data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Arrange all the four-digit numbers formed using $1, 2, 3,$ and $4$, each used exactly once, in ascending order. What is the difference between the 23rd number and the 21st number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 24749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is \\(\\sum^{100}_{i=1} \\sum^{100}_{j=1} (i+j) \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1{,}010{,}000"
} | {
"split": "train",
"index": 24750,
"solution": "We are given the double summation:\n\\[\n\\sum^{100}_{i=1} \\sum^{100}_{j=1} (i+j)\n\\]\nWe can simplify this by changing the order of summation and combining terms:\n\\[\n\\sum^{100}_{i=1} \\sum^{100}_{j=1} (i+j) = \\sum^{100}_{i=1} \\sum^{100}_{j=1} i + \\sum^{10... |
deepscaler | [
{
"role": "user",
"content": "A hotpot restaurant in Chongqing operates through three methods: dining in, takeout, and setting up a stall outside (referred to as stall). In June, the ratio of revenue from dining in, takeout, and stall for this hotpot restaurant was $3:5:2$. With the introduction of policies... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 24751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that $1^{2}+2^{2}+3^{2}+4^{2}+\\cdots+n^{2}$ is divisible by 100. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 24752,
"solution": "The sum of the first $n$ squares equals $n(n+1)(2 n+1) / 6$, so we require $n(n+1)(2 n+1)$ to be divisible by $600=24 \\cdot 25$. The three factors are pairwise relatively prime, so one of them must be divisible by 25 . The smallest $n$ for which this happens is $n... |
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. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 24753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$$\\frac {4}{5} + 9 \\frac {4}{5} + 99 \\frac {4}{5} + 999 \\frac {4}{5} + 9999 \\frac {4}{5} + 1 = \\_\\_\\_\\_\\_\\_.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11111"
} | {
"split": "train",
"index": 24754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest square number whose first five digits are 4 and the sixth digit is 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "666667"
} | {
"split": "train",
"index": 24755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A housewife saved $2.50 in buying a dress on sale. If she spent $25 for the dress, she saved about: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9 \\%"
} | {
"split": "train",
"index": 24756,
"solution": "1. **Calculate the original price of the dress**:\n The housewife saved $\\textdollar{2.50}$ and spent $\\textdollar{25}$ on the dress. Therefore, the original price of the dress before the discount was:\n \\[\n \\text{Original Price} = \\text{Amount Spent} +... |
deepscaler | [
{
"role": "user",
"content": "There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let $ N$ be the total number of intersection points of these circles. Determine the smallest possible values of $ N$ . Let's think ... | math | {
"style": "rule",
"ground_truth": "3012"
} | {
"split": "train",
"index": 24757,
"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"
} | {
"split": "train",
"index": 24758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "George now has an unfair eight-sided die. The probabilities of rolling each number from 1 to 5 are each $\\frac{1}{15}$, the probability of rolling a 6 or a 7 is $\\frac{1}{6}$ each, and the probability of rolling an 8 is $\\frac{1}{5}$. What is the expected value of the number... | math | {
"style": "rule",
"ground_truth": "4.7667"
} | {
"split": "train",
"index": 24759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the numbers from the set $\\{1,\\ 2,\\ 3,\\ldots,\\ 100\\}$ have a perfect cube factor other than one? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 24760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the coefficient of $a^2b^2$ in $(a+b)^4\\left(c+\\dfrac{1}{c}\\right)^6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 24761,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $P$ lies inside equilateral triangle $A B C$ so that $\\angle B P C=120^{\\circ}$ and $A P \\sqrt{2}=B P+C P$. $\\frac{A P}{A B}$ can be written as $\\frac{a \\sqrt{b}}{c}$, where $a, b, c$ are integers, $c$ is positive, $b$ is square-free, and $\\operatorname{gcd}(a, c)=... | math | {
"style": "rule",
"ground_truth": "255"
} | {
"split": "train",
"index": 24762,
"solution": "Let $O$ be the center of $A B C$. First, we draw in the circumcircle of $A B C$ and the circumcircle of $B O C$, labeled $\\omega_{1}$ and $\\omega_{2}$, respectively. Note that $\\omega_{1}$ is the reflection of $\\omega_{2}$ over $B C$ and that $P$ lies on $\\ome... |
deepscaler | [
{
"role": "user",
"content": "Let $a_0 = 2,$ $b_0 = 3,$ and\n\\[a_{n + 1} = \\frac{a_n^2}{b_n} \\quad \\text{and} \\quad b_{n + 1} = \\frac{b_n^2}{a_n}\\]for all $n \\ge 0.$ Then $b_8 = \\frac{3^m}{2^n}$ for some integers $m$ and $n.$ Enter the ordered pair $(m,n).$ Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "(3281,3280)"
} | {
"split": "train",
"index": 24763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a country with 15 cities, some of which are connected by airlines belonging to three different companies, it is known that even if any one of the airlines ceases operations, it will still be possible to travel between any two cities (possibly with transfers) using the remain... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 24764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be an isosceles trapezoid with $\\overline{AD}||\\overline{BC}$ whose angle at the longer base $\\overline{AD}$ is $\\dfrac{\\pi}{3}$. The diagonals have length $10\\sqrt {21}$, and point $E$ is at distances $10\\sqrt {7}$ and $30\\sqrt {7}$ from vertices $A$ and $D$... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 24765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two different numbers are selected from 1, 2, 3, 4, 5 to form a point (x, y). Find the probability that this point lies above the line x+y-5=0. 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": 24766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the subset \\( S \\) of the set \\(\\{1,2, \\cdots, 15\\}\\), if a positive integer \\( n \\) and \\( n+|S| \\) are both elements of \\( S \\), then \\( n \\) is called a \"good number\" of \\( S \\). If a subset \\( S \\) has at least one \"good number\", then \\( S \\) is... | math | {
"style": "rule",
"ground_truth": "4096"
} | {
"split": "train",
"index": 24767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the value of $$64^{-\\frac{1}{3}}+lg0.001$$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{11}{2}"
} | {
"split": "train",
"index": 24768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An integer is called \"octal\" if it is divisible by $8$ or if at least one of its digits is $8$ . \nHow many integers between $1$ and $100$ are octal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 24769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A track and field team has three sprinters. According to the statistics from their usual training sessions, the probabilities of sprinters A, B, and C finishing a 100m dash within 13 seconds (considered qualified) are $\\frac{2}{5}$, $\\frac{3}{4}$, and $\\frac{1}{3}$, respecti... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{10}"
} | {
"split": "train",
"index": 24770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $[(15^{15} \\div 15^{13})^3 \\cdot 3^2] \\div 2^3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3^8 \\cdot 5^6 \\cdot 2^{-3}"
} | {
"split": "train",
"index": 24771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If I have a $5\\times5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14400"
} | {
"split": "train",
"index": 24772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Martin is playing a game. His goal is to place tokens on an 8 by 8 chessboard in such a way that there is at most one token per square, and each column and each row contains at most 4 tokens.\n\na) How many tokens can Martin place, at most?\n\nb) If, in addition to the previous... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 24773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 4 boys and 3 girls lining up for a photo. The number of ways they can line up such that no two boys are adjacent is \\_\\_\\_\\_ (answer with a number), and the number of ways they can line up such that no two girls are adjacent is \\_\\_\\_\\_ (answer with a number).... | math | {
"style": "rule",
"ground_truth": "1440"
} | {
"split": "train",
"index": 24774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( P \\) be a regular polygon with 2006 sides. A diagonal of \\( P \\) is called good if its endpoints divide the perimeter of \\( P \\) into two parts, each having an odd number of sides of \\( P \\). The sides of \\( P \\) are also called good. Suppose \\( P \\) has been... | math | {
"style": "rule",
"ground_truth": "1003"
} | {
"split": "train",
"index": 24775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From an 8x8 chessboard, 10 squares were cut out. It is known that among the removed squares, there are both black and white squares. What is the maximum number of two-square rectangles (dominoes) that can still be guaranteed to be cut out from this board? Let's think step by st... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 24776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petya and Vasya participated in a bicycle race. All participants started at the same time and finished with different times. Petya finished immediately after Vasya and ended up in tenth place. How many people participated in the race if Vasya was fifteenth from the end? Let's t... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 24777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $3^{-1} + 3^{-2} \\pmod{25}$? Express your answer as an integer from $0$ to $24$, inclusive. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 24778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular polygon has sides of length 5 units and an exterior angle of 120 degrees. What is the perimeter of the polygon, in units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 24779,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A, B, C, and D obtained the top 4 positions in the school (no ties). They made the following statements:\n- A: \"I am neither first nor second.\"\n- B: \"I am neither second nor third.\"\n- C: \"My position is adjacent to B.\"\n- D: \"My position is adjacent to C.\"\n\nGiven th... | math | {
"style": "rule",
"ground_truth": "4123"
} | {
"split": "train",
"index": 24780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The image of the point with coordinates $(-3,-1)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(5,3)$. Find $m+b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 24781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two classmates, A and B, live in the same neighborhood and leave the neighborhood gate at the same time to go to school. Initially, A travels at a speed of 40 meters per minute, while B travels at a speed of 60 meters per minute. After A has walked half the distance, they reali... | math | {
"style": "rule",
"ground_truth": "960"
} | {
"split": "train",
"index": 24782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a_{1}, a_{2}, \\cdots, a_{n} \\) be an arithmetic sequence, and it is given that\n$$\n\\sum_{i=1}^{n}\\left|a_{i}+j\\right|=2028 \\text{ for } j=0,1,2,3. \n$$\nFind the maximum value of the number of terms \\( n \\). Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 24783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lottery. (For 7th grade, 3 points) It so happened that Absent-Minded Scientist has only 20 rubles left, but he needs to buy a bus ticket to get home. The bus ticket costs 45 rubles. Nearby the bus stop, instant lottery tickets are sold for exactly 10 rubles each. With a probabi... | math | {
"style": "rule",
"ground_truth": "0.19"
} | {
"split": "train",
"index": 24784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x=3$, $y=2x$, and $z=3y$, what is the average of $x$, $y$, and $z$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 24785,
"solution": "Since $x=3$ and $y=2x$, then $y=2 \\times 3=6$. Since $y=6$ and $z=3y$, then $z=3 \\times 6=18$. Therefore, the average of $x, y$ and $z$ is $\\frac{x+y+z}{3}=\\frac{3+6+18}{3}=9$."
} |
deepscaler | [
{
"role": "user",
"content": "Let $C$ be the coefficient of $x^2$ in the expansion of the product $(1 - x)(1 + 2x)(1 - 3x)\\cdots(1 + 14x)(1 - 15x).$ Find $|C|.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "588"
} | {
"split": "train",
"index": 24786,
"solution": "Let our polynomial be $P(x)$.\n It is clear that the coefficient of $x$ in $P(x)$ is $-1 + 2 - 3 + \\ldots + 14 - 15 = -8$, so $P(x) = 1 -8x + Cx^2 + Q(x)$, where $Q(x)$ is some polynomial divisible by $x^3$.\n Then $P(-x) = 1 + 8x + Cx^2 + Q(-x)$ and so $P(x)\\cdo... |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all two-digit positive integers whose squares end with the digits 25? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "495"
} | {
"split": "train",
"index": 24787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $x$ if $\\log_8 x = 1.75$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32\\sqrt[4]{2}"
} | {
"split": "train",
"index": 24788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $2x^2 - 5x + k = 0$ is a quadratic equation with one solution for $x$. Express $k$ as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{8}"
} | {
"split": "train",
"index": 24789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the smallest possible product when three different numbers from the set $\\{-4, -3, -1, 5, 6\\}$ are multiplied. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 24790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A 10x10 arrangement of alternating black and white squares has a black square $R$ in the second-bottom row and a white square $S$ in the top-most row. Given that a marker is initially placed at $R$ and can move to an immediately adjoining white square on the row above either to... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 24791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $\\frac{1}{4}\\%$ of 120? Express your answer as a decimal. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": ".3"
} | {
"split": "train",
"index": 24792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integers $n$ between 1 and 100 is the greatest common divisor of 15 and $n$ equal to 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 24793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number that ends with the digit 6 such that moving this digit to the front increases the number exactly fourfold. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "153846"
} | {
"split": "train",
"index": 24794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the number $74982.1035$ the value of the place occupied by the digit 9 is how many times as great as the value of the place occupied by the digit 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100,000"
} | {
"split": "train",
"index": 24795,
"solution": "1. **Identify the place values**: \n - The digit $9$ in the number $74982.1035$ is in the hundreds place, which means it represents $9 \\times 100 = 900$.\n - The digit $3$ in the number $74982.1035$ is in the thousandths place, which means it represents $3 \\t... |
deepscaler | [
{
"role": "user",
"content": "An ancient civilization has a tribe of 12 members organized hierarchically. The tribe has one main chief, two supporting chiefs (Senior and Junior), and each supporting chief has three inferior officers. If the tribe has 12 members in total, in how many ways can the leadership ... | math | {
"style": "rule",
"ground_truth": "2217600"
} | {
"split": "train",
"index": 24796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of functions $f(n),$ taking the integers to the integers, such that\n\\[f(a + b) + f(ab) = f(a) f(b) + 1\\]for all integers $a$ and $b.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 24797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be the maximum possible value of $x_1x_2 + x_2x_3 + \\cdots + x_6x_1$ where $x_1, x_2, \\dots, x_6$ is a permutation of $(1,2,3,4,5,6)$ and let $Q$ be the number of permutations for which this maximum is achieved, given the additional condition that $x_1 + x_2 + x_3 + x... | math | {
"style": "rule",
"ground_truth": "83"
} | {
"split": "train",
"index": 24798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of the expression $\\frac{1+(3 \\times 5)}{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 24799,
"solution": "Using the correct order of operations, $\\frac{1+(3 \\times 5)}{2}=\\frac{1+15}{2}=\\frac{16}{2}=8$."
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $1234 + 2341 + 3412 + 4123$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11110"
} | {
"split": "train",
"index": 24800,
"solution": "To solve the problem, we need to add the numbers $1234$, $2341$, $3412$, and $4123$. We can do this by aligning the numbers vertically and adding them column by column:\n\n\\[\n\\begin{array}{c}\n\\phantom{+}1234 \\\\\n+2341 \\\\\n+3412 \\\\\n+4123 \\\\\n\\hline\n\... |
deepscaler | [
{
"role": "user",
"content": "Sharik and Matroskin ski on a circular track, half of which is an uphill slope and the other half is a downhill slope. Their speeds are identical on the uphill slope and are four times less than their speeds on the downhill slope. The minimum distance Sharik falls behind Matros... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 24801,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of the expression given by: $$2 + \\cfrac{3}{4 + \\cfrac{5}{6}}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{76}{29}"
} | {
"split": "train",
"index": 24802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=\\sin(\\omega x)$ ($\\omega>0$) is increasing in the interval $[0, \\frac{\\pi}{3}]$ and its graph is symmetric about the point $(3\\pi, 0)$, the maximum value of $\\omega$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 24803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the first term of a geometric sequence $\\{{a_n}\\}$ is $\\frac{3}{2}$ and the sum of its first $n$ terms is $S_n$ $(n \\in \\mathbb{N^*})$, and $-2S_2, S_3, 4S_4$ form an arithmetic sequence.\n\n(I) Find the general term formula of the sequence $\\{{a_n}\\}$.\n\n(II... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 24804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of two angles of a triangle is $\\frac{6}{5}$ of a right angle, and one of these two angles is $30^{\\circ}$ larger than the other. What is the degree measure of the largest angle in the triangle? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 24805,
"solution": "1. **Identify the given information and the problem statement**: We are given that the sum of two angles of a triangle is $\\frac{6}{5}$ of a right angle, and one of these angles is $30^\\circ$ larger than the other. We need to find the degree measure of the larges... |
deepscaler | [
{
"role": "user",
"content": "\nThe circle, which has its center on the hypotenuse $AB$ of the right triangle $ABC$, touches the two legs $AC$ and $BC$ at points $E$ and $D$ respectively.\n\nFind the angle $ABC$, given that $AE = 1$ and $BD = 3$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 24806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nGiven the hyperbola \\( C_{1}: 2x^{2} - y^{2} = 1 \\) and the ellipse \\( C_{2}: 4x^{2} + y^{2} = 1 \\), let \\( M \\) and \\( N \\) be moving points on the hyperbola \\( C_{1} \\) and the ellipse \\( C_{2} \\) respectively, with \\( O \\) as the origin. If \\( O M \\) is per... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 24807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average of 12, 21 and $x$ is 18. 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": "21"
} | {
"split": "train",
"index": 24808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Our club has 20 members, 10 boys and 10 girls. In how many ways can we choose a president and a vice-president if they must be of the same gender? Assume no one can hold both offices. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 24809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integers \\( n \\) between 1 and 15 (inclusive) is \\(\\frac{n}{18}\\) a repeating decimal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 24810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All positive integers whose digits add up to 11 are listed in increasing order: $29, 38, 47, ...$. What is the eleventh number in that list? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "137"
} | {
"split": "train",
"index": 24811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "\\frac{47}{256}"
} | {
"split": "train",
"index": 24812,
"solution": "1. **Understanding the Problem:**\n We need to find the probability that no two adjacent people will stand after flipping their coins around a circular table with 8 people. Each person flips a fair coin, resulting in either heads (stand) or tails (sit).\n\n2. **T... |
deepscaler | [
{
"role": "user",
"content": "Consider the two hands of an analog clock, each of which moves with constant angular velocity. Certain positions of these hands are possible (e.g. the hour hand halfway between the 5 and 6 and the minute hand exactly at the 6), while others are impossible (e.g. the hour hand ex... | math | {
"style": "rule",
"ground_truth": "143"
} | {
"split": "train",
"index": 24813,
"solution": "143 We can look at the twelve-hour cycle beginning at midnight and ending just before noon, since during this time, the clock goes through each possible position exactly once. The minute hand has twelve times the angular velocity of the hour hand, so if the hour ha... |
deepscaler | [
{
"role": "user",
"content": "There are three pairs of real numbers $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$ that satisfy both $x^3 - 3xy^2 = 2010$ and $y^3 - 3x^2y = 2000$. Compute $\\left(1-\\frac{x_1}{y_1}\\right)\\left(1-\\frac{x_2}{y_2}\\right)\\left(1-\\frac{x_3}{y_3}\\right)$. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{100}"
} | {
"split": "train",
"index": 24814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the polar coordinate system, the equation of circle C is given by ρ = 2sinθ, and the equation of line l is given by $ρsin(θ+ \\frac {π}{3})=a$. If line l is tangent to circle C, find the value of the real number a. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "- \\frac {1}{2}"
} | {
"split": "train",
"index": 24815,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\mathbf{A} = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix},$ then its transpose is given by\n\\[\\mathbf{A}^T = \\begin{pmatrix} a & c \\\\ b & d \\end{pmatrix}.\\]Given that $\\mathbf{A}^T = \\mathbf{A}^{-1},$ find $a^2 + b^2 + c^2 + d^2.$ Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 24816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $0 < \\alpha < \\frac{\\pi}{2}$ and $\\tan \\alpha = 2$, then $\\frac{\\sin 2\\alpha + 1}{\\cos^4 \\alpha - \\sin^4 \\alpha} = \\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 24817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What are the first three digits to the right of the decimal point in the decimal representation of $\\left(10^{2003}+1\\right)^{11/8}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "375"
} | {
"split": "train",
"index": 24818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area in the plane contained by the graph of\n\\[|x + y| + |x - y| \\le 4.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 24819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for \\(x\\): \\(x\\lfloor x\\lfloor x\\lfloor x\\lfloor x\\rfloor\\rfloor\\rfloor\\rfloor=122\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{122}{41}"
} | {
"split": "train",
"index": 24820,
"solution": "This problem can be done without needless casework. (For negative values of \\(x\\), the left hand side will be negative, so we only need to consider positive values of \\(x\\).) The key observation is that for \\(x \\in[2,3), 122\\) is an extremely large value for... |
deepscaler | [
{
"role": "user",
"content": "Two divisions of an oil extraction company compete with each other for the amount of oil extracted. Depending on the availability of free tanks, each of them can extract the amount of tons defined by the following payoff matrix:\n\n$$\n\\text { Extraction }=\\left(\\begin{array... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 24821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Density is defined as the ratio of mass to volume. There are two cubes. The second cube is made from a material with twice the density of the first, and the side length of the second cube is 100% greater than the side length of the first. By what percentage is the mass of the s... | math | {
"style": "rule",
"ground_truth": "1500"
} | {
"split": "train",
"index": 24822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We marked the midpoints of all sides and diagonals of a regular 1976-sided polygon. What is the maximum number of these points that can lie on a single circle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1976"
} | {
"split": "train",
"index": 24823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of \\(\\frac{1}{1+\\frac{1}{1+\\frac{1}{2}}}\\) can be expressed as a simplified fraction. 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": 24824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Figures $I$, $II$, and $III$ are squares. The perimeter of $I$ is $12$ and the perimeter of $II$ is $24$. The perimeter of $III$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 24825,
"solution": "1. **Identify the side lengths of squares $I$ and $II$:**\n - Let the side length of square $I$ be $x$. Since the perimeter of a square is four times its side length, we have:\n \\[\n 4x = 12 \\implies x = \\frac{12}{4} = 3\n \\]\n - Let the side le... |
deepscaler | [
{
"role": "user",
"content": "What is the least possible value of\n\\[(x+1)(x+2)(x+3)(x+4)+2019\\]where $x$ is a real number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"split": "train",
"index": 24826,
"solution": "1. **Expression Simplification**:\n Start by grouping the terms in the expression \\((x+1)(x+2)(x+3)(x+4)+2019\\):\n \\[\n (x+1)(x+4)(x+2)(x+3) + 2019 = (x^2 + 5x + 4)(x^2 + 5x + 6) + 2019.\n \\]\n Here, we used the fact that \\((x+1)(x+4) = x^2 + 5x + 4\... |
deepscaler | [
{
"role": "user",
"content": "How much money should I invest at an annually compounded interest rate of $5\\%$ so that I have $\\$500,\\!000$ in ten years? Express your answer as a dollar value rounded to the nearest cent. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\$306,\\!956.63"
} | {
"split": "train",
"index": 24827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a natural number \\( N \\), if at least seven out of the nine natural numbers from 1 to 9 are factors of \\( N \\), \\( N \\) is called a \"seven-star number.\" What is the smallest \"seven-star number\" greater than 2000? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"split": "train",
"index": 24828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Through a point $P$ inside the $\\triangle ABC$ a line is drawn parallel to the base $AB$, dividing the triangle into two equal areas.\nIf the altitude to $AB$ has a length of $1$, then the distance from $P$ to $AB$ is: Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 24829,
"solution": "To solve this problem, we need to find the distance from point $P$ to the base $AB$ of $\\triangle ABC$, given that a line through $P$ parallel to $AB$ divides the triangle into two regions of equal area.\n\n1. **Understanding the Triangle and the Line through $P$:... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$ , medians $\\overline{AD}$ and $\\overline{CE}$ intersect at $P$ , $PE=1.5$ , $PD=2$ , and $DE=2.5$ . What is the area of $AEDC?$ \n[asy]\nunitsize(75);\npathpen = black; pointpen=black;\npair A = MP(\"A\", D((0,0)), dir(200));\npair B = MP(\"B\",... | math | {
"style": "rule",
"ground_truth": "13.5"
} | {
"split": "train",
"index": 24830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of digit of $\\sum_{n=0}^{99} 3^n$ .\nYou may use $\\log_{10} 3=0.4771$ .\n\n2012 Tokyo Institute of Technology entrance exam, problem 2-A Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 24831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are births in West Northland every 6 hours, deaths every 2 days, and a net immigration every 3 days. Calculate the approximate annual increase in population. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1400"
} | {
"split": "train",
"index": 24832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation of the parabola $y^{2}=4x$, and a line $l$ passing through its focus $F$ intersecting the parabola at points $A$ and $B$. If $S_{\\triangle AOF}=3S_{\\triangle BOF}$ (where $O$ is the origin), calculate the length of $|AB|$. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "\\dfrac {16}{3}"
} | {
"split": "train",
"index": 24833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It takes Mary 30 minutes to walk uphill 1 km from her home to school, but it takes her only 10 minutes to walk from school to home along the same route. What is her average speed, in km/hr, for the round trip? Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 24834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the sequence defined recursively by $u_1 = a > 0$ and\n\\[u_{n + 1} = -\\frac{1}{u_n + 1}\\]for $n \\ge 1.$ Express $u_{16}$ in terms of $a.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "a"
} | {
"split": "train",
"index": 24835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line $x+2ay-1=0$ is parallel to $\\left(a-1\\right)x+ay+1=0$, find the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 24836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the decomposition rate $v$ of a certain type of garbage approximately satisfies the relationship $v=a\\cdot b^{t}$, where $a$ and $b$ are positive constants, and the decomposition rate is $5\\%$ after $6$ months and $10\\%$ after $12$ months, calculate the time it ta... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 24837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $C(k)$ denotes the sum of all different prime divisors of a positive integer $k$. For example, $C(1)=0$, $C(2)=2, C(45)=8$. Find all positive integers $n$ such that $C(2^{n}+1)=C(n)$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "n=3"
} | {
"split": "train",
"index": 24838,
"solution": "Let $P(t)$ be the largest prime divisor of a positive integer $t>1$. Let $m$ be the largest odd divisor of $n: n=2^{k} m$. Then $2^{n}+1=2^{2^{k} m}+1=a^{m}+1$, where $a=2^{2^{k}}$. If $k>0$, that is, $n$ is even, then $C(n)=C(m)+2$ and $C(2^{n}+1)=C(a^{m}+1)$. We ... |
deepscaler | [
{
"role": "user",
"content": "In the triangle $\\triangle ABC$, let $G$ be the centroid, and let $I$ be the center of the inscribed circle. Let $\\alpha$ and $\\beta$ be the angles at the vertices $A$ and $B$, respectively. Suppose that the segment $IG$ is parallel to $AB$ and that $\\beta = 2 \\tan^{-1} (1... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 24839,
"solution": "Let $M$ and $D$ denote the midpoint of $AB$ and the foot of the altitude from $C$ to $AB$, respectively, and let $r$ be the inradius of $\\bigtriangleup ABC$. Since $C,G,M$ are collinear with $CM = 3GM$, the distance from $C$ to line $AB$ is $3$ times the distance ... |
deepscaler | [
{
"role": "user",
"content": "Find the number of different arrangements for a class to select 6 people to participate in two volunteer activities, with each activity accommodating no more than 4 people. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 24840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nSeverus Snape, the potions professor, prepared three potions, each in an equal volume of 400 ml. The first potion makes the drinker smarter, the second makes them more beautiful, and the third makes them stronger. To ensure the effect of any potion, it is sufficient to drink ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 24841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line is parameterized by a parameter $t,$ so that the vector on the line at $t = 2$ is $\\begin{pmatrix} 1 \\\\ 4 \\end{pmatrix},$ and the vector on the line at $t = 3$ is $\\begin{pmatrix} 3 \\\\ -4 \\end{pmatrix}.$ Find the vector on the line at $t = -7.$ Let's think step ... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -17 \\\\ 76 \\end{pmatrix}"
} | {
"split": "train",
"index": 24842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In quadrilateral $ABCD$, there exists a point $E$ on segment $AD$ such that $\\frac{AE}{ED}=\\frac{1}{9}$ and $\\angle BEC$ is a right angle. Additionally, the area of triangle $CED$ is 27 times more than the area of triangle $AEB$. If $\\angle EBC=\\angle EAB, \\angle ECB=\\an... | math | {
"style": "rule",
"ground_truth": "320"
} | {
"split": "train",
"index": 24843,
"solution": "Extend sides $AB$ and $CD$ to intersect at point $F$. The angle conditions yield $\\triangle BEC \\sim \\triangle AFD$, so $\\angle AFD=90^{\\circ}$. Therefore, since $\\angle BFC$ and $\\angle BEC$ are both right angles, quadrilateral $EBCF$ is cyclic and $$\\angl... |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\frac{\\sqrt{2}}{\\sqrt{5}} \\cdot \\frac{\\sqrt{3}}{\\sqrt{6}} \\cdot \\frac{\\sqrt{4}}{\\sqrt{7}}$ and rationalize the denominator of the resulting fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{35}}{35}"
} | {
"split": "train",
"index": 24844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let set $P=\\{0, 2, 4, 6, 8\\}$, and set $Q=\\{m | m=100a_1+10a_2+a_3, a_1, a_2, a_3 \\in P\\}$. Determine the 68th term of the increasing sequence of elements in set $Q$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "464"
} | {
"split": "train",
"index": 24845,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\sqrt{245}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7\\sqrt{5}"
} | {
"split": "train",
"index": 24846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many 4-letter words with at least one vowel can be constructed from the letters A, B, C, D, and E? (Note that A and E are vowels, any word is valid, not just English language words, and letters may be used more than once.) Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "544"
} | {
"split": "train",
"index": 24847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "2028"
} | {
"split": "train",
"index": 24848,
"solution": ""
} |
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