data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The regular price for a bicycle is $\\$320$. The bicycle is on sale for $20\\%$ off. The regular price for a helmet is $\\$80$. The helmet is on sale for $10\\%$ off. If Sandra bought both items on sale, what is her percentage savings on the total purchase? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "18\\%"
} | {
"split": "train",
"index": 37549,
"solution": "Since the regular price for the bicycle is $\\$320$ and the savings are $20\\%$, then the amount of money that Sandra saves on the bicycle is $\\$320 \\times 20\\%=\\$320 \\times 0.2=\\$64$. Since the regular price for the helmet is $\\$80$ and the savings are $10\... |
deepscaler | [
{
"role": "user",
"content": "The number $$316990099009901=\\frac{32016000000000001}{101}$$ is the product of two distinct prime numbers. Compute the smaller of these two primes. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4002001"
} | {
"split": "train",
"index": 37550,
"solution": "Let $x=2000$, so the numerator is $$x^{5}+x^{4}+1=\\left(x^{2}+x+1\\right)\\left(x^{3}-x+1\\right)$$ (This latter factorization can be noted by the fact that plugging in $\\omega$ or $\\omega^{2}$ into $x^{5}+x^{4}+1$ gives 0 .) Then $x^{2}+x+1=4002001$ divides the... |
deepscaler | [
{
"role": "user",
"content": "A biologist found a pond with frogs. When classifying them by their mass, he noticed the following:\n\n*The $50$ lightest frogs represented $30\\%$ of the total mass of all the frogs in the pond, while the $44$ heaviest frogs represented $27\\%$ of the total mass.*As fa... | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 37551,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all possible positive integers represented in decimal as $13 x y 45 z$, which are divisible by 792, where $x, y, z$ are unknown digits. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1380456"
} | {
"split": "train",
"index": 37552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $2^{10} \\cdot 2^{15}$ is expressed as some integer to the fifth power, what is that integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 37553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$A$ can do a piece of work in $9$ days. $B$ is $50\\%$ more efficient than $A$. The number of days it takes $B$ to do the same piece of work is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 37554,
"solution": "1. **Calculate $A$'s work rate**: If $A$ can complete the work in $9$ days, then $A$'s work rate is $\\frac{1}{9}$ of the work per day.\n\n2. **Determine $B$'s work rate**: Since $B$ is $50\\%$ more efficient than $A$, $B$'s work rate is $1.5$ times that of $A$. Th... |
deepscaler | [
{
"role": "user",
"content": "For any positive integer $n$, let $a_n$ be the $y$-coordinate of the intersection point between the tangent line of the curve $y=x^n(1-x)$ at $x=2$ and the $y$-axis in the Cartesian coordinate system. Calculate the sum of the first 10 terms of the sequence $\\{\\log_2 \\frac{a_... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 37555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We color each number in the set $S = \\{1, 2, ..., 61\\}$ with one of $25$ given colors, where it is not necessary that every color gets used. Let $m$ be the number of non-empty subsets of $S$ such that every number in the subset has the same color. What is the minimum ... | math | {
"style": "rule",
"ground_truth": "119"
} | {
"split": "train",
"index": 37556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside the square \\(ABCD\\) with side length 5, there is a point \\(X\\). The areas of triangles \\(AXB\\), \\(BXC\\), and \\(CXD\\) are in the ratio \\(1:5:9\\). Find the sum of the squares of the distances from point \\(X\\) to the sides of the square. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 37557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the radius of the circle with equation $9x^2-18x+9y^2+36y+44=0.$ 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": 37558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the mean of the data $x_1, x_2, \\ldots, x_n$ is 2, and the variance is 3, calculate the mean and variance of the data $3x_1+5, 3x_2+5, \\ldots, 3x_n+5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 37559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the greatest common divisor of all nine-digit integers formed by repeating a three-digit integer three times. For example, 256,256,256 or 691,691,691 are integers of this form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1001001"
} | {
"split": "train",
"index": 37560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A die with faces showing the numbers $0,1,2,3,4,5$ is rolled until the total sum of the rolled numbers exceeds 12. What is the most likely value of this sum? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 37561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A scout troop buys $1000$ candy bars at a price of five for $2$ dollars. They sell all the candy bars at the price of two for $1$ dollar. What was their profit, in dollars? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 37562,
"solution": "1. **Calculate the cost price of the candy bars:**\n The troop buys candy bars at a rate of 5 bars for $2 dollars. Therefore, the cost per bar is:\n \\[\n \\frac{2 \\text{ dollars}}{5 \\text{ bars}} = 0.4 \\text{ dollars per bar}\n \\]\n Since they buy 10... |
deepscaler | [
{
"role": "user",
"content": "The integer 843301 is prime. The primorial of a prime number $p$, denoted $p \\#$, is defined to be the product of all prime numbers less than or equal to $p$. Determine the number of digits in $843301 \\#$. Your score will be $$\\max \\left\\{\\left\\lfloor 60\\left(\\frac{1}{... | math | {
"style": "rule",
"ground_truth": "365851"
} | {
"split": "train",
"index": 37563,
"solution": "Remark: 843301\\#-1 is the largest known prime number of the form $p \\#-1$, where $p$ is prime."
} |
deepscaler | [
{
"role": "user",
"content": "Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to the vertex. If the sum of the numbers on the vertices is equal to $1001$, then what is the sum of the numbers written on the faces? Le... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 37564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle A B C, \\omega$ is the circumcircle, $I$ is the incenter and $I_{A}$ is the $A$-excenter. Let $M$ be the midpoint of arc $\\widehat{B A C}$ on $\\omega$, and suppose that $X, Y$ are the projections of $I$ onto $M I_{A}$ and $I_{A}$ onto $M I$, respectively. If $\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{7}"
} | {
"split": "train",
"index": 37565,
"solution": "Using Fact 5, we know that $I I_{A}$ intersects the circle $(A B C)$ at $M_{A}$, which is the center of $(I I_{A} B C X Y)$. Let $R$ be the radius of the latter circle. We have $R=\\frac{1}{\\sqrt{3}}$. We have $\\angle A I M=\\angle Y I I_{A}=\\angle Y I X=\\frac{... |
deepscaler | [
{
"role": "user",
"content": "A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "3 \\pi \\sqrt{7}"
} | {
"split": "train",
"index": 37566,
"solution": "\n1. **Understanding the Geometry of the Problem**:\n The problem states that a three-quarter sector of a circle with radius $4$ inches is rolled up to form a cone. The arc length of this sector becomes the circumference of the base of the cone, and the radius of... |
deepscaler | [
{
"role": "user",
"content": "$174 S-A B C D-S^{\\prime}$ is a regular octahedron with edge length 1. Sphere $O$ is outside this octahedron, tangent to face $S A B$, and tangent to the extended planes of faces $S B C$, $S A D$, and $S^{\\prime} A B$. Find the radius $r$ of sphere $O$. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4} \\sqrt{\\frac{2}{3}}"
} | {
"split": "train",
"index": 37567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the cards $A, 2, \\cdots, J, Q, K$ as the numbers $1, 2, \\cdots, 11, 12, 13$. If we take the 13 cards of spades and 13 cards of hearts together and randomly draw 2 cards, what is the probability that the two cards are of the same suit and the product of the two number... | math | {
"style": "rule",
"ground_truth": "2/65"
} | {
"split": "train",
"index": 37568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Marsha now has two numbers, $a$ and $b$. When she divides $a$ by $60$ she gets a remainder of $58$. When she divides $b$ by $90$ she gets a remainder of $84$. What remainder does she get when she divides $a+b$ by $30$? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 37569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The integer $n > 9$ is a root of the quadratic equation $x^2 - ax + b=0$. In this equation, the representation of $a$ in the base-$n$ system is $19$. Determine the base-$n$ representation of $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90_n"
} | {
"split": "train",
"index": 37570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The increasing sequence of positive integers $b_1,$ $b_2,$ $b_3,$ $\\dots$ has the property that\n\\[b_{n + 2} = b_{n + 1} + b_n\\]for all $n \\ge 1.$ If $b_6 = 60,$ then find $b_7.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "97"
} | {
"split": "train",
"index": 37571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a store where all items are priced in whole numbers of rubles, there are two special offers:\n1) A customer who buys at least three items at once can select one item for free, provided that its price does not exceed the minimum of the paid items' prices.\n2) A customer who b... | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 37572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when 5462 is divided by 9? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 37573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An integer $x$ is chosen so that $3 x+1$ is an even integer. Which of the following must be an odd integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7x+4"
} | {
"split": "train",
"index": 37574,
"solution": "If $x$ is an integer for which $3 x+1$ is even, then $3 x$ is odd, since it is 1 less than an even integer. If $3 x$ is odd, then $x$ must be odd (since if $x$ is even, then $3 x$ would be even). If $x$ is odd, then $7 x$ is odd (odd times odd equals odd) and so $7... |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $y=-x^{2}+3$, there exist two distinct points $A$ and $B$ on it that are symmetric about the line $x+y=0$. Find the length of the segment $|AB|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"split": "train",
"index": 37575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A container is composed of an upright hollow frustum and a hollow cylinder, each with a base radius of $12 \\,\\text{cm}$ and a height of $20 \\,\\text{cm}$. When finely granulated sand is poured into this container, it fills the frustum and partially fills the cylinder, with t... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 37576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\begin{pmatrix} 1 & -1 \\\\ 1 & 0 \\end{pmatrix}^3.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -1 & 0 \\\\ 0 & -1 \\end{pmatrix}"
} | {
"split": "train",
"index": 37577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations? \\[\\left\\{ \\begin{aligned} x+3y&=3 \\\\ \\left| |x| - |y| \\right| &= 1 \\end{aligned}\\right.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 37578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a math class, each dwarf needs to find a three-digit number without any zero digits, divisible by 3, such that when 297 is added to the number, the result is a number with the same digits in reverse order. What is the minimum number of dwarfs that must be in the class so tha... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 37579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The interior of a right, circular cone is 8 inches tall with a 2-inch radius at the opening. The interior of the cone is filled with ice cream, and the cone has a hemisphere of ice cream exactly covering the opening of the cone. What is the volume of ice cream? Express your ans... | math | {
"style": "rule",
"ground_truth": "16\\pi"
} | {
"split": "train",
"index": 37580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the maximum difference between the \\(y\\)-coordinates of the intersection points of the graphs \\(y=5-x^2+2x^3\\) and \\(y=3+2x^2+2x^3\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8\\sqrt{6}}{9}"
} | {
"split": "train",
"index": 37581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Caroline starts with the number 1, and every second she flips a fair coin; if it lands heads, she adds 1 to her number, and if it lands tails she multiplies her number by 2. Compute the expected number of seconds it takes for her number to become a multiple of 2021. Let's think... | math | {
"style": "rule",
"ground_truth": "4040"
} | {
"split": "train",
"index": 37582,
"solution": "Consider this as a Markov chain on $\\mathbb{Z} / 2021 \\mathbb{Z}$. This Markov chain is aperiodic (since 0 can go to 0) and any number can be reached from any other number (by adding 1), so it has a unique stationary distribution $\\pi$, which is uniform (since t... |
deepscaler | [
{
"role": "user",
"content": "A right cylinder with a height of 5 inches has a radius of 2 inches. What is the area of the curved surface of the cylinder, in square inches? Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20\\pi"
} | {
"split": "train",
"index": 37583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest prime factor of 2323? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 37584,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A facility has 7 consecutive parking spaces, and there are 3 different models of cars to be parked. If it is required that among the remaining 4 parking spaces, exactly 3 are consecutive, then the number of different parking methods is \\_\\_\\_\\_\\_\\_. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 37585,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\tan \\alpha =2$, find the values of the following expressions: \n$(1) \\frac{\\sin \\alpha - 3\\cos \\alpha}{\\sin \\alpha + \\cos \\alpha}$ \n$(2) 2\\sin ^{2} \\alpha - \\sin \\alpha \\cos \\alpha + \\cos ^{2} \\alpha$ Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{5}"
} | {
"split": "train",
"index": 37586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $n$ is the smallest positive integer for which there exist positive real numbers $a$ and $b$ such that\n\\[(a + bi)^n = (a - bi)^n,\\]compute $\\frac{b}{a}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 37587,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $n$ is an odd number, $\\frac{1+(-1)^{n}}{4}=$______; when $n$ is an even number, $\\frac{1+(-1)^{n}}{4}=\\_\\_\\_\\_\\_\\_$. 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": 37588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the region \\(B\\) in the complex plane consisting of all points \\(z\\) such that both \\(\\frac{z}{50}\\) and \\(\\frac{50}{\\overline{z}}\\) have real and imaginary parts between 0 and 1, inclusive. Find the area of \\(B\\). Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "2500 - 312.5 \\pi"
} | {
"split": "train",
"index": 37589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute 1 $2+2 \\cdot 3+\\cdots+(n-1) n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{(n-1) n(n+1)}{3}"
} | {
"split": "train",
"index": 37590,
"solution": "Let $S=1 \\cdot 2+2 \\cdot 3+\\cdots+(n-1) n$. We know $\\sum_{i=1}^{n} i=\\frac{n(n+1)}{2}$ and $\\sum_{i=1}^{n} i^{2}=\\frac{n(n+1)(2 n+1)}{6}$. So $S=1(1+1)+2(2+1)+\\cdots+(n-1) n=\\left(1^{2}+2^{2}+\\cdots+(n-1)^{2}\\right)+(1+2+\\cdots+(n-1))=\\frac{(n-1)(n)(2... |
deepscaler | [
{
"role": "user",
"content": "Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0,1)$. What is the probability that $\\lfloor\\log_2x\\rfloor=\\lfloor\\log_2y\\rfloor$? 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": 37591,
"solution": "1. **Understanding the Problem**: We need to find the probability that $\\lfloor\\log_2x\\rfloor=\\lfloor\\log_2y\\rfloor$ where $x$ and $y$ are chosen independently and uniformly from the interval $(0,1)$.\n\n2. **Interpreting the Floor Function and Logarithm**: T... |
deepscaler | [
{
"role": "user",
"content": "The digits from 1 to 9 are to be written in the nine cells of a $3 \\times 3$ grid, one digit in each cell.\n- The product of the three digits in the first row is 12.\n- The product of the three digits in the second row is 112.\n- The product of the three digits in the first co... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 37592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rearrange the digits of 124669 to form a different even number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 37593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What is the average of the hidden prime numbers? [as... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 37594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), points \\(M\\) and \\(N\\) are the midpoints of \\(AB\\) and \\(AC\\), respectively, and points \\(P\\) and \\(Q\\) trisect \\(BC\\). Given that \\(A\\), \\(M\\), \\(N\\), \\(P\\), and \\(Q\\) lie on a circle and \\(BC = 1\\), compute the area of triangle... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{12}"
} | {
"split": "train",
"index": 37595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point $P$ moves on the circle $x^{2}+(y-2)^{2}=1$, and point $Q$ moves on the ellipse $\\frac{x^{2}}{9}+y^{2}=1$, find the maximum value of the distance $PQ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{6}}{2} + 1"
} | {
"split": "train",
"index": 37596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many pairs $(m, n)$ with $m$ and $n$ integers satisfying $1 \\leq m \\leq 100$ and $101 \\leq n \\leq 205$ is $3^{m}+7^{n}$ divisible by 10? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2625"
} | {
"split": "train",
"index": 37597,
"solution": "The units digits of powers of 3 cycle $3,9,7,1$ and the units digits of powers of 7 cycle $7,9,3,1$. For $3^{m}+7^{n}$ to be divisible by 10, one of the following must be true: units digit of $3^{m}$ is 3 and $7^{n}$ is 7, or 9 and 1, or 7 and 3, or 1 and 9. The nu... |
deepscaler | [
{
"role": "user",
"content": "Let \\( A = \\{1, 2, \\cdots, 10\\} \\). If the equation \\( x^2 - bx - c = 0 \\) satisfies \\( b, c \\in A \\) and the equation has at least one root \\( a \\in A \\), then the equation is called a \"beautiful equation\". Find the number of \"beautiful equations\". Let's think... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 37598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $D$ and $E$ are chosen on side $BC$ of triangle $ABC$ such that $E$ is between $B$ and $D$ and $BE=1$ , $ED=DC=3$ . If $\\angle BAD=\\angle EAC=90^\\circ$ , the area of $ABC$ can be expressed as $\\tfrac{p\\sqrt q}r$ , where $p$ and $r$ are relat... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 37599,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at $4$ miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to $2$ miles per hour. After reachin... | math | {
"style": "rule",
"ground_truth": "\\frac{12}{13}"
} | {
"split": "train",
"index": 37600,
"solution": "Let's analyze the problem step by step:\n\n1. **Define the problem in terms of distance and speed:**\n Let the total distance from the trailhead to the fire tower be $2d$. Chantal walks the first half of the trail, a distance of $d$, at a speed of $4$ miles per h... |
deepscaler | [
{
"role": "user",
"content": "Compute $\\arcsin 0.$ Express your answer in radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 37601,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the radius $r$ of a circle inscribed within three mutually externally tangent circles of radii $a = 5$, $b = 10$, and $c = 20$ using the formula:\n\\[\n\\frac{1}{r} = \\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} + 2 \\sqrt{\\frac{1}{ab} + \\frac{1}{ac} + \\frac{1}{bc}}.... | math | {
"style": "rule",
"ground_truth": "1.381"
} | {
"split": "train",
"index": 37602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I randomly choose an integer \\( p \\) between \\( 1 \\) and \\( 20 \\) inclusive. What is the probability that \\( p \\) is such that there exists an integer \\( q \\) so that \\( p \\) and \\( q \\) satisfy the equation \\( pq - 6p - 3q = 3 \\)? Express your answer as a commo... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{20}"
} | {
"split": "train",
"index": 37603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can a barrel with a capacity of 10 liters be emptied using two containers with capacities of 1 liter and 2 liters? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 37604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider $13$ marbles that are labeled with positive integers such that the product of all $13$ integers is $360$ . Moor randomly picks up $5$ marbles and multiplies the integers on top of them together, obtaining a single number. What is the maximum number of different ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 37605,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product\n\\[n = f_1\\cdot f_2\\cdots f_k,\\]where $k\\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that d... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 37606,
"solution": "To solve for $D(96)$, we need to consider all possible ways to write $96$ as a product of integers greater than $1$, where the order of factors matters. We start by noting the prime factorization of $96$:\n\\[ 96 = 2^5 \\cdot 3 \\]\n\nWe will consider different cas... |
deepscaler | [
{
"role": "user",
"content": "Two children at a time can play pairball. For $90$ minutes, with only two children playing at time, five children take turns so that each one plays the same amount of time. The number of minutes each child plays is Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 37607,
"solution": "1. **Calculate Total Playing Time**: Each game involves two children playing at the same time. The total duration for which the game is played is $90$ minutes. However, since there are two children playing simultaneously, the total playing time in terms of \"child-... |
deepscaler | [
{
"role": "user",
"content": "Last summer $100$ students attended basketball camp. Of those attending, $52$ were boys and $48$ were girls. Also, $40$ students were from Jonas Middle School and $60$ were from Clay Middle School. Twenty of the girls were from Jonas Middle School. How many of the boys were... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 37608,
"solution": "1. **Organize the given data into a table**: We start by setting up a table to organize the information provided in the problem. We know the total number of students, the breakdown by gender, the breakdown by school, and the number of girls from Jonas Middle School... |
deepscaler | [
{
"role": "user",
"content": "From the point $(x, y)$, a legal move is a move to $\\left(\\frac{x}{3}+u, \\frac{y}{3}+v\\right)$, where $u$ and $v$ are real numbers such that $u^{2}+v^{2} \\leq 1$. What is the area of the set of points that can be reached from $(0,0)$ in a finite number of legal moves? Let'... | math | {
"style": "rule",
"ground_truth": "\\frac{9 \\pi}{4}"
} | {
"split": "train",
"index": 37609,
"solution": "We claim that the set of points is the disc with radius $\\frac{3}{2}$ centered at the origin, which clearly has area $\\frac{9 \\pi}{4}$. First, we show that the set is contained in this disc. This is because if we are currently at a distance of $r$ from the origi... |
deepscaler | [
{
"role": "user",
"content": "Suppose that 7 boys and 13 girls line up in a row. Let $S$ be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row $\\text{GBBGGGBGBGGGBGBGGBGG}$ we have that $S=12$. The average value of $S$ (if all possible orders of... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 37610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\[f(x) =\n\\begin{cases}\n2x + 9 &\\text{if }x<-2, \\\\\n5-2x&\\text{if }x\\ge -2.\n\\end{cases}\n\\]Find $f(3).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 37611,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the value of $p$ such that the equation\n\\[\\frac{2x + 3}{px - 2} = x\\]\nhas exactly one solution. 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": 37612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two different points, $C$ and $D$, lie on the same side of line $AB$ so that $\\triangle ABC$ and $\\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\\tfrac mn$, where $m$ and $n$ are relatively p... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 37613,
"solution": "Let $a = \\angle{CAB}$. By Law of Cosines, \\[\\cos a = \\frac{17^2+9^2-10^2}{2*9*17} = \\frac{15}{17}\\] \\[\\sin a = \\sqrt{1-\\cos^2 a} = \\frac{8}{17}\\] \\[\\tan a = \\frac{8}{15}\\] \\[A = \\frac{1}{2}* 9*\\frac{9}{2}\\tan a = \\frac{54}{5}\\] And $54+5=\\box... |
deepscaler | [
{
"role": "user",
"content": "March 12th is Tree Planting Day. A school organizes 65 high school students and their parents to participate in the \"Plant a tree, green a land\" tree planting activity as a family unit. The activity divides the 65 families into two groups, A and B. Group A is responsible for ... | math | {
"style": "rule",
"ground_truth": "\\frac{12}{5}"
} | {
"split": "train",
"index": 37614,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equation $\\frac{2}{x}=\\frac{1}{x+1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 37615,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\\mathcal{T}.$ Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "2520"
} | {
"split": "train",
"index": 37616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the region $A$ in the complex plane that consists of all points $z$ such that both $\\frac{z}{40}$ and $\\frac{40}{\\overline{z}}$ have real and imaginary parts between $0$ and $1$, inclusive. What is the integer that is nearest the area of $A$? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "572"
} | {
"split": "train",
"index": 37617,
"solution": "Let $z=a+bi \\implies \\frac{z}{40}=\\frac{a}{40}+\\frac{b}{40}i$. Since $0\\leq \\frac{a}{40},\\frac{b}{40}\\leq 1$ we have the inequality \\[0\\leq a,b \\leq 40\\]which is a square of side length $40$.\n Also, $\\frac{40}{\\overline{z}}=\\frac{40}{a-bi}=\\frac{40... |
deepscaler | [
{
"role": "user",
"content": "Given that $a, b > 0$, $2^a = 3^b = m$, and $a, ab, b$ form an arithmetic sequence, find $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"split": "train",
"index": 37618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the following expression:\n\\[\n\\frac{(1 + 15) \\left( 1 + \\dfrac{15}{2} \\right) \\left( 1 + \\dfrac{15}{3} \\right) \\dotsm \\left( 1 + \\dfrac{15}{17} \\right)}{(1 + 17) \\left( 1 + \\dfrac{17}{2} \\right) \\left( 1 + \\dfrac{17}{3} \\right) \\dotsm \\left( 1 + \\d... | math | {
"style": "rule",
"ground_truth": "496"
} | {
"split": "train",
"index": 37619,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( x \\) is a multiple of \\( 7200 \\), what is the greatest common divisor of \\( f(x)=(5x+6)(8x+3)(11x+9)(4x+12) \\) and \\( x \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 37620,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be an isosceles trapezoid with $AD=BC$ and $AB<CD.$ Suppose that the distances from $A$ to the lines $BC,CD,$ and $BD$ are $15,18,$ and $10,$ respectively. Let $K$ be the area of $ABCD.$ Find $\\sqrt2 \\cdot K.$\nDiagram\n[asy] /* Made by MRENTHUSIASM */ size(250); p... | math | {
"style": "rule",
"ground_truth": "567"
} | {
"split": "train",
"index": 37621,
"solution": "Draw the distances in terms of $B$, as shown in the diagram. By similar triangles, $\\triangle{AEC}\\sim\\triangle{BIC}$. As a result, let $AB=u$, then $BC=AD=\\frac{6}{5}u$ and $2AC=3BC$. The triangle $ABC$ is $6-5-9$ which $\\cos(\\angle{ABC})=-\\frac{1}{3}$. By ... |
deepscaler | [
{
"role": "user",
"content": "Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\\sqrt{21}$ and $\\sqrt{31}$. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive inte... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 37622,
"solution": "Let us inscribe a tetrahedron $ACB'D'$ in given parallelepiped so that its edges coincide with the diagonals of the faces of the parallelepiped. Note that the three edges outgoing from the vertex $B'$ have the same length $b$, and the three edges at the base have a... |
deepscaler | [
{
"role": "user",
"content": "For each permutation $b_1, b_2, b_3, \\ldots, b_8$ of the integers $1, 2, 3, \\ldots, 8$, form the sum\n\\[|b_1 - b_2| + |b_3 - b_4| + |b_5 - b_6| + |b_7 - b_8|.\\]\nThe average value of all such sums can be written in the form $\\dfrac{p}{q}$, where $p$ and $q$ are relatively ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 37623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\vec{a}$ and $\\vec{b}$ satisfy $|\\vec{a}|=|\\vec{b}|=2$, and $\\vec{b}$ is perpendicular to $(2\\vec{a}+\\vec{b})$, find the angle between vector $\\vec{a}$ and $\\vec{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 37624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Paint is to be mixed so that the ratio of red paint to white paint is 3 to 2. If Ben wants to make 30 cans of the mixture and all cans hold the same volume of paint, how many cans of red paint will he need? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 37625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A tetrahedron $PQRS$ has edges of lengths $8, 14, 19, 28, 37,$ and $42$ units. If the length of edge $PQ$ is $42$, determine the length of edge $RS$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 37626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the enclosure dimensions are 15 feet long, 8 feet wide, and 7 feet tall, with each wall and floor being 1 foot thick, determine the total number of one-foot cubical blocks used to create the enclosure. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "372"
} | {
"split": "train",
"index": 37627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six teams play in a soccer tournament where each team faces every other team exactly once. Each match results in a win or a loss, with no ties allowed. Winners receive one point; losers receive none. In the first match, Team $A$ defeats Team $B$. Assuming each team has an equal... | math | {
"style": "rule",
"ground_truth": "419"
} | {
"split": "train",
"index": 37628,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain hyperbola, the center is at $(-2,0),$ one focus is at $(-2 + \\sqrt{34},0),$ and one vertex is at $(-5,0).$ The equation of this hyperbola can be written as\n\\[\\frac{(x - h)^2}{a^2} - \\frac{(y - k)^2}{b^2} = 1.\\]Find $h + k + a + b.$ Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 37629,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = x - 3$ and $g(x) = x/2$. Compute \\[f(g^{-1}(f^{-1}(g(f^{-1}(g(f(23))))))).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 37630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $2^x+ 2^x+ 2^x+ 2^x= 512$, 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": "7"
} | {
"split": "train",
"index": 37631,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of five-digit positive integers, $n$, that satisfy the following conditions:\n (a) the number $n$ is divisible by $5,$\n (b) the first and last digits of $n$ are equal, and\n (c) the sum of the digits of $n$ is divisible by $5.$ Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 37632,
"solution": "The number takes a form of $5\\text{x,y,z}5$, in which $5|(x+y+z)$. Let $x$ and $y$ be arbitrary digits. For each pair of $x,y$, there are exactly two values of $z$ that satisfy the condition of $5|(x+y+z)$. Therefore, the answer is $10\\times10\\times2=\\boxed{200... |
deepscaler | [
{
"role": "user",
"content": "In a circular arrangement of 101 natural numbers, it is known that among any 5 consecutive numbers, there are at least two even numbers. What is the minimum number of even numbers that can be among the listed numbers? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 37633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cagney can frost a cupcake every 18 seconds and Lacey can frost a cupcake every 40 seconds. Lacey starts working 1 minute after Cagney starts. Calculate the number of cupcakes that they can frost together in 6 minutes. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 37634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Real numbers $x, y, z$ satisfy $$x+x y+x y z=1, \\quad y+y z+x y z=2, \\quad z+x z+x y z=4$$ The largest possible value of $x y z$ is $\\frac{a+b \\sqrt{c}}{d}$, where $a, b, c, d$ are integers, $d$ is positive, $c$ is square-free, and $\\operatorname{gcd}(a, b, d)=1$. Find $10... | math | {
"style": "rule",
"ground_truth": "5272"
} | {
"split": "train",
"index": 37635,
"solution": "Solution 1: Let $p=x y z$ and $q=(x+1)(y+1)(z+1)$. Then, we get $$p q=[x(1+y)] \\cdot[y(1+z)] \\cdot[z(1+x)]=(1-p)(2-p)(4-p)$$ Additionally, note that $$q-p=x y+y z+z x+x+y+z+1=(x+x y)+(y+y z)+(z+x z)+1=8-3 p$$ Therefore, we have $q=8-2 p$. Substituting this into o... |
deepscaler | [
{
"role": "user",
"content": "Jennifer plans a profit of 20% on the selling price of an item, and her expenses are 10% of the selling price. There is also a sales tax of 5% on the selling price of the item. The item sells for $10.00. Calculate the rate of markup on cost of this item. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "53.85\\%"
} | {
"split": "train",
"index": 37636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\prod_{k = 1}^{12} \\prod_{j = 1}^{10} (e^{2 \\pi ji/11} - e^{2 \\pi ki/13}).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 37637,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first n terms of the sequence $\\{a_n\\}$ is $S_n$. If the terms of the sequence $\\{a_n\\}$ are arranged according to the following rule: $$\\frac {1}{2}, \\frac {1}{3}, \\frac {2}{3}, \\frac {1}{4}, \\frac {2}{4}, \\frac {3}{4}, \\frac {1}{5}, \\frac {2}{5}, \\... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{7}"
} | {
"split": "train",
"index": 37638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two types of electronic toy cars, Type I and Type II, each running on the same two circular tracks. Type I completes a lap every 5 minutes, while Type II completes a lap every 3 minutes. At a certain moment, both Type I and Type II cars start their 19th lap simultaneo... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 37639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=\\sin 3x$, determine the horizontal shift required to obtain the graph of the function $y=\\sin \\left(3x+\\frac{\\pi }{4}\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{12}"
} | {
"split": "train",
"index": 37640,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Thirty-six 6-inch wide square posts are evenly spaced with 6 feet between adjacent posts to enclose a square field. What is the outer perimeter, in feet, of the fence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "236"
} | {
"split": "train",
"index": 37641,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "January 1st of a certain non-leap year fell on a Saturday. How many Fridays are there in this year? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 37642,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "1. Given that ${(3x-2)^{6}}={a_{0}}+{a_{1}}(2x-1)+{a_{2}}{(2x-1)^{2}}+ \\cdots +{a_{6}}{(2x-1)^{6}}$, find the value of $\\dfrac{{a_{1}}+{a_{3}}+{a_{5}}}{{a_{0}}+{a_{2}}+{a_{4}}+{a_{6}}}$.\n2. A group of 6 volunteers is to be divided into 4 teams, with 2 teams of 2 people and t... | math | {
"style": "rule",
"ground_truth": "\\dfrac{ \\sqrt{2}}{2}"
} | {
"split": "train",
"index": 37643,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian plane, let $A=(0,0), B=(200,100)$, and $C=(30,330)$. Compute the number of ordered pairs $(x, y)$ of integers so that $\\left(x+\\frac{1}{2}, y+\\frac{1}{2}\\right)$ is in the interior of triangle $A B C$. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "31480"
} | {
"split": "train",
"index": 37644,
"solution": "We use Pick's Theorem, which states that in a lattice polygon with $I$ lattice points in its interior and $B$ lattice points on its boundary, the area is $I+B / 2-1$. Also, call a point center if it is of the form $\\left(x+\\frac{1}{2}, y+\\frac{1}{2}\\right)$ for... |
deepscaler | [
{
"role": "user",
"content": "In the number $2 * 0 * 1 * 6 * 0 *$, each of the 5 asterisks needs to be replaced by any digit from $0, 1, 2, 3, 4, 5, 6, 7, 8$ (digits can repeat) so that the resulting 10-digit number is divisible by 45. In how many ways can this be done? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "1458"
} | {
"split": "train",
"index": 37645,
"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 $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 37646,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $cos({α-\\frac{π}{6}})=\\frac{1}{3}$, find $sin({2α+\\frac{π}{6}})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{9}"
} | {
"split": "train",
"index": 37647,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the absolute value of the difference of single-digit integers \\( C \\) and \\( D \\) such that in base \\( 5 \\):\n$$ \\begin{array}{c@{}c@{\\;}c@{}c@{}c@{}c}\n& & & D & D & C_5 \\\\\n& & & \\mathbf{3} & \\mathbf{2} & D_5 \\\\\n& & + & C & \\mathbf{2} & \\mathbf{4_5} \\\\... | math | {
"style": "rule",
"ground_truth": "1_5"
} | {
"split": "train",
"index": 37648,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.