data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Six male middle school students from a school participated in a pull-up physical fitness test and scored respectively: $8$, $5$, $2$, $5$, $6$, $4$. Calculate the variance of this dataset. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"split": "train",
"index": 33549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( F \\) be the left focus of the ellipse \\( E: \\frac{x^{2}}{3}+y^{2}=1 \\). A line \\( l \\) with a positive slope passes through point \\( F \\) and intersects \\( E \\) at points \\( A \\) and \\( B \\). Through points \\( A \\) and \\( B \\), lines \\( AM \\) and \\(... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"split": "train",
"index": 33550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The $y$-intercepts, $P$ and $Q$, of two perpendicular lines intersecting at the point $A(6,8)$ have a sum of zero. What is the area of $\\triangle APQ$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 33551,
"solution": "1. **Identify the properties of the lines**: Given that the lines are perpendicular and intersect at $A(6,8)$, we can denote the equations of the lines as $y = m_1x + b_1$ and $y = m_2x + b_2$. Since the lines are perpendicular, their slopes satisfy $m_1 \\cdot m_2... |
deepscaler | [
{
"role": "user",
"content": "Let $f$ be defined by \\[f(x) = \\left\\{\n\\begin{array}{cl}\n2-x & \\text{ if } x \\leq 1, \\\\\n2x-x^2 & \\text{ if } x>1.\n\\end{array}\n\\right.\\]Calculate $f^{-1}(-3)+f^{-1}(0)+f^{-1}(3)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 33552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the sum $25^2 - 23^2 + 21^2 - 19^2 + ... + 3^2 - 1^2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1196"
} | {
"split": "train",
"index": 33553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $R = (8,6)$. The lines whose equations are $8y = 15x$ and $10y = 3x$ contain points $P$ and $Q$, respectively, such that $R$ is the midpoint of $\\overline{PQ}$. The length of $PQ$ equals $\\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 33554,
"solution": "[asy] pointpen = black; pathpen = black+linewidth(0.7); pair R = (8,6), P = (32,60)/7, Q= (80,24)/7; D((0,0)--MP(\"x\",(13,0),E),EndArrow(6)); D((0,0)--MP(\"y\",(0,10),N),EndArrow(6)); D((0,0)--(10/(15/8),10),EndArrow(6)); D((0,0)--(13,13 * 3/10),EndArrow(6)); D(D(... |
deepscaler | [
{
"role": "user",
"content": "Four friends — Alex, Betty, Clara, and Dave — participated in a relay race by running in pairs, with one pair sitting out each race. Dave ran in 8 races, which was more than any other friend, and Betty ran in 3 races, which was fewer than any other friend. Determine the total n... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 33555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square of side length $1$ and a circle of radius $\\frac{\\sqrt{3}}{3}$ share the same center. What is the area inside the circle, but outside the square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{9} - \\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 33556,
"solution": "1. **Identify the geometric setup**: We have a square with side length $1$ and a circle with radius $\\frac{\\sqrt{3}}{3}$ sharing the same center. We need to find the area inside the circle but outside the square.\n\n2. **Calculate the diagonal of the square**: Th... |
deepscaler | [
{
"role": "user",
"content": "A wire of length $80$cm is randomly cut into three segments. The probability that each segment is no less than $20$cm is $\\_\\_\\_\\_\\_\\_\\_.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"split": "train",
"index": 33557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the roots of the cubic equation \\(27x^3 - 81x^2 + 63x - 14 = 0\\) be in geometric progression. Find the difference between the square of the largest root and the square of the smallest root. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 33558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the 2009 Stanford Olympics, Willy and Sammy are two bikers. The circular race track has two\nlanes, the inner lane with radius 11, and the outer with radius 12. Willy will start on the inner lane,\nand Sammy on the outer. They will race for one complete lap, measured by the ... | math | {
"style": "rule",
"ground_truth": "265 - 132\\sqrt{3}"
} | {
"split": "train",
"index": 33559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a bag there are $1007$ black and $1007$ white balls, which are randomly numbered $1$ to $2014$ . In every step we draw one ball and put it on the table; also if we want to, we may choose two different colored balls from the table and put them in a different bag. If we... | math | {
"style": "rule",
"ground_truth": "1014049"
} | {
"split": "train",
"index": 33560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sarah stands at $(0,0)$ and Rachel stands at $(6,8)$ in the Euclidean plane. Sarah can only move 1 unit in the positive $x$ or $y$ direction, and Rachel can only move 1 unit in the negative $x$ or $y$ direction. Each second, Sarah and Rachel see each other, independently pick a... | math | {
"style": "rule",
"ground_truth": "\\frac{63}{64}"
} | {
"split": "train",
"index": 33561,
"solution": "We make the following claim: In a game with $n \\times m$ grid where $n \\leq m$ and $n \\equiv m(\\bmod 2)$, the probability that Sarah wins is $\\frac{1}{2^{n}}$ under optimal play. Proof: We induct on $n$. First consider the base case $n=0$. In this case Rachel ... |
deepscaler | [
{
"role": "user",
"content": "Find the number of ordered pairs of positive integers $(m,n)$ such that ${m^2n = 20 ^{20}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "231"
} | {
"split": "train",
"index": 33562,
"solution": "In this problem, we want to find the number of ordered pairs $(m, n)$ such that $m^2n = 20^{20}$. Let $x = m^2$. Therefore, we want two numbers, $x$ and $n$, such that their product is $20^{20}$ and $x$ is a perfect square. Note that there is exactly one valid $n$ ... |
deepscaler | [
{
"role": "user",
"content": "Tetrahedron $ABCD$ has $AD=BC=28$, $AC=BD=44$, and $AB=CD=52$. For any point $X$ in space, suppose $f(X)=AX+BX+CX+DX$. The least possible value of $f(X)$ can be expressed as $m\\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any pri... | math | {
"style": "rule",
"ground_truth": "682"
} | {
"split": "train",
"index": 33563,
"solution": "Set $a=BC=28$, $b=CA=44$, $c=AB=52$. Let $O$ be the point which minimizes $f(X)$.\n$\\boxed{\\textrm{Claim 1: } O \\textrm{ is the gravity center } \\ \\tfrac {1}{4}(\\vec A + \\vec B + \\vec C + \\vec D)}$\n$\\textrm{Proof:}$ Let $M$ and $N$ denote the midpoints o... |
deepscaler | [
{
"role": "user",
"content": "Compute the number of ways to color the vertices of a regular heptagon red, green, or blue (with rotations and reflections distinct) such that no isosceles triangle whose vertices are vertices of the heptagon has all three vertices the same color. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "294"
} | {
"split": "train",
"index": 33564,
"solution": "Number the vertices 1 through 7 in order. Then, the only way to have three vertices of a regular heptagon that do not form an isosceles triangle is if they are vertices $1,2,4$, rotated or reflected. Thus, it is impossible for have four vertices in the heptagon of ... |
deepscaler | [
{
"role": "user",
"content": "The Ivanov family consists of three people: a father, a mother, and a daughter. Today, on the daughter's birthday, the mother calculated the sum of the ages of all family members and got 74 years. It is known that 10 years ago, the total age of the Ivanov family members was 47 ... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 33565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The operation $\\Diamond$ is defined by $a\\Diamond b=ab^2-b+1$. What is the value of $(-1)\\Diamond 6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-41"
} | {
"split": "train",
"index": 33566,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two identical test tubes were filled with 200 ml of a liquid substance each. From the first test tube, $1 / 4$ of the content was poured out and replaced with the same amount of water. This procedure was repeated 3 more times, each time pouring out a quarter of the content and ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 33567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=x^{3}+x+1$. Suppose $g$ is a cubic polynomial such that $g(0)=-1$, and the roots of $g$ are the squares of the roots of $f$. Find $g(9)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "899"
} | {
"split": "train",
"index": 33568,
"solution": "Let $a, b, c$ be the zeros of $f$. Then $f(x)=(x-a)(x-b)(x-c)$. Then, the roots of $g$ are $a^{2}, b^{2}, c^{2}$, so $g(x)=k(x-a^{2})(x-b^{2})(x-c^{2})$ for some constant $k$. Since $a b c=-f(0)=-1$, we have $k=k a^{2} b^{2} c^{2}=-g(0)=1$. Thus, $g(x^{2})=(x^{2}-a... |
deepscaler | [
{
"role": "user",
"content": "What is $3\\cdot 9+4\\cdot 10+11\\cdot 3+3\\cdot 8$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "124"
} | {
"split": "train",
"index": 33569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the maximum value of the sum\n\\[S = \\sum_{n=1}^\\infty \\frac{n}{2^n} (a_1 a_2 \\cdots a_n)^{1/n}\\]\nover all sequences $a_1, a_2, a_3, \\cdots$ of nonnegative real numbers satisfying\n\\[\\sum_{k=1}^\\infty a_k = 1.\\] Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"split": "train",
"index": 33570,
"solution": "The answer is $2/3$. \n\nBy AM-GM, we have\n\\begin{align*}\n2^{n+1}(a_1\\cdots a_n)^{1/n} &= \\left((4a_1)(4^2a_2)\\cdots (4^na_n)\\right)^{1/n}\\\\\n& \\leq \\frac{\\sum_{k=1}^n (4^k a_k)}{n}.\n\\end{align*}\nThus\n\\begin{align*}\n2S &\\leq \\sum_{n=1}^\\infty \... |
deepscaler | [
{
"role": "user",
"content": "A triangle has one side of length $13$, and the angle opposite this side is $60^{\\circ}$. The ratio of the other two sides is $4:3$. Calculate the area of this triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39 \\sqrt{3}"
} | {
"split": "train",
"index": 33571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=104$. What is $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 33572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The center of a semicircle, inscribed in a right triangle such that its diameter lies on the hypotenuse, divides the hypotenuse into segments of 30 and 40. Find the length of the arc of the semicircle that is enclosed between the points where it touches the legs. Let's think st... | math | {
"style": "rule",
"ground_truth": "12\\pi"
} | {
"split": "train",
"index": 33573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(PQR\\), the point \\(S\\) is on \\(PQ\\) so that the ratio of the length of \\(PS\\) to the length of \\(SQ\\) is \\(2: 3\\). The point \\(T\\) lies on \\(SR\\) so that the area of triangle \\(PTR\\) is 20 and the area of triangle \\(SQT\\) is 18. What is the are... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 33574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Select three digits from 1, 3, 5, 7, 9, and two digits from 0, 2, 4, 6, 8 to form a five-digit number without any repeating digits. How many such numbers can be formed? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11040"
} | {
"split": "train",
"index": 33575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petya's bank account contains $500. The bank allows only two types of transactions: withdrawing $300 or adding $198. What is the maximum amount Petya can withdraw from the account, if he has no other money? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 33576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$(1+x^2)(1-x^3)$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1+x^2-x^3-x^5"
} | {
"split": "train",
"index": 33577,
"solution": "To solve the problem, we need to expand the expression $(1+x^2)(1-x^3)$ using the distributive property (also known as the FOIL method for binomials).\n\n1. **Distribute each term in the first binomial to each term in the second binomial:**\n \\[\n (1+x^2)(1-x^... |
deepscaler | [
{
"role": "user",
"content": "Kolya, an excellent student in the 7th-8th grade, found the sum of the digits of all the numbers from 0 to 2012 and added them all together. What number did he get? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28077"
} | {
"split": "train",
"index": 33578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct arrangements of the letters in the word \"balloon\" are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1260"
} | {
"split": "train",
"index": 33579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let's call a number \"remarkable\" if it has exactly 4 different natural divisors, among which there are two such that neither is a multiple of the other. How many \"remarkable\" two-digit numbers exist? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 33580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The positive integers $A, B, C$, and $D$ form an arithmetic and geometric sequence as follows: $A, B, C$ form an arithmetic sequence, while $B, C, D$ form a geometric sequence. If $\\frac{C}{B} = \\frac{7}{3}$, what is the smallest possible value of $A + B + C + D$? Let's think... | math | {
"style": "rule",
"ground_truth": "76"
} | {
"split": "train",
"index": 33581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve\n\\[\\arctan \\frac{1}{x} + \\arctan \\frac{1}{x^3} = \\frac{\\pi}{4}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1 + \\sqrt{5}}{2}"
} | {
"split": "train",
"index": 33582,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many ... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 33583,
"solution": "1. **Define Variables:**\n Let $r$ be the number of marbles Ringo has and $p$ be the number of marbles Paul has.\n\n2. **Set Up Congruences:**\n From the problem, we know:\n \\[ r \\equiv 4 \\pmod{6} \\]\n \\[ p \\equiv 3 \\pmod{6} \\]\n These congruences... |
deepscaler | [
{
"role": "user",
"content": "What is the result if you add the largest odd two-digit number to the smallest even three-digit number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "199"
} | {
"split": "train",
"index": 33584,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For which values of \\( a \\) do the quadratic trinomials \\( x^{2} + a x + 1 \\) and \\( x^{2} + x + a \\) have a common root? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 33585,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the numbers 3, 0, 4, 8, and a decimal point to form decimals, the largest three-digit decimal is \\_\\_\\_\\_\\_\\_, the smallest decimal is \\_\\_\\_\\_\\_\\_, and their difference is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "8.082"
} | {
"split": "train",
"index": 33586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the degree measure of the supplement of the complement of a 42-degree angle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "132"
} | {
"split": "train",
"index": 33587,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Solve the inequality $|x+1| + 2|x-1| < 3x + 5$.\n(2) Given $a, b \\in [0, 1]$, find the maximum value of $ab + (1-a-b)(a+b)$. 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": 33588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three distinct integers are chosen uniformly at random from the set $$ \\{2021, 2022, 2023, 2024, 2025, 2026, 2027, 2028, 2029, 2030\\}. $$ Compute the probability that their arithmetic mean is an integer. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "7/20"
} | {
"split": "train",
"index": 33589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the chord cut by the line $ax-by+2=0$ on the circle $x^{2}+y^{2}+2x-4y+1=0$ is $4$, find the minimum value of $\\dfrac{2}{a}+\\dfrac{3}{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4+2\\sqrt{3}"
} | {
"split": "train",
"index": 33590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ be an acute isosceles triangle with circumcircle $\\omega$. The tangents to $\\omega$ at vertices $B$ and $C$ intersect at point $T$. Let $Z$ be the projection of $T$ onto $BC$. Assume $BT = CT = 20$, $BC = 24$, and $TZ^2 + 2BZ \\cdot CZ = 478$. Find $BZ \\... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 33591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $m+\\frac{1}{m}=8$, then what is the value of $m^{2}+\\frac{1}{m^{2}}+4$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 33592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the common rational root \\( k \\) of the following polynomial equations which is not integral:\n\\[45x^4 + ax^3 + bx^2 + cx + 8 = 0\\]\n\\[8x^5 + dx^4 + ex^3 + fx^2 + gx + 45 = 0\\]\nThis root \\( k \\) is assumed to be a negative non-integer. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"split": "train",
"index": 33593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A farmer had an enclosure with a fence 50 rods long, which could only hold 100 sheep. Suppose the farmer wanted to expand the enclosure so that it could hold twice as many sheep.\n\nHow many additional rods will the farmer need? Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 33594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain intelligence station has four different kinds of passwords $A$, $B$, $C$, and $D$. Each week, one of these passwords is used, and each week a password is chosen uniformly at random from the three passwords that were not used the previous week. Given that password $A$ ... | math | {
"style": "rule",
"ground_truth": "61/243"
} | {
"split": "train",
"index": 33595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the coin denominations 1 cent, 5 cents, 10 cents, and 50 cents, determine the smallest number of coins Lisa would need so she could pay any amount of money less than a dollar. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 33596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If I choose four cards from a standard $52$-card deck, with replacement, what is the probability that I will end up with one card from each suit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{32}"
} | {
"split": "train",
"index": 33597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is half of the absolute value of the difference of the squares of 18 and 16? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 33598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten chairs are evenly spaced around a round table and numbered clockwise from $1$ through $10$. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seating arrangements are pos... | math | {
"style": "rule",
"ground_truth": "480"
} | {
"split": "train",
"index": 33599,
"solution": "1. **Fixing the first man's position**: We can fix the first man in any of the 10 seats. This is a common strategy in circular arrangements to avoid equivalent rotations being counted multiple times. Thus, we have 10 choices for the first man.\n\n2. **Seating the o... |
deepscaler | [
{
"role": "user",
"content": "At 9:00, a pedestrian set off on a journey. An hour later, a cyclist set off from the same starting point. At 10:30, the cyclist caught up with the pedestrian and continued ahead, but after some time, the bicycle broke down. After repairing the bike, the cyclist resumed the jou... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 33600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A garden can be watered by any of three sprinklers X, Y, or Z. Sprinklers X and Y together take 5 hours to water the garden. Sprinklers X and Z together take 6 hours to water the garden. Sprinklers Y and Z together take 7 hours to water the garden. How many hours does it take s... | math | {
"style": "rule",
"ground_truth": "3.93"
} | {
"split": "train",
"index": 33601,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $ \\frac {\\pi}{2} < \\alpha < \\pi$ and $0 < \\beta < \\frac {\\pi}{2}$, with $\\tan \\alpha= -\\frac {3}{4}$ and $\\cos (\\beta-\\alpha)= \\frac {5}{13}$, find the value of $\\sin \\beta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {63}{65}"
} | {
"split": "train",
"index": 33602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the statements:\n$\\textbf{(1)}\\ p\\wedge \\sim q\\wedge r \\qquad\\textbf{(2)}\\ \\sim p\\wedge \\sim q\\wedge r \\qquad\\textbf{(3)}\\ p\\wedge \\sim q\\wedge \\sim r \\qquad\\textbf{(4)}\\ \\sim p\\wedge q\\wedge r$\nwhere $p,q$, and $r$ are propositions. How many ... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 33603,
"solution": "To solve this problem, we need to evaluate each statement to see if it implies the truth of $(p \\rightarrow q) \\rightarrow r$. We start by understanding the implication $p \\rightarrow q$, which is logically equivalent to $\\sim p \\vee q$. Then, we evaluate $(p ... |
deepscaler | [
{
"role": "user",
"content": "Compute the number of complex numbers $z$ with $|z|=1$ that satisfy $$1+z^{5}+z^{10}+z^{15}+z^{18}+z^{21}+z^{24}+z^{27}=0$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 33604,
"solution": "Let the polynomial be $f(z)$. One can observe that $$f(z)=\\frac{1-z^{15}}{1-z^{5}}+z^{15} \\frac{1-z^{15}}{1-z^{3}}=\\frac{1-z^{20}}{1-z^{5}}+z^{18} \\frac{1-z^{12}}{1-z^{3}}$$ so all primitive 15th roots of unity are roots, along with -1 and $\\pm i$. To show tha... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given vectors $\\overrightarrow{m}=(\\cos A,\\cos B)$ and $\\overrightarrow{n}=(a,2c-b)$, and $\\overrightarrow{m} \\parallel \\overrightarrow{n}$. \n(Ⅰ) Find the... | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}"
} | {
"split": "train",
"index": 33605,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is equilateral with side length $12$ . Point $D$ is the midpoint of side $\\overline{BC}$ . Circles $A$ and $D$ intersect at the midpoints of side $AB$ and $AC$ . Point $E$ lies on segment $\\overline{AD}$ and circle $E$ is tangent to circles ... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3} - 6"
} | {
"split": "train",
"index": 33606,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the line $l: x+ \\sqrt {2}y=4 \\sqrt {2}$ and the ellipse $C: mx^{2}+ny^{2}=1$ ($n>m>0$) have exactly one common point $M[2 \\sqrt {2},2]$. \n(1) Find the equation of the ellipse $C$; \n(2) Let the left and right vertices of the ellipse $C$ be $A$ and $B$, respectively,... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 33607,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the numbers from the set $\\{1,\\ 2,\\ 3,\\ldots,\\ 100\\}$ have a perfect square factor other than one? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 33608,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $(b_1, b_2, ..., b_{12})$ is a list of the first 12 positive integers, where for each $2 \\leq i \\leq 12$, either $b_i + 1$, $b_i - 1$, or both appear somewhere in the list before $b_i$, and all even integers precede any of their immediate consecutive odd integers, find ... | math | {
"style": "rule",
"ground_truth": "2048"
} | {
"split": "train",
"index": 33609,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $\\left\\{a_{n}\\right\\}$ with real terms, and the sum of the first $n$ terms is $S_{n}$. If $S_{10} = 10$ and $S_{30} = 70$, then $S_{40}$ is equal to: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 33610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be real numbers. One of the roots of\n\\[x^3 + ax^2 - x + b = 0\\]is $1 - 2i.$ Enter the ordered pair $(a,b).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(1,15)"
} | {
"split": "train",
"index": 33611,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be positive real numbers such that $a + 3b = 2.$ Find the minimum value of\n\\[\\frac{2}{a} + \\frac{4}{b}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 33612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides corresponding to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is known that $B= \\frac{\\pi}{4}$ and $\\cos A-\\cos 2A=0$.\n\n$(1)$ Find the angle $C$.\n\n$(2)$ If $b^{2}+c^{2}=a-bc+2$, find the area of $\\triangle AB... | math | {
"style": "rule",
"ground_truth": "1- \\frac{ \\sqrt{3}}{3}"
} | {
"split": "train",
"index": 33613,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the values of $a$, $b$, and $c$ such that the equation $\\sin^2 x + \\sin^2 3x + \\sin^2 4x + \\sin^2 5x = 3$ can be reduced to an equivalent form involving $\\cos ax \\cos bx \\cos cx = 0$ for some positive integers $a$, $b$, and $c$, and then find $a + b + c$. Let's thin... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 33614,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten points are spaced evenly along the perimeter of a rectangle measuring $3 \\times 2$ units. What is the probability that the two points are one unit apart? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"split": "train",
"index": 33615,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the price of a stamp is 45 cents, what is the maximum number of stamps that could be purchased with $50? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "111"
} | {
"split": "train",
"index": 33616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three three-digit numbers, with all digits except zero being used in their digits, sum up to 1665. In each number, the first digit was swapped with the last digit. What is the sum of the new numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "1665"
} | {
"split": "train",
"index": 33617,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many $n=2,3,4,\\ldots,109,110$ is the base-$n$ number $432143_n$ a multiple of $11$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 33618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $ABCD$ is inscribed in a circle. Square $EFGH$ has vertices $E$ and $F$ on $\\overline{CD}$ and vertices $G$ and $H$ on the circle. If the area of square $ABCD$ is $1$, then the area of square $EFGH$ can be expressed as $\\frac {m}{n}$ where $m$ and $n$ are relatively pr... | math | {
"style": "rule",
"ground_truth": "251"
} | {
"split": "train",
"index": 33619,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $a$ varies inversely with $b^2$. If $a=9$ when $b=2$, find the value of $a$ when $b=3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 33620,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with incircle tangent to the perpendicular bisector of $B C$. If $B C=A E=$ 20, where $E$ is the point where the $A$-excircle touches $B C$, then compute the area of $\\triangle A B C$. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "100 \\sqrt{2}"
} | {
"split": "train",
"index": 33621,
"solution": "Let the incircle and $B C$ touch at $D$, the incircle and perpendicular bisector touch at $X, Y$ be the point opposite $D$ on the incircle, and $M$ be the midpoint of $B C$. Recall that $A, Y$, and $E$ are collinear by homothety at $A$. Additionally, we have $M D=M... |
deepscaler | [
{
"role": "user",
"content": "Suppose two distinct integers are chosen from between 1 and 29, inclusive. What is the probability that their product is neither a multiple of 2 nor 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{45}{406}"
} | {
"split": "train",
"index": 33622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the perimeter of pentagon $ABCDE$ in this diagram? [asy]\npair cis(real r,real t) { return (r*cos(t),r*sin(t)); }\npair a=(0,0);\npair b=cis(1,-pi/2);\npair c=cis(sqrt(2),-pi/4);\npair d=cis(sqrt(3),-pi/4+atan(1/sqrt(2)));\npair e=cis(2,-pi/4+atan(1/sqrt(2))+atan(1/sqrt... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 33623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a geometric sequence $\\{a_n\\}$ consists of positive terms, and $(a_3, \\frac{1}{2}a_5,a_4)$ form an arithmetic sequence, find the value of $\\frac{a_3+a_5}{a_4+a_6}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}-1}{2}"
} | {
"split": "train",
"index": 33624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integers $x$ is the number $x^4 - 53x^2 + 150$ negative? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 33625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At Rachelle's school, an A counts $4$ points, a B $3$ points, a C $2$ points, and a D $1$ point. Her GPA in the four classes she is taking is computed as the total sum of points divided by 4. She is certain that she will get A's in both Mathematics and Science and at least a C ... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{24}"
} | {
"split": "train",
"index": 33626,
"solution": "1. **Calculate the probability of getting each grade in English and History:**\n - For English:\n - Probability of A: $\\frac{1}{6}$\n - Probability of B: $\\frac{1}{4}$\n - Probability of C: $1 - \\frac{1}{6} - \\frac{1}{4} = 1 - \\frac{5}{12} = \\fr... |
deepscaler | [
{
"role": "user",
"content": "What is the smallest value that the sum of the digits of the number $3n^2+n+1,$ $n\\in\\mathbb{N}$ can take? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 33627,
"solution": "\nWe need to find the smallest value of the sum of the digits of the expression \\( 3n^2 + n + 1 \\) for \\( n \\in \\mathbb{N} \\).\n\nTo determine this, we will explore values of \\( n \\) and calculate the resulting expression \\( 3n^2 + n + 1 \\), then find the... |
deepscaler | [
{
"role": "user",
"content": "Doug constructs a square window using $8$ equal-size panes of glass. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 33628,
"solution": "1. **Identify the dimensions of each pane**: Given that the ratio of the height to the width of each pane is $5:2$, let the height of each pane be $5x$ inches and the width be $2x$ inches.\n\n2. **Calculate the total dimensions of the window**: The window is constr... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, which form a geometric progression. Also, $(2a-c)\\cos B = b\\cos C$.\n\n(Ⅰ) Find the magnitude of angle $B$; \n(Ⅱ) Calculate $\\frac{1}{\\tan A} + \\frac{1}{\\tan C}$. Let'... | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{3}}{3}"
} | {
"split": "train",
"index": 33629,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $x^{3}=4$, solve for $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{4}"
} | {
"split": "train",
"index": 33630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the slope of the line $2y = -3x + 6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac32"
} | {
"split": "train",
"index": 33631,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of increasing sequences of positive integers $b_1 \\le b_2 \\le b_3 \\le \\cdots \\le b_{15} \\le 3005$ such that $b_i-i$ is odd for $1\\le i \\le 15$. Express your answer as ${p \\choose q}$ for some integers $p > q$ and find the remainder when $p$ is divide... | math | {
"style": "rule",
"ground_truth": "509"
} | {
"split": "train",
"index": 33632,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation of the ellipse shown below can be written as\n\\[\\frac{(x - h)^2}{a^2} + \\frac{(y - k)^2}{b^2} = 1.\\]Find $h + k + a + b.$\n\n[asy]\nunitsize(0.3 cm);\n\nint i, n = 10;\n\nfor (i = -n; i <= n; ++i) {\n draw((i,-n)--(i,n),gray(0.7));\n draw((-n,i)--(n,i),gray(0... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 33633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two straight lines, each of which passes through four points of the form $(1,0,a), (b,1,0), (0,c,1),$ and $(6d,6d,-d),$ where $a,b,c,$ and $d$ are real numbers, not necessarily in that order. Enter all possible values of $d,$ separated by commas. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}, \\frac{1}{8}"
} | {
"split": "train",
"index": 33634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the binomial coefficient of only the fourth term in the expansion of $(x^{2} - \\frac {1}{2x})^{n}$ is the largest, then the sum of all the coefficients in the expansion is $\\boxed{\\text{answer}}$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{64}"
} | {
"split": "train",
"index": 33635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the phase shift of the graph of $y = 3 \\sin \\left( x - \\frac{\\pi}{5} \\right).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{5}"
} | {
"split": "train",
"index": 33636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a math lesson, each gnome needs to find a three-digit number without zero digits, divisible by 3, such that when 297 is added to it, the resulting number consists of the same digits but in reverse order. What is the minimum number of gnomes that must be in the lesson so that... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 33637,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive four-digit number divisible by 8 which has three odd and one even digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1032"
} | {
"split": "train",
"index": 33638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An iterative average of the numbers 2, 3, 4, 6, and 7 is computed by arranging the numbers in some order. Find the difference between the largest and smallest possible values that can be obtained using this procedure. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{4}"
} | {
"split": "train",
"index": 33639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cuckoo clock strikes the number of times corresponding to the current hour (for example, at 19:00, it strikes 7 times). One morning, Max approached the clock when it showed 9:05. He started turning the minute hand until it moved forward by 7 hours. How many times did the cuck... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 33640,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point \\( M \\) is the midpoint of the hypotenuse \\( AC \\) of right triangle \\( ABC \\). Points \\( P \\) and \\( Q \\) on lines \\( AB \\) and \\( BC \\) respectively are such that \\( AP = PM \\) and \\( CQ = QM \\). Find the measure of angle \\( \\angle PQM \\) if \\( \\a... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 33641,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A round-robin tennis tournament is organized where each player is supposed to play every other player exactly once. However, the tournament is scheduled to have one rest day during which no matches will be played. If there are 10 players in the tournament, and the tournament wa... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 33642,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sqrt {3} \\overrightarrow{a}+ \\overrightarrow{b}+2 \\overrightarrow{c}= \\overrightarrow{0}$, and $| \\overrightarrow{a}|=| \\overrightarrow{b}|=| \\overrightarrow{c}|=1$, find the value of $\\overrightarrow{a}\\cdot ( \\overrightarrow{b}+ \\overrightarrow{c})$. ... | math | {
"style": "rule",
"ground_truth": "-\\dfrac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 33643,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that every high school in the town of Pythagoras sent a team of 3 students to a math contest, and Andrea's score was the median among all students, and hers was the highest score on her team, and Andrea's teammates Beth and Carla placed 40th and 75th, respectively, calcul... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 33644,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Equilateral $\\triangle ABC$ has side length $\\sqrt{111}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to $\\triangle ABC$, with $BD_1 = BD_2 = \\sqrt{11}$. Find $\\sum_{k=1}^4(CE_k)^2$. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "677"
} | {
"split": "train",
"index": 33645,
"solution": "This method uses complex numbers with $A$ as the origin. Let $A=0$, $B=\\sqrt{111}$, $C = \\sqrt{111}\\theta$, where $\\theta = e^{i \\pi/3} = \\frac{1}{2} + \\frac{\\sqrt{3}}{2}i$.\n Also, let $x$ be $D_1$ or $D_2$. Then\n$|x|=\\sqrt{111}, |x-\\sqrt{111}|=\\sqrt{1... |
deepscaler | [
{
"role": "user",
"content": "The equation\n\\[(x - \\sqrt[3]{13})(x - \\sqrt[3]{53})(x - \\sqrt[3]{103}) = \\frac{1}{3}\\]has three distinct solutions $r,$ $s,$ and $t.$ Calculate the value of $r^3 + s^3 + t^3.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "170"
} | {
"split": "train",
"index": 33646,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two positive numbers that may be inserted between $3$ and $9$ such that the first three are in geometric progression while the last three are in arithmetic progression. The sum of those two positive numbers is Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "11\\frac{1}{4}"
} | {
"split": "train",
"index": 33647,
"solution": "1. **Define the variables and equations:** Let $a$ and $b$ be the two numbers to be inserted between $3$ and $9$. We know that $3, a, b$ are in geometric progression and $a, b, 9$ are in arithmetic progression. Let $r$ be the common ratio of the geometric progressi... |
deepscaler | [
{
"role": "user",
"content": "Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $1500$ are not factorial tails? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 33648,
"solution": ""
} |
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