data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Larry now only likes numbers that end with two digits making the whole number divisible by 4. He has some favorite numbers like 120, 1156, and 504. How many different pairs of the last two digits are possible in numbers that Larry likes? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 21849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $(x,y)$ is randomly and uniformly chosen inside the rectangle with vertices (0,0), (0,3), (4,3), and (4,0). What is the probability that $x + 2y < 6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{2}{3}"
} | {
"split": "train",
"index": 21850,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are several pairs of integers $ (a, b) $ satisfying $ a^2 - 4a + b^2 - 8b = 30 $ . Find the sum of the sum of the coordinates of all such points. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 21851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate: $-{1^{2023}}+|{\\sqrt{3}-2}|-3\\tan60°$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 - 4\\sqrt{3}"
} | {
"split": "train",
"index": 21852,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "My school's Chess Club has 24 members. It needs to select 3 officers: president, secretary, and treasurer. Each person can hold at most one office. Two of the members, Alice and Bob, will only serve together as officers. In how many ways can the club choose its officers? Let's ... | math | {
"style": "rule",
"ground_truth": "9372"
} | {
"split": "train",
"index": 21853,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $w, x, y$, and $z$ be positive real numbers such that $0 \\neq \\cos w \\cos x \\cos y \\cos z$, $2 \\pi =w+x+y+z$, $3 \\tan w =k(1+\\sec w)$, $4 \\tan x =k(1+\\sec x)$, $5 \\tan y =k(1+\\sec y)$, $6 \\tan z =k(1+\\sec z)$. Find $k$. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "\\sqrt{19}"
} | {
"split": "train",
"index": 21854,
"solution": "From the identity $\\tan \\frac{u}{2}=\\frac{\\sin u}{1+\\cos u}$, the conditions work out to $3 \\tan \\frac{w}{2}=4 \\tan \\frac{x}{2}=5 \\tan \\frac{y}{2}=6 \\tan \\frac{z}{2}=k$. Let $a=\\tan \\frac{w}{2}, b=\\tan \\frac{x}{2}, c=\\tan \\frac{y}{2}$, and $d=\\t... |
deepscaler | [
{
"role": "user",
"content": "Three different numbers are chosen at random from the list \\(1, 3, 5, 7, 9, 11, 13, 15, 17, 19\\). The probability that one of them is the mean of the other two is \\(p\\). What is the value of \\(\\frac{120}{p}\\) ? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 21855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An isosceles triangle with a base of $\\sqrt{2}$ has medians intersecting at a right angle. What is the area of this triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 21856,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The four-digit numeral $3AA1$ is divisible by 9. What digit does $A$ represent? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 21857,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The school committee has organized a \"Chinese Dream, My Dream\" knowledge speech competition. There are 4 finalists, and each contestant can choose any one topic from the 4 backup topics to perform their speech. The number of scenarios where exactly one of the topics is not se... | math | {
"style": "rule",
"ground_truth": "324"
} | {
"split": "train",
"index": 21858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( n \\) is a positive integer, \\( P \\) is a prime number, and \\( pn \\) has exactly 8 positive divisors, arrange them in ascending order as \\( 1=d_{1}<d_{2}< \\cdots <d_{8}=pn \\). Additionally, let \\( d_{17p-d_{3}}=\\left(d_{1}+d_{2}+d_{3}\\right)\\left(d_{3}... | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"split": "train",
"index": 21859,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A group of 8 boys and 8 girls was paired up randomly. Find the probability that there is at least one pair with two girls. Round your answer to the nearest hundredth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.98"
} | {
"split": "train",
"index": 21860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among all victims of zombie bites, 10% are prescribed the experimental drug Undetenin to treat them. Overall, 4% of the human population suffer an adverse reaction to Undetenin. Out of all the patients being treated with Undetenin, 2% suffer an adverse reaction to the drug. Wha... | math | {
"style": "rule",
"ground_truth": "0.05"
} | {
"split": "train",
"index": 21861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Both cities are in the same time zone. If her flight took $h$ hours and $m$ minutes, with $0 < m < 60$, what is $h + m$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 21862,
"solution": "1. **Calculate the time from 10:34 AM to noon:**\n - From 10:34 AM to 11:34 AM is 1 hour.\n - From 11:34 AM to 12:00 PM is $60 - 34 = 26$ minutes.\n - Therefore, the time from 10:34 AM to noon is $1$ hour and $26$ minutes.\n\n2. **Calculate the time from noon... |
deepscaler | [
{
"role": "user",
"content": "If a worker receives a 30% cut in wages, calculate the percentage raise he needs to regain his original pay. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42.857\\%"
} | {
"split": "train",
"index": 21863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an isosceles triangle with a vertex angle of 36°, the ratio of the base to the leg is equal to . 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": 21864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\{a_{n}\\}$ be a geometric sequence, and let $S_{n}$ be the sum of the first n terms of $\\{a_{n}\\}$. Given that $S_{2}=2$ and $S_{6}=4$, calculate the value of $S_{4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1+\\sqrt{5}"
} | {
"split": "train",
"index": 21865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\(0<\\theta<\\pi\\), a complex number \\(z_{1}=1-\\cos \\theta+i \\sin \\theta\\) and \\(z_{2}=a^{2}+a i\\), where \\(a \\in \\mathbb{R}\\), it is known that \\(z_{1} z_{2}\\) is a pure imaginary number, and \\(\\bar{a}=z_{1}^{2}+z_{2}^{2}-2 z_{1} z_{2}\\). Determine the... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 21866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each face of a regular tetrahedron is labeled with one of the numbers $1, 2, 3, 4$. Four identical regular tetrahedrons are simultaneously rolled onto a table. What is the probability that the product of the four numbers on the faces touching the table is divisible by 4? Let's ... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{16}"
} | {
"split": "train",
"index": 21867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integers $x$ is the number $x^4-51x^2+50$ negative? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 21868,
"solution": "1. **Substitute and Simplify**: Let $x^2 = u$. Then the expression $x^4 - 51x^2 + 50$ becomes $u^2 - 51u + 50$.\n\n2. **Factorization**: Factor the quadratic expression:\n \\[\n u^2 - 51u + 50 = (u - 1)(u - 50)\n \\]\n This factorization is possible because... |
deepscaler | [
{
"role": "user",
"content": "Express $43210_{6}-3210_{7}$ in base 10. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4776"
} | {
"split": "train",
"index": 21869,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $ABCD$, the sides $AB$ and $CD$ are equal. The length of the base $BC$ is 10 units, and the height from $AB$ to $CD$ is 5 units. The base $AD$ is 22 units. Calculate the perimeter of $ABCD$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{61} + 32"
} | {
"split": "train",
"index": 21870,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $MNO$, $\\tan{M}=\\frac{5}{4}$, $OM=8$, and $\\angle O = 90^\\circ$. Find $MN$. Express your answer in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{41}"
} | {
"split": "train",
"index": 21871,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The integers $r$ and $k$ are randomly selected, where $-3 < r < 6$ and $1 < k < 8$. What is the probability that the division $r \\div k$ is an integer value? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 21872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, where $AB = BC > AC$, let $AH$ and $AM$ be the altitude and median to side $BC$, respectively. Given $\\frac{S_{\\triangle AMH}}{S_{\\triangle ABC}} = \\frac{3}{8}$, determine the value of $\\cos \\angle BAC$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 21873,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the set \\( I = \\{1, 2, \\cdots, n\\} (n \\geqslant 3) \\). If two non-empty proper subsets \\( A \\) and \\( B \\) of \\( I \\) satisfy \\( A \\cap B = \\varnothing \\) and \\( A \\cup B = I \\), then \\( A \\) and \\( B \\) are called a partition of \\( I \\). If for any... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 21874,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cone is sliced into five pieces by planes parallel to its base. All of these pieces have the same height. What is the ratio of the volume of the third-largest piece to the volume of the largest piece? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{61}"
} | {
"split": "train",
"index": 21875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with a positive common difference $d > 0$, and $a_2$, $a_5 - 1$, $a_{10}$ form a geometric sequence. If $a_1 = 5$, and $S_n$ is the sum of the first $n$ terms of the sequence $\\{a_n\\}$, determine the minimum value of $$\\frac{2S_{n} + ... | math | {
"style": "rule",
"ground_truth": "\\frac{20}{3}"
} | {
"split": "train",
"index": 21876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\frac{2016!^{2}}{2015!2017!}$. Here $n$ ! denotes $1 \\times 2 \\times \\cdots \\times n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2016}{2017}"
} | {
"split": "train",
"index": 21877,
"solution": "$\\frac{2016!^{2}}{2015!2017!}=\\frac{2016!}{2015!} \\frac{2016!}{2017!}=\\frac{2016}{1} \\frac{1}{2017}=\\frac{2016}{2017}$"
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of two numbers is 15 and their product is 36, find the sum of their reciprocals and the sum of their squares. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "153"
} | {
"split": "train",
"index": 21878,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $ M$ is taken on the perpendicular bisector of the side $ AC$ of an acute-angled triangle $ ABC$ so that $ M$ and $ B$ are on the same side of $ AC$ . If $ \\angle BAC\\equal{}\\angle MCB$ and $ \\angle ABC\\plus{}\\angle MBC\\equal{}180^{\\circ}$ , find ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 21879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the 2009 fractions of the form $\\frac{2}{n(n+2)}$ if the values of $n$ are the positive integers from 1 through 2009? Express your answer as a decimal to the nearest thousandth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.499"
} | {
"split": "train",
"index": 21880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a\\star b = a^b+ab$. If $a$ and $b$ are positive integers greater than or equal to 2 and $a\\star b =15$, find $a+b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 21881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence {a<sub>n</sub>} is a decreasing geometric sequence and satisfies the conditions $$a_{2}a_{7}= \\frac {1}{2}$$ and $$a_{3}+a_{6}= \\frac {9}{4}$$, find the maximum value of a<sub>1</sub>a<sub>2</sub>a<sub>3</sub>…a<sub>2n</sub>. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 21882,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A factory packs its products in cubic boxes. In one store, they put $512$ of these cubic boxes together to make a large $8\\times 8 \\times 8$ cube. When the temperature goes higher than a limit in the store, it is necessary to separate the $512$ set of boxes using horizo... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 21883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the coefficient of \\( x^{1992} \\) in the power series \\( (1 + x)^{\\alpha} = 1 + \\alpha x + \\dots \\) be \\( C(\\alpha) \\). Find \\( \\int_{0}^{1} C(-y-1) \\sum_{k=1}^{1992} \\frac{1}{y+k} \\, dy \\). Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "1992"
} | {
"split": "train",
"index": 21884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of all three-digit numbers that, when divided by 7 give a remainder of 5, when divided by 5 give a remainder of 2, and when divided by 3 give a remainder of 1, calculate the sum of these numbers. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "4436"
} | {
"split": "train",
"index": 21885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an isosceles trapezoid, the longer base \\(AB\\) is 24 units, the shorter base \\(CD\\) is 12 units, and each of the non-parallel sides has a length of 13 units. What is the length of the diagonal \\(AC\\)? Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "\\sqrt{457}"
} | {
"split": "train",
"index": 21886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\alpha$ is an angle in the second quadrant, simplify the function $f(\\alpha) = \\frac{\\tan(\\alpha - \\pi) \\cos(2\\pi - \\alpha) \\sin(-\\alpha + \\frac{3\\pi}{2})}{\\cos(-\\alpha - \\pi) \\tan(\\pi + \\alpha)}$. Then, if $\\cos(\\alpha + \\frac{\\pi}{2}) = -\\f... | math | {
"style": "rule",
"ground_truth": "-\\frac{2\\sqrt{6}}{5}"
} | {
"split": "train",
"index": 21887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pedestrian reported to a traffic officer the number of a car whose driver grossly violated traffic rules. This number is expressed as a four-digit number, where the unit digit is the same as the tens digit, and the hundreds digit is the same as the thousands digit. Moreover, ... | math | {
"style": "rule",
"ground_truth": "7744"
} | {
"split": "train",
"index": 21888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Apartment rentals in Fairview run $\\$$0.90 per square foot. Jillian has determined that she can afford $\\$$630 per month for rent. What is the largest apartment, in square feet, she should consider at the given rate? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "700"
} | {
"split": "train",
"index": 21889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive integer $n$, let $d(n)$ be the number of all positive divisors of $n$. Find all positive integers $n$ such that $d(n)^3=4n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2, 128, 2000"
} | {
"split": "train",
"index": 21890,
"solution": "\nTo solve the problem, we need to find all positive integers \\( n \\) such that the cube of the number of divisors of \\( n \\), denoted \\( d(n)^3 \\), is equal to \\( 4n \\). The equation we need to solve is:\n\\[\nd(n)^3 = 4n.\n\\]\n\nFirst, recall that for a ... |
deepscaler | [
{
"role": "user",
"content": "Triangle \\( ABC \\) has a right angle at \\( B \\). Point \\( D \\) lies on side \\( BC \\) such that \\( 3 \\angle BAD = \\angle BAC \\). Given \\( AC = 2 \\) and \\( CD = 1 \\), compute \\( BD \\). Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"split": "train",
"index": 21891,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\\mathcal{T}.$ Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "227.052227052227"
} | {
"split": "train",
"index": 21892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sorcerer is concocting a healing potion. In order to do so, he must have one of three roots, and one of five minerals. However, one of the minerals is incompatible with two of the roots. Otherwise, he does not foresee any complications with other combinations of roots and min... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 21893,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mady has an infinite number of balls and empty boxes available to her. The empty boxes, each capable of holding four balls, are arranged in a row from left to right. At the first step, she places a ball in the first box (the leftmost box) of the row. At each subsequent step, sh... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 21894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive four-digit integers of the form $\\_\\_50$ are divisible by 50? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 21895,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Foma and Erema were traveling along a straight road to Moscow in a cart at a constant speed.\n\n- At 12:00, Foma asked: \"How many versts to Moscow?\"\n- Erema replied: \"82.\"\n- At 13:00, Foma asked: \"How many versts to Moscow?\"\n- Erema replied: \"71.\"\n- At 15:00, Foma a... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 21896,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You have a length of string and 7 beads in the 7 colors of the rainbow. You place the beads on the string as follows - you randomly pick a bead that you haven't used yet, then randomly add it to either the left end or the right end of the string. What is the probability that, a... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5040}"
} | {
"split": "train",
"index": 21897,
"solution": "The threading method does not depend on the colors of the beads, so at the end all configurations are equally likely. Since there are $7!=5040$ configurations in total, the probability of any particular configuration is $\\frac{1}{5040}$."
} |
deepscaler | [
{
"role": "user",
"content": "If $\\alpha$ , $\\beta$ , and $\\gamma$ are the roots of $x^3 - x - 1 = 0$ , compute $\\frac{1+\\alpha}{1-\\alpha} + \\frac{1+\\beta}{1-\\beta} + \\frac{1+\\gamma}{1-\\gamma}$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 21898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $g(x)$ satisfies\n\\[ g(x + g(x)) = 5g(x) \\]\nfor all $x$, and $g(1) = 5$. Find $g(26)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 21899,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain clothing wholesale market sells a type of shirt, with a purchase price of $50$ yuan per shirt. It is stipulated that the selling price of each shirt is not lower than the purchase price. According to a market survey, the monthly sales volume $y$ (in units) and the sel... | math | {
"style": "rule",
"ground_truth": "19500"
} | {
"split": "train",
"index": 21900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the two foci of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ ($a>0$, $b>0$), an isosceles right triangle $MF_1F_2$ is constructed with $F_1$ as the right-angle vertex. If the midpoint of the side $MF_1$ lies on the hyperbola, calculate ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} + 1}{2}"
} | {
"split": "train",
"index": 21901,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cheetah takes strides of 2 meters each, while a fox takes strides of 1 meter each. The time it takes for the cheetah to run 2 strides is the same as the time it takes for the fox to run 3 strides. If the distance between the cheetah and the fox is 30 meters, how many meters m... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 21902,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a group of 21 persons, every two person communicate with different radio frequency. It's possible for two person to not communicate (means there's no frequency occupied to connect them). Only one frequency used by each couple, and it's unique for every couple. In every 3 per... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 21903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(x\\), \\(y\\), and \\(z\\) be positive real numbers such that \\(x + y + z = 3.\\) Find the maximum value of \\(x^3 y^3 z^2.\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4782969}{390625}"
} | {
"split": "train",
"index": 21904,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $r$: $$\\frac{r+3}{r-2} = \\frac{r-1}{r+1}.$$Express your answer as a fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{7}"
} | {
"split": "train",
"index": 21905,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain product was bought in the fall, and 825 rubles were paid for it. In the fall, the price per kilogram of this product was 1 ruble cheaper than in the spring. Therefore, for the same amount in the spring, 220 kg less was bought. How much does 1 kg of the product cost in... | math | {
"style": "rule",
"ground_truth": "550"
} | {
"split": "train",
"index": 21906,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The amount $2.5$ is split into two nonnegative real numbers uniformly at random, for instance, into $2.143$ and $.357$, or into $\\sqrt{3}$ and $2.5-\\sqrt{3}$. Then each number is rounded to its nearest integer, for instance, $2$ and $0$ in the first case above, $2$ and $1$ in... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 21907,
"solution": "Let's denote the two parts into which $2.5$ is split as $x$ and $2.5 - x$. We need to find the probability that the sum of the nearest integers to $x$ and $2.5 - x$ equals $3$.\n\n1. **Identify the rounding conditions**: \n - $x$ rounds to $0$ if $x < 0.5$.\n -... |
deepscaler | [
{
"role": "user",
"content": "To open the safe, you need to enter a code — a number consisting of seven digits: twos and threes. The safe will open if there are more twos than threes, and the code is divisible by both 3 and 4. Create a code that opens the safe. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "2222232"
} | {
"split": "train",
"index": 21908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers $n\\leq100$ satisfy $\\left\\lfloor n\\pi\\right\\rfloor=\\left\\lfloor\\left(n-1\\right)\\pi\\right\\rfloor+3$ ? Here $\\left\\lfloor x\\right\\rfloor$ is the greatest integer less than or equal to $x$ ; for example, $\\left\\lfloor\\pi\\right... | math | {
"style": "rule",
"ground_truth": "86"
} | {
"split": "train",
"index": 21909,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\\begin{cases}x=3\\cos \\alpha \\\\ y=\\sin \\alpha\\end{cases}$ (where $\\alpha$ is the parameter), and in the polar coordinate system with the origin as the pole and the positive $x$-axis as th... | math | {
"style": "rule",
"ground_truth": "\\frac{18 \\sqrt{2}}{5}"
} | {
"split": "train",
"index": 21910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the real numbers \\( x \\) and \\( y \\) satisfy the equation \\( 2x^2 + 3xy + 2y^2 = 1 \\), find the minimum value of \\( x + y + xy \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{9}{8}"
} | {
"split": "train",
"index": 21911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $m$ and $n$ are positive integers with $\\sqrt{7+\\sqrt{48}}=m+\\sqrt{n}$. What is the value of $m^{2}+n^{2}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 21912,
"solution": "Suppose that $\\sqrt{7+\\sqrt{48}}=m+\\sqrt{n}$. Squaring both sides, we obtain $7+\\sqrt{48}=(m+\\sqrt{n})^{2}$. Since $(m+\\sqrt{n})^{2}=m^{2}+2m\\sqrt{n}+n$, then $7+\\sqrt{48}=(m^{2}+n)+2m\\sqrt{n}$. Let's make the assumption that $m^{2}+n=7$ and $2m\\sqrt{n}=\... |
deepscaler | [
{
"role": "user",
"content": "The function\n$$\n\\begin{aligned}\ny= & |x-1|+|2x-1|+|3x-1|+ \\\\\n& |4x-1|+|5x-1|\n\\end{aligned}\n$$\n\nachieves its minimum value when the variable $x$ equals ______. 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": 21913,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The general formula of the sequence \\\\(\\{a_n\\}\\) is \\\\(a_n=n\\cos \\frac{n\\pi}{2}\\\\), and the sum of its first \\\\(n\\\\) terms is \\\\(S_n\\\\). Find \\\\(S_{2019}\\\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1010"
} | {
"split": "train",
"index": 21914,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $x$ and $y$ satisfy $\\frac{x-y}{x+y}=9$ and $\\frac{xy}{x+y}=-60$. What is the value of $(x+y)+(x-y)+xy$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-150"
} | {
"split": "train",
"index": 21915,
"solution": "The first equation $\\frac{x-y}{x+y}=9$ gives $x-y=9x+9y$ and so $-8x=10y$ or $-4x=5y$. The second equation $\\frac{xy}{x+y}=-60$ gives $xy=-60x-60y$. Multiplying this equation by 5 gives $5xy=-300x-300y$ or $x(5y)=-300x-60(5y)$. Since $5y=-4x$, then $x(-4x)=-300x-... |
deepscaler | [
{
"role": "user",
"content": "The number $2^{1993}+3^{1993}$ is a multiple of $5.$ What is the units digit of the quotient $\\frac{2^{1993}+3^{1993}}{5}?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 21916,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The positive difference between the two roots of the quadratic equation $3x^2 - 7x - 8 = 0$ can be written as $\\frac{\\sqrt{m}}{n}$, where $n$ is an integer and $m$ is an integer not divisible by the square of any prime number. Find $m + n$. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "148"
} | {
"split": "train",
"index": 21917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the base-16 number $B1234_{16}$ is written in base 2, how many base-2 digits (bits) does it have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 21918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $y = f(x)$ is an even function defined on $\\mathbb{R}$, and when $x \\geq 0$, $f(x) = \\log_2(x+2) - 3$. Find the values of $f(6)$ and $f(f(0))$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 21919,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The only prime factors of an integer $n$ are 2 and 3. If the sum of the divisors of $n$ (including itself) is $1815$ , find $n$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "648"
} | {
"split": "train",
"index": 21920,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $15\\cdot\\frac{16}{9}\\cdot\\frac{-45}{32}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{25}{6}"
} | {
"split": "train",
"index": 21921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the polynomial $p(x),$ with real coefficients, such that\n\\[p(x^3) - p(x^3 - 2) = [p(x)]^2 + 12\\]for all real numbers $x.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6x^3 - 6"
} | {
"split": "train",
"index": 21922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $x^5 - 5x$ when $x = 4$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1004"
} | {
"split": "train",
"index": 21923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the product of the sums of the squares and the cubes of the roots of the equation \\[x\\sqrt{x} - 8x + 9\\sqrt{x} - 1 = 0,\\] given that all roots are real and nonnegative. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13754"
} | {
"split": "train",
"index": 21924,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Susie pays for $4$ muffins and $3$ bananas. Calvin spends twice as much paying for $2$ muffins and $16$ bananas. A muffin is how many times as expensive as a banana? 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": 21925,
"solution": "1. **Define Variables:**\n Let $m$ represent the cost of one muffin and $b$ represent the cost of one banana.\n\n2. **Set Up Equations:**\n According to the problem, Susie's total cost for $4$ muffins and $3$ bananas is:\n \\[\n 4m + 3b\n \\]\n Calvin s... |
deepscaler | [
{
"role": "user",
"content": "Let point $P$ be a moving point on the curve $C\\_1$: $(x-2)^2 + y^2 = 4$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis. Rotate point $P$ counterclockwise by $90^{{∘}}$ around the pol... | math | {
"style": "rule",
"ground_truth": "3 - \\sqrt{3}"
} | {
"split": "train",
"index": 21926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Take a clay sphere of radius 13, and drill a circular hole of radius 5 through its center. Take the remaining \"bead\" and mold it into a new sphere. What is this sphere's radius? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 21927,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram below shows \\( \\triangle ABC \\), which is isosceles with \\( AB = AC \\) and \\( \\angle A = 20^\\circ \\). The point \\( D \\) lies on \\( AC \\) such that \\( AD = BC \\). The segment \\( BD \\) is constructed as shown. Determine \\( \\angle ABD \\) in degrees.... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 21928,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A, B, C$, and $D$ lie on a line in that order such that $\\frac{A B}{B C}=\\frac{D A}{C D}$. If $A C=3$ and $B D=4$, find $A D$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 21929,
"solution": "Let $B C=x$, then the equation becomes $\\frac{3-x}{x}=\\frac{7-x}{4-x}$. This simplifies to a quadratic equation with solutions $x=1$ and $x=6$. Since $x<3$, we have $x=1$ and $A D=7-x=6$."
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive value of $m$ so that the equation $15x^2 - mx + 315 = 0$ has integral solutions? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 21930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When a positive integer is expressed in base 7, it is $AB_7$, and when it is expressed in base 5, it is $BA_5$. What is the positive integer in decimal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 21931,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a number $x$ is randomly taken from the interval $[0,2π]$, the probability that the value of $\\sin x$ is between $0$ and $\\frac{\\sqrt{3}}{2}$ is _______. 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": 21932,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m$ be the number of five-element subsets that can be chosen from the set of the first $14$ natural numbers so that at least two of the five numbers are consecutive. Find the remainder when $m$ is divided by $1000$. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "750"
} | {
"split": "train",
"index": 21933,
"solution": "We can use complementary counting. We can choose a five-element subset in ${14\\choose 5}$ ways. We will now count those where no two numbers are consecutive. We will show a bijection between this set, and the set of 10-element strings that contain 5 $A$s and 5 $B$... |
deepscaler | [
{
"role": "user",
"content": "A cone and a sphere are placed inside a cylinder without overlapping. All bases are aligned and rest on the same flat surface. The cylinder has a radius of 6 cm and a height of 15 cm. The cone has a radius of 6 cm and a height of 7.5 cm. The sphere has a radius of 6 cm. Find th... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{138}"
} | {
"split": "train",
"index": 21934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square \\(ABCD\\) has a side-length of 2, and \\(M\\) is the midpoint of \\(BC\\). The circle \\(S\\) inside the quadrilateral \\(AMCD\\) touches the three sides \\(AM\\), \\(CD\\), and \\(DA\\). What is its radius? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "3 - \\sqrt{5}"
} | {
"split": "train",
"index": 21935,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Into how many parts can a plane be divided by four lines? Consider all possible cases and make a drawing for each case. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 21936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A natural number is called lucky if all its digits are equal to 7. For example, 7 and 7777 are lucky, but 767 is not. João wrote down the first twenty lucky numbers starting from 7, and then added them. What is the remainder of that sum when divided by 1000? Let's think step by... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 21937,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A spiral staircase turns $270^\\circ$ as it rises 10 feet. The radius of the staircase is 3 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "17.3"
} | {
"split": "train",
"index": 21938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nGiven a positive integer \\(N\\) (written in base 10), define its integer substrings to be integers that are equal to strings of one or more consecutive digits from \\(N\\), including \\(N\\) itself. For example, the integer substrings of 3208 are \\(3, 2, 0, 8, 32, 20, 320, ... | math | {
"style": "rule",
"ground_truth": "88,888,888"
} | {
"split": "train",
"index": 21939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jack and Jill are going swimming at a pool that is one mile from their house. They leave home simultaneously. Jill rides her bicycle to the pool at a constant speed of $10$ miles per hour. Jack walks to the pool at a constant speed of $4$ miles per hour. How many minutes before... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 21940,
"solution": "1. **Calculate Jill's travel time:**\n - We use the formula for distance, \\(d = rt\\), where \\(d\\) is the distance, \\(r\\) is the rate, and \\(t\\) is the time.\n - For Jill, \\(d = 1\\) mile, \\(r = 10\\) miles per hour. Plugging in the values, we get:\n ... |
deepscaler | [
{
"role": "user",
"content": "Given that one of the children is a boy and the probability of having a boy or a girl is equal, calculate the probability that the other child is a girl. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 21941,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the country of Draconia, there are red, green, and blue dragons. Each dragon has three heads, each of which always tells the truth or always lies. Furthermore, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of ... | math | {
"style": "rule",
"ground_truth": "176"
} | {
"split": "train",
"index": 21942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A polynomial $p(x)$ is called self-centered if it has integer coefficients and $p(100) = 100.$ If $p(x)$ is a self-centered polynomial, what is the maximum number of integer solutions $k$ to the equation $p(k) = k^3$? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 21943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the sum:\n\\[\\sum_{1 \\le a < b < c} \\frac{1}{3^a 5^b 7^c}.\\]\n(The sum is taken over all triples $(a,b,c)$ of positive integers such that $1 \\le a < b < c.$) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{21216}"
} | {
"split": "train",
"index": 21944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathrm{O}$ be the intersection point of the diagonals of a convex quadrilateral $A B C D$, and let $P, Q, R$, and $S$ be the centroids of triangles $A O B$, $B O C$, $C O D$, and $D O A$, respectively. Find the ratio of the areas of the quadrilateral $P Q R S$ to that of... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"split": "train",
"index": 21945,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x)$ is an even function on $(-\\infty, +\\infty)$, if for $x \\geq 0$, we have $f(x+2) = -f(x)$, and when $x \\in [0, 2)$, $f(x) = \\log_{2}(x+1)$, find the value of $f(-2010) - f(2009)$. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 21946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a finite sequence \\( B = (b_1, b_2, \\dots, b_{50}) \\) of numbers, the Cesaro sum is defined as\n\\[\n\\frac{S_1 + \\cdots + S_{50}}{50},\n\\]\nwhere \\( S_k = b_1 + \\cdots + b_k \\) and \\( 1 \\leq k \\leq 50 \\).\n\nIf the Cesaro sum of the 50-term sequence \\( (b_1, \... | math | {
"style": "rule",
"ground_truth": "492"
} | {
"split": "train",
"index": 21947,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alli rolls a fair $8$-sided die twice. What is the probability of rolling numbers that differ by $3$ in her first two rolls? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 21948,
"solution": ""
} |
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