data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The polynomial $x^3 - 2004 x^2 + mx + n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of $n$ are possible? Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "250500"
} | {
"split": "train",
"index": 36149,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cat and Claire are having a conversation about Cat’s favorite number. Cat says, “My favorite number is a two-digit perfect square!”\nClaire asks, “If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there an... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 36150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A person bequeathed an amount of money, slightly less than 1500 dollars, to be distributed as follows. His five children and the notary received amounts such that the square root of the eldest son's share, half of the second son's share, the third son's share minus 2 dollars, t... | math | {
"style": "rule",
"ground_truth": "1464"
} | {
"split": "train",
"index": 36151,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\(\\alpha, \\beta \\geq 0\\) and \\(\\alpha + \\beta \\leq 2\\pi\\), find the minimum value of \\(\\sin \\alpha + 2 \\cos \\beta\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{5}"
} | {
"split": "train",
"index": 36152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Before the soccer match between the teams \"North\" and \"South\", five predictions were made:\n\na) There will be no draw;\n\nb) Goals will be scored against \"South\";\n\nc) \"North\" will win;\n\nd) \"North\" will not lose;\n\ne) Exactly 3 goals will be scored in the match.\... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"split": "train",
"index": 36153,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$A_{2n}^{n+3} + A_{4}^{n+1} = \\boxed{\\text{\\_\\_\\_\\_\\_\\_\\_\\_}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "744"
} | {
"split": "train",
"index": 36154,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the product $\\frac{3}{2} \\cdot \\frac{4}{3} \\cdot \\frac{5}{4} \\cdot \\frac{6}{5} \\cdot \\ldots \\cdot \\frac{a}{b} = 9$, what is the sum of $a$ and $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 36155,
"solution": "1. **Identify the pattern in the product sequence**: \n The given product sequence is $\\dfrac{3}{2}\\cdot \\dfrac{4}{3}\\cdot \\dfrac{5}{4}\\cdot \\dfrac{6}{5}\\cdot \\ldots\\cdot \\dfrac{a}{b}$. Notice that each fraction $\\dfrac{n+1}{n}$ cancels out the $n$ in... |
deepscaler | [
{
"role": "user",
"content": "Simplify $2a(2a^2 + a) - a^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4a^3 + a^2"
} | {
"split": "train",
"index": 36156,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $y=y_{1}+y_{2}$, where $y_{1}$ is directly proportional to $(x+1)$, and $y_{2}$ is inversely proportional to $(x+1)$. When $x=0$, $y=-5$; when $x=2$, $y=-7$. Find the value of $x$ when $y=5$. 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": 36157,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jessica enjoys vanilla ice cream, and she visits her favorite ice cream shop every day for a week. However, sometimes the shop serves only chocolate ice cream, so each day the shop has a 3/4 chance of serving vanilla ice cream. What is the probability that the shop serves vanil... | math | {
"style": "rule",
"ground_truth": "\\frac{945}{16384}"
} | {
"split": "train",
"index": 36158,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f$ be a mapping from set $A = \\{a, b, c, d\\}$ to set $B = \\{0, 1, 2\\}$. \n(1) How many different mappings $f$ are there? \n(2) If it is required that $f(a) + f(b) + f(c) + f(d) = 4$, how many different mappings $f$ are there? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 36159,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two complex numbers ${z_1}$ and ${z_2}$ that correspond to points in the complex plane that are symmetric about the origin, and ${z_1 = 2 - i}$, determine the value of the complex number $\\frac{z_1}{z_2}$. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 36160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a painting club meeting, 7 friends are present. They need to create two separate teams: one team of 4 members, and another team of 2 members for different competitions. How many distinct ways can they form these teams? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 36161,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ria writes down the numbers $1,2,\\cdots, 101$ in red and blue pens. The largest blue number is equal to the number of numbers written in blue and the smallest red number is equal to half the number of numbers in red. How many numbers did Ria write with red pen? Let's think s... | math | {
"style": "rule",
"ground_truth": "68"
} | {
"split": "train",
"index": 36162,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $3 \\times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90^{\\circ}$ clockwise about its center, and every white square i... | math | {
"style": "rule",
"ground_truth": "\\frac{49}{512}"
} | {
"split": "train",
"index": 36163,
"solution": "1. **Identify the invariant**: The center square remains unchanged after a $90^\\circ$ rotation. Therefore, for the entire grid to be black after the process, the center square must initially be black. The probability of the center square being black is $\\frac{1}{... |
deepscaler | [
{
"role": "user",
"content": "The Student council has 24 members: 12 boys and 12 girls. A 5-person committee is selected at random. What is the probability that the committee includes at least one boy and at least one girl? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{455}{472}"
} | {
"split": "train",
"index": 36164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a press conference before a championship game, ten players from four teams will be taking questions. The teams are as follows: three Celtics, three Lakers, two Warriors, and two Nuggets. If teammates insist on sitting together and one specific Warrior must sit at the end of ... | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 36165,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a_n\\}$ satisfies $a_1=3$, and for any integer $n$, always has $(a_{n+1}-1)(1-a_n)=2a_n$, calculate the sum of the first 2018 terms of the sequence $\\{a_n\\}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "589"
} | {
"split": "train",
"index": 36166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c,$ where $a, b,$ and $c$ are integers in $\\{-20,-19,-18,\\ldots,18,19,20\\},$ such that there is a unique integer $m \\not= 2$ with $p(m) = p(2).$ Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "738"
} | {
"split": "train",
"index": 36167,
"solution": "$p(x)-p(2)$ is a cubic with at least two integral real roots, therefore it has three real roots, which are all integers.\n There are exactly two distinct roots, so either $p(x)=p(2)+(x-2)^2(x-m)$ or $p(x)=p(2)+(x-2)(x-m)^2$, with $m\\neq 2$.\n In the first case $p(... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest two-digit number \\( N \\) such that the sum of digits of \\( 10^N - N \\) is divisible by 170. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 36168,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle in which (${BL}$is the angle bisector of ${\\angle{ABC}}$ $\\left( L\\in AC \\right)$, ${AH}$ is an altitude of$\\vartriangle ABC$ $\\left( H\\in BC \\right)$ and ${M}$is the midpoint of the side ${AB}$. It is known that the midpoints of the segments ${... | math | {
"style": "rule",
"ground_truth": "60^\\circ"
} | {
"split": "train",
"index": 36169,
"solution": "\nGiven a triangle \\(\\triangle ABC\\) with the following properties: \n\n- \\( BL \\) is the angle bisector of \\(\\angle ABC\\), with \\( L \\) on \\( AC \\).\n- \\( AH \\) is the altitude from \\( A \\) to \\( BC \\), with \\( H \\) on \\( BC \\).\n- \\( M \\) ... |
deepscaler | [
{
"role": "user",
"content": "If $\\odot$ and $\\nabla$ represent different positive integers less than 20, and $\\odot \\times \\odot \\times \\odot = \\nabla$, what is the value of $\\nabla \\times \\nabla$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 36170,
"solution": "If $\\odot = 2$, then $\\nabla = \\odot \\times \\odot \\times \\odot = 2 \\times 2 \\times 2 = 8$, which is possible. Thus, $\\odot = 2$ and so $\\nabla = 8$. This means that $\\nabla \\times \\nabla = 8 \\times 8 = 64$."
} |
deepscaler | [
{
"role": "user",
"content": "Each of the 33 warriors either always lies or always tells the truth. It is known that each warrior has exactly one favorite weapon: a sword, a spear, an axe, or a bow.\n\nOne day, Uncle Chernomor asked each warrior four questions:\n- Is your favorite weapon a sword?\n- Is your... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 36171,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $1990-1980+1970-1960+\\cdots -20+10$, calculate the sum. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 36172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 6 class officers, among which there are 3 boys and 3 girls.\n\n(1) Now, 3 people are randomly selected to participate in the school's voluntary labor. Calculate the probability that at least 2 of the selected people are girls.\n\n(2) If these 6 people stand in a row f... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 36173,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$, $y$, and $z$ be nonnegative real numbers such that $x + y + z = 8$. Find the maximum value of\n\\[\n\\sqrt{3x + 2} + \\sqrt{3y + 2} + \\sqrt{3z + 2}.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{10}"
} | {
"split": "train",
"index": 36174,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest prime factor of $15! + 18!$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 36175,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( x, y, z \\in [0, 1] \\), find the maximum value of \\( M = \\sqrt{|x-y|} + \\sqrt{|y-z|} + \\sqrt{|z-x|} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + 1"
} | {
"split": "train",
"index": 36176,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the point on the line\n\\[y = \\frac{x + 5}{2}\\]that is closest to the point $(6,1).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{21}{5}, \\frac{23}{5} \\right)"
} | {
"split": "train",
"index": 36177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A parabola has focus $(3,3)$ and directrix $3x + 7y = 21.$ Express the equation of the parabola in the form\n\\[ax^2 + bxy + cy^2 + dx + ey + f = 0,\\]where $a,$ $b,$ $c,$ $d,$ $e,$ $f$ are integers, $a$ is a positive integer, and $\\gcd(|a|,|b|,|c|,|d|,|e|,|f|) = 1.$ Let's th... | math | {
"style": "rule",
"ground_truth": "49x^2 - 42xy + 9y^2 - 222x - 54y + 603 = 0"
} | {
"split": "train",
"index": 36178,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $i^6+i^{16}+i^{-26}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 36179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.3 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (... | math | {
"style": "rule",
"ground_truth": "\\text{B}"
} | {
"split": "train",
"index": 36180,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C D E F G H$ be an equilateral octagon with $\\angle A \\cong \\angle C \\cong \\angle E \\cong \\angle G$ and $\\angle B \\cong \\angle D \\cong \\angle F \\cong$ $\\angle H$. If the area of $A B C D E F G H$ is three times the area of $A C E G$, then $\\sin B$ can be... | math | {
"style": "rule",
"ground_truth": "405"
} | {
"split": "train",
"index": 36181,
"solution": "Assume $A C=1$. Note that from symmetry, it can be seen that all angles in $A C E G$ must be equal. Further, by similar logic all sides must be equal which means that $A C E G$ is a square. Additionally, as $A B=B C, A B C$ is an isosceles triangle, which means the... |
deepscaler | [
{
"role": "user",
"content": "Solve the equation \\(\\frac{15}{x\\left(\\sqrt[3]{35-8 x^{3}}\\right)}=2x+\\sqrt[3]{35-8 x^{3}}\\). Write the sum of all obtained solutions as the answer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"split": "train",
"index": 36182,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sixth graders were discussing how old their principal is. Anya said, \"He is older than 38 years.\" Borya said, \"He is younger than 35 years.\" Vova: \"He is younger than 40 years.\" Galya: \"He is older than 40 years.\" Dima: \"Borya and Vova are right.\" Sasha: \"You are... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 36183,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the data set $(4.7)$, $(4.8)$, $(5.1)$, $(5.4)$, $(5.5)$, calculate the variance of the data set. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.1"
} | {
"split": "train",
"index": 36184,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit integers $abcd$, with $a \\neq 0$, have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$, where $a=4$, $b=6$, $c=9$, and $d=2$. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 36185,
"solution": "We are given a four-digit number $abcd$ and need to find how many such numbers satisfy $ab < bc < cd$ where these are two-digit numbers forming an increasing arithmetic sequence. \n\n#### Step 1: Define the two-digit numbers\nThe two-digit numbers are:\n- $ab = 10a... |
deepscaler | [
{
"role": "user",
"content": "There are $10000$ trees in a park, arranged in a square grid with $100$ rows and $100$ columns. Find the largest number of trees that can be cut down, so that sitting on any of the tree stumps one cannot see any other tree stump.\n Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "2500"
} | {
"split": "train",
"index": 36186,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Andreas, Boyu, Callista, and Diane each randomly choose an integer from 1 to 9, inclusive. Each of their choices is independent of the other integers chosen and the same integer can be chosen by more than one person. The probability that the sum of their four integers is even i... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"split": "train",
"index": 36187,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Distribute 5 students into two dormitories, A and B, with each dormitory accommodating at least 2 students. Find the number of distinct arrangements. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 36188,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many pairs $(m,n)$ of integers satisfy the equation $m+n=mn$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 36189,
"solution": "To find the number of integer pairs $(m,n)$ that satisfy the equation $m+n=mn$, we can start by rearranging the equation:\n\n\\[ m+n = mn \\]\n\nSubtracting $m+n$ from both sides gives:\n\n\\[ mn - m - n = 0 \\]\n\nAdding 1 to both sides, we can factorize the left-... |
deepscaler | [
{
"role": "user",
"content": "A six-digit palindrome is a positive integer with respective digits $abcdcba$, where $a$ is non-zero. Let $T$ be the sum of all six-digit palindromes. Calculate the sum of the digits of $T$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 36190,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A club has 15 members and needs to choose 2 members to be co-presidents. In how many ways can the club choose its co-presidents? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 36191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest five-digit number divisible by 4 that can be formed with the digits 1, 2, 3, 4, and 9? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13492"
} | {
"split": "train",
"index": 36192,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cowboy is 5 miles north of a stream which flows due west. He is also 10 miles east and 6 miles south of his cabin. He wishes to water his horse at the stream and then return home. Determine the shortest distance he can travel to accomplish this.\nA) $5 + \\sqrt{256}$ miles\nB... | math | {
"style": "rule",
"ground_truth": "5 + \\sqrt{356}"
} | {
"split": "train",
"index": 36193,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $(3-2x)^{5}=a_{0}+a_{1}x+a_{2}x^{2}+…+a_{5}x^{5}$, find the value of $a_{0}+a_{1}+2a_{2}+…+5a_{5}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "233"
} | {
"split": "train",
"index": 36194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the probability that athlete A cannot run the first leg and athlete B cannot run the last leg in a 4x100 meter relay race selection from 6 short-distance runners, including athletes A and B, to form a team of 4 runners. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{10}"
} | {
"split": "train",
"index": 36195,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the coefficients $p$ and $q$ are integers and the roots $\\alpha_{1}$ and $\\alpha_{2}$ are irrational, a quadratic trinomial $x^{2} + px + q$ is called an irrational quadratic trinomial. Determine the minimum sum of the absolute values of the roots among all irratio... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 36196,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a real number $x$ let $\\lfloor x\\rfloor$ be the greatest integer less than or equal to $x$, and define $\\{x\\} = x - \\lfloor x \\rfloor$ to be the fractional part of $x$. For example, $\\{3\\} = 0$ and $\\{4.56\\} = 0.56$. Define $f(x)=x\\{x\\}$, and let $N$ be the numb... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 36197,
"solution": "To solve $f(f(f(x)))=17$, we need to solve $f(x) = y$ where $f(f(y))=17$, and to solve that we need to solve $f(y) = z$ where $f(z) = 17$.\n It is clear to see for some integer $a \\geq 17$ there is exactly one value of $z$ in the interval $[a, a+1)$ where $f(z) = ... |
deepscaler | [
{
"role": "user",
"content": "Given that the base area of a cone is $\\pi$, and the lateral area is twice the base area, determine the surface area of the circumscribed sphere of the cone. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16\\pi}{3}"
} | {
"split": "train",
"index": 36198,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The altitude drawn to the base of an isosceles triangle is $8$, and the perimeter $32$. The area of the triangle is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 36199,
"solution": "1. **Identify the properties of the triangle**: The triangle is isosceles, and the altitude splits it into two congruent right triangles. Let's denote the base of the isosceles triangle as $2b$ and the length of the equal sides as $s$. The altitude is given as $8$.... |
deepscaler | [
{
"role": "user",
"content": "In a circle with radius 7, two intersecting chords $PQ$ and $RS$ are given such that $PQ=10$ and $RS$ bisects $PQ$ at point $T$. It is also given that $PQ$ is the only chord starting at $P$ which is bisected by $RS$. Determine the sine of the central angle subtended by the mino... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 36200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, one at the rate of $3$ feet per second, the other at $2$ feet per second. They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they p... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 36201,
"solution": "1. **Calculate the time taken by each swimmer to swim the length of the pool:**\n - The first swimmer swims at $3$ feet per second, so the time to swim $90$ feet is:\n \\[\n \\frac{90 \\text{ feet}}{3 \\text{ feet/second}} = 30 \\text{ seconds}\n \\]\... |
deepscaler | [
{
"role": "user",
"content": "What is the 10th term of an arithmetic sequence of 20 terms with the first term being 7 and the last term being 67? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{673}{19}"
} | {
"split": "train",
"index": 36202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A subset $M$ of $\\{1, 2, . . . , 2006\\}$ has the property that for any three elements $x, y, z$ of $M$ with $x < y < z$ , $x+ y$ does not divide $z$ . Determine the largest possible size of $M$ . Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "1004"
} | {
"split": "train",
"index": 36203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a regular quadrilateral prism $ABCDA'A'B'C'D'$ with vertices on the same sphere, $AB = 1$ and $AA' = \\sqrt{2}$, the spherical distance between points $A$ and $C$ is ( ). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 36204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four cubes with edge lengths $2$, $3$, $4$, and $5$ are stacked with their bottom faces on the $xy$-plane, and one vertex at the origin $(0,0,0)$. The stack sequence follows the increasing order of cube sizes from the bottom. If point $X$ is at $(0,0,0)$ and point $Y$ is at the... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 36205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 36 students at the Multiples Obfuscation Program, including a singleton, a pair of identical twins, a set of identical triplets, a set of identical quadruplets, and so on, up to a set of identical octuplets. Two students look the same if and only if they are from the ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{17}"
} | {
"split": "train",
"index": 36206,
"solution": "Let $X$ and $Y$ be the students Nithya encounters during the day. The number of pairs $(X, Y)$ for which $X$ and $Y$ look the same is $1 \\cdot 1+2 \\cdot 2+\\ldots+8 \\cdot 8=204$, and these pairs include all the ones in which $X$ and $Y$ are identical. As $X$ and... |
deepscaler | [
{
"role": "user",
"content": "Read the material: Calculate $\\frac{1}{30}÷(\\frac{2}{3}-\\frac{1}{10}+\\frac{1}{6}-\\frac{2}{5})$. Analysis: It is very cumbersome to calculate the result of $\\frac{2}{3}-\\frac{1}{10}+\\frac{1}{6}-\\frac{2}{5}$ using a common denominator. The following method can be used fo... | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{29}"
} | {
"split": "train",
"index": 36207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Mr. Lee's classroom, there are six more boys than girls among a total of 36 students. What is the ratio of the number of boys to the number of girls? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{5}"
} | {
"split": "train",
"index": 36208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\) on side \\(AB\\) points \\(E\\) and \\(F\\) lie. The area of triangle \\(AEC\\) is \\(1 \\text{ cm}^2\\), the area of triangle \\(EFC\\) is \\(3 \\text{ cm}^2\\), and the area of triangle \\(FBC\\) is \\(2 \\text{ cm}^2\\). Point \\(T\\) is the centroid of... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 36209,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the minimum number of digits to the right of the decimal point needed to express the fraction $\\frac{987654321}{2^{30} \\cdot 5^3}$ as a decimal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 36210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $P$ lies in the same plane as a given square of side $1$. Let the vertices of the square, taken counterclockwise, be $A, B, C$ and $D$. Also, let the distances from $P$ to $A, B$ and $C$, respectively, be $u, v$ and $w$. What is the greatest distance that $P$ can be fro... | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{2}"
} | {
"split": "train",
"index": 36211,
"solution": "1. **Position the Square and Define Coordinates:**\n Place the square in the $xy$-plane such that $A$ is at the origin $(0,0)$, $B$ at $(1,0)$, $C$ at $(1,1)$, and $D$ at $(0,1)$.\n\n2. **Translate the Given Condition:**\n We are given that $u^2 + v^2 = w^2$, w... |
deepscaler | [
{
"role": "user",
"content": "Given a moving circle P that is internally tangent to the circle M: (x+1)²+y²=8 at the fixed point N(1,0).\n(1) Find the trajectory equation of the moving circle P's center.\n(2) Suppose the trajectory of the moving circle P's center is curve C. A and B are two points on curve ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 36212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A set containing three real numbers can be represented as $\\{a, \\frac{b}{a}, 1\\}$, and also as $\\{a^2, a+b, 0\\}$. Find the value of $a+b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 36213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which of the following divisions is not equal to a whole number: $\\frac{60}{12}$, $\\frac{60}{8}$, $\\frac{60}{5}$, $\\frac{60}{4}$, $\\frac{60}{3}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"split": "train",
"index": 36214,
"solution": "Since $\\frac{60}{8}=60 \\div 8=7.5$, then this choice is not equal to a whole number. Note as well that $\\frac{60}{12}=5, \\frac{60}{5}=12, \\frac{60}{4}=15$, and $\\frac{60}{3}=20$ are all whole numbers."
} |
deepscaler | [
{
"role": "user",
"content": "Let $A_{1} A_{2} A_{3}$ be a triangle. Construct the following points:\n\n- $B_{1}, B_{2}$, and $B_{3}$ are the midpoints of $A_{1} A_{2}, A_{2} A_{3}$, and $A_{3} A_{1}$, respectively.\n- $C_{1}, C_{2}$, and $C_{3}$ are the midpoints of $A_{1} B_{1}, A_{2} B_{2}$, and $A_{3} B... | math | {
"style": "rule",
"ground_truth": "25/49"
} | {
"split": "train",
"index": 36215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $x \\gt -1$, $y \\gt 0$, and $x+2y=1$, find the minimum value of $\\frac{1}{x+1}+\\frac{1}{y}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3+2\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 36216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of increasing sequences of positive integers $a_1 \\le a_2 \\le a_3 \\le \\cdots \\le a_8 \\le 1023$ such that $a_i-i$ is even for $1\\le i \\le 8$. The answer can be expressed as $\\binom{m}{n}$ for some $m > n$. Compute the remainder when $m$ is divided b... | math | {
"style": "rule",
"ground_truth": "515"
} | {
"split": "train",
"index": 36217,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each integer $x$ with $1 \\leq x \\leq 10$, a point is randomly placed at either $(x, 1)$ or $(x,-1)$ with equal probability. What is the expected area of the convex hull of these points? Note: the convex hull of a finite set is the smallest convex polygon containing it. Le... | math | {
"style": "rule",
"ground_truth": "\\frac{1793}{128}"
} | {
"split": "train",
"index": 36218,
"solution": "Let $n=10$. Given a random variable $X$, let $\\mathbb{E}(X)$ denote its expected value. If all points are collinear, then the convex hull has area zero. This happens with probability $\\frac{2}{2^{n}}$ (either all points are at $y=1$ or all points are at $y=-1$ ).... |
deepscaler | [
{
"role": "user",
"content": "We define a number as an ultimate mountain number if it is a 4-digit number and the third digit is larger than the second and fourth digit but not necessarily the first digit. For example, 3516 is an ultimate mountain number. How many 4-digit ultimate mountain numbers are there... | math | {
"style": "rule",
"ground_truth": "204"
} | {
"split": "train",
"index": 36219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( S = \\{1, 2, 3, 4, \\ldots, 50\\} \\). A 3-element subset \\(\\{a, b, c\\}\\) of \\(S\\) is said to be good if \\(a + b + c\\) is divisible by 3. Determine the number of 3-element subsets of \\(S\\) which are good. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "6544"
} | {
"split": "train",
"index": 36220,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right triangle ABC with sides 9, 12, and 15, a small circle with center Q and radius 2 rolls around the inside of the triangle, always remaining tangent to at least one side of the triangle. When Q first returns to its original position, through what distance has Q travele... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 36221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest three-digit palindrome whose product with 101 is not a five-digit palindrome. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "505"
} | {
"split": "train",
"index": 36222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when $7^{2010}$ is divided by $100$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 36223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of nonempty subsets $S \\subseteq\\{-10,-9,-8, \\ldots, 8,9,10\\}$ that satisfy $|S|+\\min (S)$. $\\max (S)=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "335"
} | {
"split": "train",
"index": 36224,
"solution": "Since $\\min (S) \\cdot \\max (S)<0$, we must have $\\min (S)=-a$ and $\\max (S)=b$ for some positive integers $a$ and $b$. Given $a$ and $b$, there are $|S|-2=a b-2$ elements left to choose, which must come from the set $\\{-a+1,-a+2, \\ldots, b-2, b-1\\}$, which ... |
deepscaler | [
{
"role": "user",
"content": "Find a costant $C$ , such that $$ \\frac{S}{ab+bc+ca}\\le C $$ where $a,b,c$ are the side lengths of an arbitrary triangle, and $S$ is the area of the triangle. \n(The maximal number of points is given for the best possible constant, with proof.) Let's think step b... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4\\sqrt{3}}"
} | {
"split": "train",
"index": 36225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The distance between every two utility poles along the road is 50 meters. Xiao Wang travels at a constant speed in a car, and sees 41 utility poles in 2 minutes after seeing the first pole. How many meters does the car travel per hour? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "60000"
} | {
"split": "train",
"index": 36226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\cos \\left( \\frac {π}{4}-α \\right) = \\frac {3}{5}$, and $\\sin \\left( \\frac {5π}{4}+β \\right) = - \\frac {12}{13}$, with $α \\in \\left( \\frac {π}{4}, \\frac {3π}{4} \\right)$ and $β \\in (0, \\frac {π}{4})$, find the value of $\\sin (α+β)$. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\frac {56}{65}"
} | {
"split": "train",
"index": 36227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A larger square contains two non-overlapping shapes: a circle with diameter $2$ and a rectangle with side lengths $2$ and $4$. Find the smallest possible side length of the larger square such that these shapes can fit without overlapping, and then, find the area of the square $... | math | {
"style": "rule",
"ground_truth": "\\sqrt{8 - \\pi}"
} | {
"split": "train",
"index": 36228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square with a side length of 2 rotates around one of its sides, which is the axis of rotation. What is the volume of the cylinder obtained from this rotation? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8\\pi"
} | {
"split": "train",
"index": 36229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each pack of hot dogs contains 10 hot dogs (no buns), but each pack of hot dog buns contains 8 buns. Phil buys a number of these packs for a barbecue. After the barbecue, Phil finds that he has 4 hot dogs left over. What is the SECOND smallest number of packs of hot dogs he ... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 36230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What common fraction is exactly half-way between $\\frac{2}{3}$ and $\\frac{4}{5}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{15}"
} | {
"split": "train",
"index": 36231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bus at a certain station departs punctually at 7:00 and 7:30 in the morning. Student Xiao Ming arrives at the station to catch the bus between 6:50 and 7:30, and his arrival time is random. The probability that he waits for less than 10 minutes for the bus is ______. Let's th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 36232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be a set of positive integers satisfying the following two conditions: - For each positive integer $n$, at least one of $n, 2 n, \\ldots, 100 n$ is in $S$. - If $a_{1}, a_{2}, b_{1}, b_{2}$ are positive integers such that $\\operatorname{gcd}\\left(a_{1} a_{2}, b_{1} b_... | math | {
"style": "rule",
"ground_truth": "396"
} | {
"split": "train",
"index": 36233,
"solution": "The optimal value of $r$ is $\\frac{1}{252}$. This is attained by letting $S$ be the set of integers $n$ for which $\\nu_{2}(n) \\equiv 4 \\bmod 5$ and $\\nu_{3}(n) \\equiv 1 \\bmod 2$. Let $S$ be a set of positive integers satisfying the two conditions. For each p... |
deepscaler | [
{
"role": "user",
"content": "A triangular corner with side lengths $DB=EB=1$ is cut from equilateral triangle ABC of side length $3$. The perimeter of the remaining quadrilateral is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 36234,
"solution": "1. **Identify the Similar Triangles**: \n - $\\triangle DBE$ is cut from $\\triangle ABC$. Since $\\triangle ABC$ is equilateral, all its angles are $60^\\circ$. \n - $\\triangle DBE$ is also equilateral because $DB = EB$ and each angle in $\\triangle DBE$ is $... |
deepscaler | [
{
"role": "user",
"content": "In $\\vartriangle ABC$ points $D, E$ , and $F$ lie on side $\\overline{BC}$ such that $\\overline{AD}$ is an angle bisector of $\\angle BAC$ , $\\overline{AE}$ is a median, and $\\overline{AF}$ is an altitude. Given that $AB = 154$ and $AC = 128$ , and $9 \\ti... | math | {
"style": "rule",
"ground_truth": "94"
} | {
"split": "train",
"index": 36235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On February 1, it was $16.2^{\\circ} \\mathrm{C}$ outside Jacinta's house at 3:00 p.m. On February 2, it was $-3.6^{\\circ} \\mathrm{C}$ outside Jacinta's house at 2:00 a.m. If the temperature changed at a constant rate between these times, what was the rate at which the temper... | math | {
"style": "rule",
"ground_truth": "1.8^{\\circ} \\mathrm{C} / \\mathrm{h}"
} | {
"split": "train",
"index": 36236,
"solution": "The total decrease in temperature between these times is $16.2^{\\circ} \\mathrm{C}-\\left(-3.6^{\\circ} \\mathrm{C}\\right)=19.8^{\\circ} \\mathrm{C}$. The length of time between 3:00 p.m. one day and 2:00 a.m. the next day is 11 hours, since it is 1 hour shorter ... |
deepscaler | [
{
"role": "user",
"content": "In a game of Chomp, two players alternately take bites from a 5-by-7 grid of unit squares. To take a bite, a player chooses one of the remaining squares, then removes (\"eats\") all squares in the quadrant defined by the left edge (extended upward) and the lower edge (extended ... | math | {
"style": "rule",
"ground_truth": "792"
} | {
"split": "train",
"index": 36237,
"solution": "By drawing possible examples of the subset, one can easily see that making one subset is the same as dividing the game board into two parts.\n One can also see that it is the same as finding the shortest route from the upper left hand corner to the lower right hand... |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB=AC$ and $D$ is a point on $\\overline{AC}$ so that $\\overline{BD}$ bisects angle $ABC$. If $BD=BC$, what is the measure, in degrees, of angle $A$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 36238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f\\left(\\alpha \\right)=\\frac{\\mathrm{sin}\\left(\\pi -\\alpha \\right)\\mathrm{cos}\\left(2\\pi -\\alpha \\right)\\mathrm{cos}\\left(-\\alpha +\\frac{3\\pi }{2}\\right)}{\\mathrm{cos}\\left(\\frac{\\pi }{2}-\\alpha \\right)\\mathrm{sin}\\left(-\\pi -\\alpha \\right)}... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{6}}{5}"
} | {
"split": "train",
"index": 36239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The base of an oblique prism is a parallelogram with sides 3 and 6 and an acute angle of $45^{\\circ}$. The lateral edge of the prism is 4 and is inclined at an angle of $30^{\\circ}$ to the base plane. Find the volume of the prism. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "18\\sqrt{6}"
} | {
"split": "train",
"index": 36240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\sin \\left(\\frac{\\pi }{3}+\\alpha \\right)=\\frac{1}{4}$, then calculate $\\cos \\left(\\frac{\\pi }{3}-2\\alpha \\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{8}"
} | {
"split": "train",
"index": 36241,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $(9 \\times 10^8) \\div (3 \\times 10^3)$. (Your answer should be a single number, with no products or quotients.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "300,\\!000"
} | {
"split": "train",
"index": 36242,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{p}$ be the projection of $\\mathbf{v}$ onto $\\mathbf{w},$ and let $\\mathbf{q}$ be the projection of $\\mathbf{p}$ onto $\\mathbf{v}.$ If $\\frac{\\|\\mathbf{p}\\|}{\\|\\mathbf{v}\\|} = \\frac{5}{7},$ then find $\\frac{\\|\\mathbf{q}\\|}{\\|\\mathbf{v}\\|}.$ Let... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{49}"
} | {
"split": "train",
"index": 36243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin\\left(\\theta - \\frac{\\pi}{6}\\right) = \\frac{1}{4}$ with $\\theta \\in \\left( \\frac{\\pi}{6}, \\frac{2\\pi}{3}\\right)$, calculate the value of $\\cos\\left(\\frac{3\\pi}{2} + \\theta\\right)$. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15} + \\sqrt{3}}{8}"
} | {
"split": "train",
"index": 36244,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $3.5 \\times 0.3 + 1.2 \\times 0.4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.53"
} | {
"split": "train",
"index": 36245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the region defined by the equation $x^2 + y^2 - 3 = 6y - 18x + 9$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "102\\pi"
} | {
"split": "train",
"index": 36246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $e > 0$ be a given real number. Find the least value of $f(e)$ (in terms of $e$ only) such that the inequality $a^{3}+ b^{3}+ c^{3}+ d^{3} \\leq e^{2}(a^{2}+b^{2}+c^{2}+d^{2}) + f(e)(a^{4}+b^{4}+c^{4}+d^{4})$ holds for all real numbers $a, b, c, d$ . Let's think st... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4e^2}"
} | {
"split": "train",
"index": 36247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\\dots a_n$ of the\nsequence $1$, $2$, $\\dots$ , $n$ satisfying\n$$a_1 \\le 2a_2\\le 3a_3 \\le \\dots \\le na_n$$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "F_{n+1}"
} | {
"split": "train",
"index": 36248,
"solution": "\nConsider the problem of counting the number of permutations of the sequence \\(1, 2, \\ldots, n\\) that satisfy the inequality:\n\n\\[\na_1 \\le 2a_2 \\le 3a_3 \\le \\cdots \\le na_n.\n\\]\n\nTo solve this, we relate the problem to a known sequence, specifically,... |
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