$ \\_\\_\\_\\_\\_\\_ and $\\tan (A+ \\frac{\\pi}{4})=$ \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 15390,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that angle $\\alpha$ is in the fourth quadrant, and the x-coordinate of the point where its terminal side intersects the unit circle is $\\frac{1}{3}$.\n(1) Find the value of $\\tan \\alpha$;\n(2) Find the value of $\\frac{\\sin^{2}\\alpha - \\sqrt{2}\\sin \\alpha\\cos \\... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{5}"
} | {
"index": 15391,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that angle DEF is a right angle and the sides of triangle DEF are the diameters of semicircles, the area of the semicircle on segment DE equals $18\\pi$, and the arc of the semicircle on segment DF has length $10\\pi$. Determine the radius of the semicircle on segment EF.... | math | {
"style": "rule",
"ground_truth": "\\sqrt{136}"
} | {
"index": 15392,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that angles $\\angle A, \\angle B, \\angle C$ are the interior angles of triangle $ABC$, and vector $\\alpha=\\left(\\cos \\frac{A-B}{2}, \\sqrt{3} \\sin \\frac{A+B}{2}\\right)$ with $|\\alpha|=\\sqrt{2}$. If when $\\angle C$ is maximized, there exists a moving point $M$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3} + \\sqrt{2}}{4}"
} | {
"index": 15393,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that at the beginning of the year, 40% of students answered \"Yes\", 40% answered \"No\", and 20% were undecided, while at the end of the year, 60% answered \"Yes\", 30% answered \"No\", and 10% remained undecided, find the difference between the maximum and minimum possi... | math | {
"style": "rule",
"ground_truth": "60\\%"
} | {
"index": 15394,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that b is an even number between 1 and 11 (inclusive) and c is any natural number, determine the number of quadratic equations x^{2} + bx + c = 0 that have two distinct real roots."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 15395,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that bag A contains 3 white balls and 5 black balls, and bag B contains 4 white balls and 6 black balls, calculate the probability that the number of white balls in bag A does not decrease after a ball is randomly taken from bag A and put into bag B, and a ball is then ra... | math | {
"style": "rule",
"ground_truth": "\\frac{35}{44}"
} | {
"index": 15396,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that both $\\alpha$ and $\\beta$ are acute angles, and $\\sin \\alpha = \\frac{3}{5}$, $\\tan (\\alpha - \\beta) = -\\frac{1}{3}$.\n\n(1) Find the value of $\\sin (\\alpha - \\beta)$;\n(2) Find the value of $\\cos \\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{10}}{50}"
} | {
"index": 15397,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that both $m$ and $n$ are non-negative integers, find the number of \"simple\" ordered pairs $(m, n)$ with a value of 2019."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 15398,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that both $α$ and $β$ are acute angles, $\\cos α= \\frac {1}{7}$, and $\\cos (α+β)=- \\frac {11}{14}$, find the value of $\\cos β$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"index": 15399,
"solution": ""
} |
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