data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of the parameter \\( b \\) for which the inequality \\( b \\sqrt{b}\\left(x^{2}-10 x+25\\right)+\\frac{\\sqrt{b}}{\\left(x^{2}-10 x+25\\right)} \\leq \\frac{1}{5} \\cdot \\sqrt[4]{b^{3}} \\cdot \\left| \\sin \\frac{\\pi x}{10} \\right| \\) has at least on... | math | {
"style": "rule",
"ground_truth": "1/10000"
} | {
"split": "train",
"index": 33349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Expand $(1+0.1)^{500}$ by the binomial theorem and calculate each term as\n\\[{500 \\choose k}(0.1)^k\\] for $k = 0, 1, 2, \\ldots, 500.$ Determine the value of $k$ for which the term is the largest. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 33350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are four points that are $7$ units from the line $y=20$ and $15$ units from the point $(10,20)$. What is the sum of the $x$- and $y$-coordinates of all four of these points? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 33351,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many $3$-digit squares are palindromes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 33352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parametric equation of line $l$ is $\\begin{cases} & x=1+3t \\\\ & y=2-4t \\end{cases}$ (where $t$ is the parameter), calculate the cosine of the inclination angle of line $l$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{5}"
} | {
"split": "train",
"index": 33353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of different ways to schedule volleyball, basketball, and table tennis competitions in 4 different gyms, given that each sport must be held in only one gym and that no more than 2 sports can take place in the same gym. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 33354,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "While exploring a cave, Carl comes across a collection of $5$-pound rocks worth $\\$14$ each, $4$-pound rocks worth $\\$11$ each, and $1$-pound rocks worth $\\$2$ each. There are at least $20$ of each size. He can carry at most $18$ pounds. What is the maximum value, in dollars... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 33355,
"solution": "To determine the maximum value of rocks Carl can carry, we first calculate the value per pound for each type of rock:\n- **5-pound rocks:** $\\frac{\\$14}{5} = \\$2.80$ per pound\n- **4-pound rocks:** $\\frac{\\$11}{4} = \\$2.75$ per pound\n- **1-pound rocks:** $\\... |
deepscaler | [
{
"role": "user",
"content": "A right square pyramid with base edges of length $8\\sqrt{2}$ units each and slant edges of length 10 units each is cut by a plane that is parallel to its base and 3 units above its base. What is the volume, in cubic units, of the new pyramid that is cut off by this plane? [asy... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 33356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Several students are competing in a series of three races. A student earns $5$ points for winning a race, $3$ points for finishing second and $1$ point for finishing third. There are no ties. What is the smallest number of points that a student must earn in the three races t... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 33357,
"solution": "To solve this problem, we need to determine the minimum number of points a student must earn in three races to ensure they have more points than any other student. We will analyze the distribution of points and the possible outcomes.\n\n1. **Point Distribution per ... |
deepscaler | [
{
"role": "user",
"content": "Given that $x$ is a multiple of $2520$, what is the greatest common divisor of $g(x) = (4x+5)(5x+2)(11x+8)(3x+7)$ and $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "280"
} | {
"split": "train",
"index": 33358,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the three internal angles are $A$, $B$, and $C$. If $\\dfrac{\\sqrt{3}\\cos A + \\sin A}{\\sqrt{3}\\sin A - \\cos A} = \\tan(-\\dfrac{7}{12}\\pi)$, find the maximum value of $2\\cos B + \\sin 2C$. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{2}"
} | {
"split": "train",
"index": 33359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle has sides with lengths of $18$, $24$, and $30$. Calculate the length of the shortest altitude. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 33360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the curve $y=x^{n+1}$ (where $n$ is a positive integer) and its tangent at the point (1,1). Let the x-coordinate of the intersection point between this tangent and the x-axis be $x_n$.\n(Ⅰ) Let $a_n = \\log{x_n}$. Find the value of $a_1 + a_2 + \\ldots + a_9$.\n(Ⅱ) Def... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 33361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bus arrives randomly sometime between 2:00 and 3:00 and waits for 15 minutes before leaving. If Carla arrives randomly between 2:00 and 3:00, what is the probability that the bus will still be there when Carla arrives? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{32}"
} | {
"split": "train",
"index": 33362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider all permutations of the numbers $1, 2, \\cdots, 8$ as eight-digit numbers. How many of these eight-digit numbers are multiples of 11? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4608"
} | {
"split": "train",
"index": 33363,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$A B$ is a diameter of circle $O . X$ is a point on $A B$ such that $A X=3 B X$. Distinct circles $\\omega_{1}$ and $\\omega_{2}$ are tangent to $O$ at $T_{1}$ and $T_{2}$ and to $A B$ at $X$. The lines $T_{1} X$ and $T_{2} X$ intersect $O$ again at $S_{1}$ and $S_{2}$. What is... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 33364,
"solution": "Since the problem only deals with ratios, we can assume that the radius of $O$ is 1. As we have proven in Problem 5, points $S_{1}$ and $S_{2}$ are midpoints of arc $A B$. Since $A B$ is a diameter, $S_{1} S_{2}$ is also a diameter, and thus $S_{1} S_{2}=2$. Let $O... |
deepscaler | [
{
"role": "user",
"content": "Find all prime numbers $ p,q,r$, such that $ \\frac{p}{q}\\minus{}\\frac{4}{r\\plus{}1}\\equal{}1$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(7, 3, 2), (5, 3, 5), (3, 2, 7)"
} | {
"split": "train",
"index": 33365,
"solution": "\nWe are tasked with finding all prime numbers \\( p, q, r \\) that satisfy the equation:\n\n\\[\n\\frac{p}{q} - \\frac{4}{r+1} = 1.\n\\]\n\nFirst, we rearrange the equation to find a common denominator:\n\n\\[\n\\frac{p}{q} - \\frac{4}{r+1} = 1 \\implies \\frac{p(... |
deepscaler | [
{
"role": "user",
"content": "Two circles \\( C_{1} \\) and \\( C_{2} \\) touch each other externally and the line \\( l \\) is a common tangent. The line \\( m \\) is parallel to \\( l \\) and touches the two circles \\( C_{1} \\) and \\( C_{3} \\). The three circles are mutually tangent. If the radius of ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 33366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive real number $c$ such that for all nonnegative real numbers $x, y,$ and $z$, the following inequality holds:\n\\[\\sqrt[3]{xyz} + c |x - y + z| \\ge \\frac{x + y + z}{3}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 33367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $(a_1,a_2,a_3,\\ldots,a_{15})$ be a permutation of $(1,2,3,\\ldots,15)$ for which\n$a_1>a_2>a_3>a_4>a_5>a_6>a_7 \\mathrm{\\ and \\ } a_7<a_8<a_9<a_{10}<a_{11}<a_{12}<a_{13}<a_{14}<a_{15}.$\nAn example of such a permutation is $(7,6,5,4,3,2,1,8,9,10,11,12,13,14,15).$ Find th... | math | {
"style": "rule",
"ground_truth": "3003"
} | {
"split": "train",
"index": 33368,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five distinct points are arranged on a plane such that the segments connecting them form lengths $a$, $a$, $a$, $a$, $a$, $b$, $b$, $2a$, and $c$. The shape formed by these points is no longer restricted to a simple polygon but could include one bend (not perfectly planar). Wha... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 33369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\(x \\geqslant 1\\), the minimum value of the function \\(y=f(x)= \\frac {4x^{2}-2x+16}{2x-1}\\) is \\_\\_\\_\\_\\_\\_, and the corresponding value of \\(x\\) is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {5}{2}"
} | {
"split": "train",
"index": 33370,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Assume that $a$, $b$, $c$, and $d$ are positive integers such that $a^5 = b^4$, $c^3 = d^2$, and $c - a = 19$. Determine $d - b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "757"
} | {
"split": "train",
"index": 33371,
"solution": "It follows from the givens that $a$ is a perfect fourth power, $b$ is a perfect fifth power, $c$ is a perfect square and $d$ is a perfect cube. Thus, there exist integers $s$ and $t$ such that $a = t^4$, $b = t^5$, $c = s^2$ and $d = s^3$. So $s^2 - t^4 = 19$. We c... |
deepscaler | [
{
"role": "user",
"content": "In parallelogram \\(ABCD\\), \\(OE = EF = FD\\). The area of the parallelogram is 240 square centimeters. The area of the shaded region is _______ square centimeters. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 33372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of \\(y^2 + 2xy + 60|x| = 900\\) partitions the plane into several regions. What is the area of the bounded region? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1800"
} | {
"split": "train",
"index": 33373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three positive reals \\( x \\), \\( y \\), and \\( z \\) are such that\n\\[\n\\begin{array}{l}\nx^{2}+2(y-1)(z-1)=85 \\\\\ny^{2}+2(z-1)(x-1)=84 \\\\\nz^{2}+2(x-1)(y-1)=89\n\\end{array}\n\\]\nCompute \\( x + y + z \\). Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 33374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ is a positive integer such that $1^{x+2} + 2^{x+1} + 3^{x-1} + 4^x = 1170$, what is the value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 33375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Line $l_{1}$: $mx+2y-3=0$ is parallel to line $l_{2}$: $3x+\\left(m-1\\right)y+m-6=0$. Find the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 33376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the following equations: \n(1) $\\frac {1}{2}x - 2 = 4 + \\frac {1}{3}x$ \n(2) $\\frac {x-1}{4} - 2 = \\frac {2x-3}{6}$ \n(3) $\\frac {1}{3}[x - \\frac {1}{2}(x-1)] = \\frac {2}{3}(x - \\frac {1}{2})$ \n(4) $\\frac {x}{0.7} - \\frac {0.17-0.2x}{0.03} = 1$ Let's think ... | math | {
"style": "rule",
"ground_truth": "\\frac {14}{17}"
} | {
"split": "train",
"index": 33377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We define $|\\begin{array}{l}{a}&{c}\\\\{b}&{d}\\end{array}|=ad-bc$. For example, $|\\begin{array}{l}{1}&{3}\\\\{2}&{4}\\end{array}|=1\\times 4-2\\times 3=4-6=-2$. If $x$ and $y$ are integers, and satisfy $1 \\lt |\\begin{array}{l}{2}&{y}\\\\{x}&{3}\\end{array}| \\lt 3$, then t... | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 33378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two people are flipping a coin: one flipped it 10 times, and the other flipped it 11 times. Find the probability that the second person gets more heads than the first person. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 33379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T$ be the set of points $(x, y)$ in the Cartesian plane that satisfy\n\\[\\big|\\big| |x|-3\\big|-1\\big|+\\big|\\big| |y|-3\\big|-1\\big|=2.\\]\nWhat is the total length of all the lines that make up $T$? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "32\\sqrt{2}"
} | {
"split": "train",
"index": 33380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equations: \n(1) $2x^2-3x-2=0$; \n(2) $2x^2-3x-1=0$ (using the method of completing the square). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3-\\sqrt{17}}{4}"
} | {
"split": "train",
"index": 33381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jill has 8 red marbles and 4 blue marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 8 marbles. What is the probability that exactly four of the marbles that she removes are red? Express your ... | math | {
"style": "rule",
"ground_truth": "0.171"
} | {
"split": "train",
"index": 33382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One night, 21 people exchanged phone calls $n$ times. It is known that among these people, there are $m$ people $a_{1}, a_{2}, \\cdots, a_{m}$ such that $a_{i}$ called $a_{i+1}$ (for $i=1,2, \\cdots, m$ and $a_{m+1}=a_{1}$), and $m$ is an odd number. If no three people among th... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 33383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many interior intersection points are there on a 12 by 12 grid of squares? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"split": "train",
"index": 33384,
"solution": "A 12 by 12 grid of squares will have 11 interior vertical lines and 11 interior horizontal lines. Each of the 11 interior vertical lines intersects each of the 11 interior horizontal lines and creates an interior intersection point. Therefore, the number of interio... |
deepscaler | [
{
"role": "user",
"content": "In how many ways can you arrange the digits of 11250 to get a five-digit number that is a multiple of 2? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 33385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any subset \\( S \\subseteq \\{1, 2, \\ldots, 15\\} \\), a number \\( n \\) is called an \"anchor\" for \\( S \\) if \\( n \\) and \\( n+|S| \\) are both members of \\( S \\), where \\( |S| \\) denotes the number of members of \\( S \\). Find the average number of anchors o... | math | {
"style": "rule",
"ground_truth": "13/8"
} | {
"split": "train",
"index": 33386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "6 * cos(18°) + 2 * cos(36°) + 4 * cos(54°) + ... + 20 * cos(360°) = ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 33387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression \\(\\frac{\\operatorname{tg}\\left(\\frac{5}{4} \\pi - 4 \\alpha\\right) \\sin^{2}\\left(\\frac{5}{4} \\pi + 4 \\alpha\\right)}{1 - 2 \\cos^{2} 4 \\alpha}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 33388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We draw two lines $(\\ell_1) , (\\ell_2)$ through the orthocenter $H$ of the triangle $ABC$ such that each one is dividing the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "60^\\circ, 60^\\circ, 60^\\circ"
} | {
"split": "train",
"index": 33389,
"solution": "\nWe are given that the lines \\((\\ell_1)\\) and \\((\\ell_2)\\) pass through the orthocenter \\(H\\) of triangle \\(ABC\\) and each line divides the triangle into two figures of equal area and equal perimeters. We need to determine the angles of the triangle.\n\n... |
deepscaler | [
{
"role": "user",
"content": "To encourage residents to conserve water, a city charges residents for domestic water use in a tiered pricing system. The table below shows partial information on the tiered pricing for domestic water use for residents in the city, each with their own water meter:\n\n| Water Sa... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 33390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α$ is an angle in the third quadrant and $\\cos 2α=-\\frac{3}{5}$, find $\\tan (\\frac{π}{4}+2α)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{7}"
} | {
"split": "train",
"index": 33391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c,$ $d$ be real numbers such that\n\\[\\frac{(a - b)(c - d)}{(b - c)(d - a)} = \\frac{3}{7}.\\] Find the product of all possible values of\n\\[\\frac{(a - c)(b - d)}{(a - b)(c - d)}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{3}"
} | {
"split": "train",
"index": 33392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When 2007 bars of soap are packed into \\( N \\) boxes, where \\( N \\) is a positive integer, there is a remainder of 5. How many possible values of \\( N \\) are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 33393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ and $c$ be positive real numbers. Find the minimum value of\n\\[\\frac{a + b}{c} + \\frac{a + c}{b} + \\frac{b + c}{a}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 33394,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a group of monkeys transporting peaches from location $A$ to location $B$. Every 3 minutes a monkey departs from $A$ towards $B$, and it takes 12 minutes for a monkey to complete the journey. A rabbit runs from $B$ to $A$. When the rabbit starts, a monkey has just arri... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 33395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A club has 12 members, and wishes to pick a president, a vice-president, a secretary, and a treasurer. However, the president and vice-president must have been members of the club for at least 3 years. If 4 of the existing members meet this criterion, in how many ways can these... | math | {
"style": "rule",
"ground_truth": "1080"
} | {
"split": "train",
"index": 33396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a real number $a$, let $\\lfloor a \\rfloor$ denote the greatest integer less than or equal to $a$. Let $\\mathcal{R}$ denote the region in the coordinate plane consisting of points $(x,y)$ such that $\\lfloor x \\rfloor ^2 + \\lfloor y \\rfloor ^2 = 25$. The region $\\math... | math | {
"style": "rule",
"ground_truth": "132"
} | {
"split": "train",
"index": 33397,
"solution": "The desired region consists of 12 boxes, whose lower-left corners are integers solutions of $x^2 + y^2 = 25$, namely $(\\pm5,0), (0,\\pm5), (\\pm3,\\pm4), (\\pm4,\\pm3).$ Since the points themselves are symmetric about $(0,0)$, the boxes are symmetric about $\\left... |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\\subseteq\\{1,2,\\ldots,2n\\}$, for integers $i,j$ with $0\\leq i,j\\leq n$, such that:\n[list]\n[*] for all $0\\leq i,j\\leq n$, the set $S_{i,j}$ has $i+j$ elements; and... | math | {
"style": "rule",
"ground_truth": "(2n)! \\cdot 2^{n^2}"
} | {
"split": "train",
"index": 33398,
"solution": "\nTo solve this problem, we need to determine the number of ways to choose the sets \\( S_{i,j} \\) such that they satisfy the given conditions. First, consider a fixed set \\(\\{1, 2, \\ldots, 2n\\}\\). We construct nested sets \\( S_{i,j} \\) with \\( i + j \\) e... |
deepscaler | [
{
"role": "user",
"content": "Let \\( p, q, \\) and \\( r \\) be the roots of the equation \\( x^3 - 15x^2 + 25x - 10 = 0 \\). Find the value of \\( (1+p)(1+q)(1+r) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 33399,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three people, A, B, and C, start from point $A$ to point $B$. A starts at 8:00, B starts at 8:20, and C starts at 8:30. They all travel at the same speed. Ten minutes after C starts, the distance from A to point $B$ is exactly half the distance from B to point $B$. At this time... | math | {
"style": "rule",
"ground_truth": "2418"
} | {
"split": "train",
"index": 33400,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain school holds a men's table tennis team competition. The final match adopts a points system. The two teams in the final play three matches in sequence, with the first two matches being men's singles matches and the third match being a men's doubles match. Each particip... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 33401,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average of the numbers $1, 2, 3,\\dots, 50,$ and $x$ is $51x$. What is $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{51}{104}"
} | {
"split": "train",
"index": 33402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The slope angle of the tangent line to the curve $y= \\frac{1}{2}x^{2}$ at the point $(1, \\frac{1}{2})$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 33403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the ellipse $C\\_2$ passes through the two foci and the two endpoints of the minor axis of the ellipse $C\\_1$: $\\frac{x^{2}}{14} + \\frac{y^{2}}{9} = 1$, find the eccentricity of the ellipse $C\\_2$. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 33404,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$ be a real number and $c \\neq 0$ an integer such that $c-0.1<x^{p}\\left(\\frac{1-(1+x)^{10}}{1+(1+x)^{10}}\\right)<c+0.1$ for all (positive) real numbers $x$ with $0<x<10^{-100}$. Find the ordered pair $(p, c)$. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "(-1, -5)"
} | {
"split": "train",
"index": 33405,
"solution": "We are essentially studying the rational function $f(x):=\\frac{1-(1+x)^{10}}{1+(1+x)^{10}}=\\frac{-10 x+O\\left(x^{2}\\right)}{2+O(x)}$. Intuitively, $f(x) \\approx \\frac{-10 x}{2}=-5 x$ for \"small nonzero $x$ \". So $g(x):= x^{p} f(x) \\approx-5 x^{p+1}$ for \"... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest possible side of a square in which five circles of radius $1$ can be placed, so that no two of them have a common interior point. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 + 2\\sqrt{2}"
} | {
"split": "train",
"index": 33406,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( s(n) \\) denote the sum of the digits of the natural number \\( n \\). Solve the equation \\( n + s(n) = 2018 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2008"
} | {
"split": "train",
"index": 33407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Either increasing the radius of a cylinder by 4 inches or the height by 10 inches results in the same volume. The original height of the cylinder is 5 inches. What is the original radius in inches? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 + 2\\sqrt{3}"
} | {
"split": "train",
"index": 33408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $X \\sim B(4, p)$, and $P(X=2)=\\frac{8}{27}$, find the probability of success in one trial. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 33409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a tribal council meeting, 60 people spoke in turn. Each of them said only one phrase. The first three speakers all said the same thing: \"I always tell the truth!\" The next 57 speakers also said the same phrase: \"Among the previous three speakers, exactly two of them told ... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 33410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If for any three distinct numbers $a$, $b$, and $c$ we define $f(a,b,c)=\\frac{c+a}{c-b}$, then $f(1,-2,-3)$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 33411,
"solution": "1. We are given the function \\( f(a, b, c) = \\frac{c+a}{c-b} \\) and need to evaluate \\( f(1, -2, -3) \\).\n\n2. Substitute \\( a = 1 \\), \\( b = -2 \\), and \\( c = -3 \\) into the function:\n \\[\n f(1, -2, -3) = \\frac{-3 + 1}{-3 - (-2)}.\n \\]\n\n3. S... |
deepscaler | [
{
"role": "user",
"content": "Find the number of triangulations of a general convex 7-gon into 5 triangles by 4 diagonals that do not intersect in their interiors. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 33412,
"solution": "Define the Catalan numbers by $C(n)=\\frac{1}{n+1}\\binom{2 n}{n}$. The current solution is the $C$ (number of triangles) $=C(5)=42$."
} |
deepscaler | [
{
"role": "user",
"content": "Quadrilateral $ALEX,$ pictured below (but not necessarily to scale!)\ncan be inscribed in a circle; with $\\angle LAX = 20^{\\circ}$ and $\\angle AXE = 100^{\\circ}:$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 33413,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle with angles \\( A, B, C \\) satisfies the following conditions:\n\\[\n\\frac{\\sin A + \\sin B + \\sin C}{\\cos A + \\cos B + \\cos C} = \\frac{12}{7},\n\\]\nand\n\\[\n\\sin A \\sin B \\sin C = \\frac{12}{25}.\n\\]\n\nGiven that \\( \\sin C \\) takes on three possibl... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 33414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The angle that has the same terminal side as $- \\frac{\\pi}{3}$ is $\\frac{\\pi}{3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{3}"
} | {
"split": "train",
"index": 33415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six positive numbers, not exceeding 3, satisfy the equations \\(a + b + c + d = 6\\) and \\(e + f = 2\\). What is the minimum value of the expression\n\n$$\n\\left(\\sqrt{a^{2}+4}+\\sqrt{b^{2}+e^{2}}+\\sqrt{c^{2}+f^{2}}+\\sqrt{d^{2}+4}\\right)^{2}\n$$ Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 33416,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If 640 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 33417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any positive integer $n$ , let $a_n=\\sum_{k=1}^{\\infty}[\\frac{n+2^{k-1}}{2^k}]$ , where $[x]$ is the largest integer that is equal or less than $x$ . Determine the value of $a_{2015}$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2015"
} | {
"split": "train",
"index": 33418,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the limit of the function:\n\n$$\\lim_{x \\rightarrow \\frac{1}{3}} \\frac{\\sqrt[3]{\\frac{x}{9}}-\\frac{1}{3}}{\\sqrt{\\frac{1}{3}+x}-\\sqrt{2x}}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2 \\sqrt{2}}{3 \\sqrt{3}}"
} | {
"split": "train",
"index": 33419,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Hexagon $ABCDEF$ is divided into five rhombuses, $\\mathcal{P, Q, R, S,}$ and $\\mathcal{T,}$ as shown. Rhombuses $\\mathcal{P, Q, R,}$ and $\\mathcal{S}$ are congruent, and each has area $\\sqrt{2006}.$ Let $K$ be the area of rhombus $\\mathcal{T}$. Given that $K$ is a positiv... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 33420,
"solution": "Let $x$ denote the common side length of the rhombi. Let $y$ denote one of the smaller interior angles of rhombus $\\mathcal{P}$. Then $x^2\\sin(y)=\\sqrt{2006}$. We also see that $K=x^2\\sin(2y) \\Longrightarrow K=2x^2\\sin y \\cdot \\cos y \\Longrightarrow K = 2\... |
deepscaler | [
{
"role": "user",
"content": "In the coordinate plane, a parallelogram $O A B C$ is drawn such that its center is at the point $\\left(\\frac{19}{2}, \\frac{15}{2}\\right)$, and the points $A, B,$ and $C$ have natural number coordinates. Find the number of such parallelograms. (Here, $O$ denotes the origin ... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 33421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $(\\sin A + \\sin B)(a-b) = c(\\sin C - \\sqrt{3}\\sin B)$. \n$(1)$ Find the measure of angle $A$; \n$(2)$ If $\\cos \\angle ABC = -\\frac{1}{7}$, $D$ is a poi... | math | {
"style": "rule",
"ground_truth": "7\\sqrt{3}"
} | {
"split": "train",
"index": 33422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "My three friends and I have dinner together every weekend. Each weekend, two of us cook and the other two clean up afterwards. How many different ways are there for us to choose who cooks and who cleans? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 33423,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( a, b, c \\) are the three real roots of the equation\n\\[ x^{3} - x^{2} - x + m = 0, \\]\nthen the minimum value of \\( m \\) is _____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{27}"
} | {
"split": "train",
"index": 33424,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $1011_2 + 101_2 - 1100_2 + 1101_2$. Express your answer in base 2. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10001_2"
} | {
"split": "train",
"index": 33425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bug starts at a vertex of a square. On each move, it randomly selects one of the three vertices where it is not currently located and crawls along a side of the square to that vertex. Given that the probability that the bug moves to its starting vertex on its eighth move is \... | math | {
"style": "rule",
"ground_truth": "2734"
} | {
"split": "train",
"index": 33426,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set $A=\\{2,3,4,8,9,16\\}$, if $a\\in A$ and $b\\in A$, the probability that the event \"$\\log_{a}b$ is not an integer but $\\frac{b}{a}$ is an integer\" occurs is $\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{18}"
} | {
"split": "train",
"index": 33427,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On Arbor Day, 20 students in a class planted trees along one side of a straight road, with each person planting one tree, and the distance between two adjacent trees being 10 meters. Initially, it is required to place all the saplings next to one of the tree pits so that the to... | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 33428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $x-y$ if a town has 2017 houses, 1820 have a dog, 1651 have a cat, 1182 have a turtle, $x$ is the largest possible number of houses that have a dog, a cat, and a turtle, and $y$ is the smallest possible number of houses that have a dog, a cat, and a turtle? Let's think ... | math | {
"style": "rule",
"ground_truth": "563"
} | {
"split": "train",
"index": 33429,
"solution": "Since there are 1182 houses that have a turtle, then there cannot be more than 1182 houses that have a dog, a cat, and a turtle. \n Since there are more houses with dogs and more houses with cats than there are with turtles, it is possible that all 1182 houses that... |
deepscaler | [
{
"role": "user",
"content": "Eugene, Brianna, and Katie are going on a run. Eugene runs at a rate of 4 miles per hour. If Brianna runs $\\frac{2}{3}$ as fast as Eugene, and Katie runs $\\frac{7}{5}$ as fast as Brianna, how fast does Katie run? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\frac{56}{15}"
} | {
"split": "train",
"index": 33430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The difference between two perfect squares is 221. What is the smallest possible sum of the two perfect squares? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24421"
} | {
"split": "train",
"index": 33431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $0.4 \\cdot 0.6$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.24"
} | {
"split": "train",
"index": 33432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a + 3 = (b-1)^2$ and $b + 3 = (a-1)^2$. Assuming $a \\neq b$, determine the value of $a^2 + b^2$.\nA) 5\nB) 10\nC) 15\nD) 20\nE) 25 Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 33433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of \\( 333 + 33 + 3 \\) is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "369"
} | {
"split": "train",
"index": 33434,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the function $y=g(x)$ is shown below. For all $x > 5$, it is true that $g(x) > 0.5$. If $g(x) = \\frac{x^2}{Dx^2 + Ex + F}$, where $D, E,$ and $F$ are integers, then find $D+E+F$. Assume the function has vertical asymptotes at $x = -3$ and $x = 4$ and a horizontal ... | math | {
"style": "rule",
"ground_truth": "-24"
} | {
"split": "train",
"index": 33435,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate and simplify \n(1) $(1\\frac{1}{2})^0 - (1-0.5^{-2}) \\div \\left(\\frac{27}{8}\\right)^{\\frac{2}{3}}$\n(2) $\\sqrt{2\\sqrt{2\\sqrt{2}}}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2^{\\frac{7}{8}}"
} | {
"split": "train",
"index": 33436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Maria invested $10,000 for 3 years at an annual interest rate of 5 percent compounded annually. Liam invested $10,000 for the same period of time, at the same interest rate, but the interest was compounded semi-annually. Calculate the difference in the amount earned by Liam's i... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 33437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the numbers \\(\\sqrt{5}, 2.1, \\frac{7}{3}, 2.0 \\overline{5}, 2 \\frac{1}{5}\\) are arranged in order from smallest to largest, the middle number is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\frac{1}{5}"
} | {
"split": "train",
"index": 33438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the units digit of $9^{8^7}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 33439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A frog is positioned at the origin of the coordinate plane. From the point $(x, y)$, the frog can jump to any of the points $(x + 1, y)$, $(x + 2, y)$, $(x, y + 1)$, or $(x, y + 2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0, 0)$ and ends at... | math | {
"style": "rule",
"ground_truth": "556"
} | {
"split": "train",
"index": 33440,
"solution": "Casework Solution: x-distribution: 1-1-1-1 (1 way to order) y-distribution: 1-1-1-1 (1 way to order) $\\dbinom{8}{4} = 70$ ways total\n x-distribution: 1-1-1-1 (1 way to order) y-distribution: 1-1-2 (3 ways to order) $\\dbinom{7}{3} \\times 3= 105$ ways total\n x-d... |
deepscaler | [
{
"role": "user",
"content": "Solve the equation using the completing the square method: $2x^{2}-4x-1=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2-\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 33441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The points $(9, -5)$ and $(-3, -1)$ are the endpoints of a diameter of a circle. What is the sum of the coordinates of the center of the circle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 33442,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer $n$ such that\n$\\frac 1{\\sin 45^\\circ\\sin 46^\\circ}+\\frac 1{\\sin 47^\\circ\\sin 48^\\circ}+\\cdots+\\frac 1{\\sin 133^\\circ\\sin 134^\\circ}=\\frac 1{\\sin n^\\circ}.$ Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 33443,
"solution": "Let S be the sum of the sequence. We begin the same as in Solution 1 to get $S\\sin(1)=\\cot(45)-\\cot(46)+\\cot(47)-\\cot(48)+...+\\cot(133)-\\cot(134)$. Observe that this \"almost telescopes,\" if only we had some extra terms. Consider adding the sequence $\\frac... |
deepscaler | [
{
"role": "user",
"content": "Let $a, b$, and $c$ be the 3 roots of $x^{3}-x+1=0$. Find $\\frac{1}{a+1}+\\frac{1}{b+1}+\\frac{1}{c+1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 33444,
"solution": "We can substitute $x=y-1$ to obtain a polynomial having roots $a+1, b+1, c+1$, namely, $(y-1)^{3}-(y-1)+1=y^{3}-3 y^{2}+2 y+1$. The sum of the reciprocals of the roots of this polynomial is, by Viete's formulas, $\\frac{2}{-1}=-2$."
} |
deepscaler | [
{
"role": "user",
"content": "On an algebra quiz, $10\\%$ of the students scored $70$ points, $35\\%$ scored $80$ points, $30\\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 33445,
"solution": "1. **Calculate the percentage of students scoring 100 points**:\n Given that $10\\%$ scored $70$ points, $35\\%$ scored $80$ points, and $30\\%$ scored $90$ points, the percentage of students scoring $100$ points is:\n \\[\n 100\\% - (10\\% + 35\\% + 30\\%) =... |
deepscaler | [
{
"role": "user",
"content": "Sarah subscribes to a virtual fitness class platform that charges a monthly membership fee plus a per-class fee. If Sarah paid a total of $30.72 in February for 4 classes, and $54.72 in March for 8 classes, with the monthly membership fee increasing by 10% from February to Marc... | math | {
"style": "rule",
"ground_truth": "7.47"
} | {
"split": "train",
"index": 33446,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During the first eleven days, 700 people responded to a survey question. Each respondent chose exactly one of the three offered options. The ratio of the frequencies of each response was \\(4: 7: 14\\). On the twelfth day, more people participated in the survey, which changed t... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 33447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $y=g(x)$, defined on a limited domain shown, is conceptualized through the function $g(x) = \\frac{(x-6)(x-4)(x-2)(x)(x+2)(x+4)(x+6)}{945} - 2.5$. If each horizontal grid line represents a unit interval, determine the sum of all integers $d$ for which the equation ... | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 33448,
"solution": ""
} |
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