data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Find the sum $$\\frac{3^1}{9^1 - 1} + \\frac{3^2}{9^2 - 1} + \\frac{3^3}{9^3 - 1} + \\frac{3^4}{9^4 - 1} + \\cdots.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 28249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three different natural numbers $x$, $y$ and $z$ each have exactly three natural-number factors. How many factors does $x^{2}y^{3}z^{4}$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "315"
} | {
"split": "train",
"index": 28250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ zeros.\n\n[hide=\"Note\"]Note. Here (and generally in MathLinks) natural numbers supposed to be positive.[/hide] Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "n=7920, 7921, 7922, 7923, 7924"
} | {
"split": "train",
"index": 28251,
"solution": "\nTo find the least possible value of the natural number \\( n \\) such that \\( n! \\) ends in exactly 1987 zeros, we need to determine the number of trailing zeros of a factorial. The number of trailing zeros of \\( n! \\) is given by the sum of the floor divisio... |
deepscaler | [
{
"role": "user",
"content": "The function $f(n)$ is defined on the positive integers such that $f(f(n)) = 2n$ and $f(4n + 1) = 4n + 3$ for all positive integers $n.$ Find $f(1000).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1016"
} | {
"split": "train",
"index": 28252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that θ is an acute angle and $\\sqrt {2}$sinθsin($θ+ \\frac {π}{4}$)=5cos2θ, find the value of tanθ. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {5}{6}"
} | {
"split": "train",
"index": 28253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a new school $40$ percent of the students are freshmen, $30$ percent are sophomores, $20$ percent are juniors, and $10$ percent are seniors. All freshmen are required to take Latin, and $80$ percent of the sophomores, $50$ percent of the juniors, and $20$ percent of the seni... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 28254,
"solution": "We see that $40\\% \\cdot 100\\% + 30\\% \\cdot 80\\% + 20\\% \\cdot 50\\% + 10\\% \\cdot 20\\% = 76\\%$ of students are learning Latin. In addition, $30\\% \\cdot 80\\% = 24\\%$ of students are sophomores learning Latin. Thus, our desired probability is $\\dfrac{2... |
deepscaler | [
{
"role": "user",
"content": "I have fifteen books, of which I want to bring two to read on vacation. However, out of these, there are three specific books that cannot be paired together (let's say they are volumes of the same series). How many different pairs can I choose? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "102"
} | {
"split": "train",
"index": 28255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nine fair coins are flipped independently and placed in the cells of a 3 by 3 square grid. Let $p$ be the probability that no row has all its coins showing heads and no column has all its coins showing tails. If $p=\\frac{a}{b}$ for relatively prime positive integers $a$ and $b... | math | {
"style": "rule",
"ground_truth": "8956"
} | {
"split": "train",
"index": 28256,
"solution": "Consider the probability of the complement. It is impossible for some row to have all heads and some column to have tails, since every row intersects every column. Let $q$ be the probability that some row has all heads. By symmetry, $q$ is also the probability that... |
deepscaler | [
{
"role": "user",
"content": "Find the number of integers \\( n \\) that satisfy\n\\[ 15 < n^2 < 120. \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 28257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A newly designed car travels 4.2 kilometers further per liter of gasoline than an older model. The fuel consumption for the new car is 2 liters less per 100 kilometers. How many liters of gasoline does the new car consume per 100 kilometers? If necessary, round your answer to t... | math | {
"style": "rule",
"ground_truth": "5.97"
} | {
"split": "train",
"index": 28258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$:\n\n$$x^2 + 4x + 3 = -(x + 3)(x + 5).$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 28259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many real numbers $a^{}_{}$ does the quadratic equation $x^2 + ax^{}_{} + 6a=0$ have only integer roots for $x^{}_{}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 28260,
"solution": "Let $x^2 + ax + 6a = (x - s)(x - r)$. Vieta's yields $s + r = - a, sr = 6a$. \\begin{eqnarray*}sr + 6s + 6r &=& 0\\\\ sr + 6s + 6r + 36 &=& 36\\\\ (s + 6)(r + 6) &=& 36 \\end{eqnarray*}\nWithout loss of generality let $r \\le s$.\n The possible values of $(r + 6,s ... |
deepscaler | [
{
"role": "user",
"content": "During a recent campaign for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went north, on the third day west, on the fourth day south, on the fifth day east, etc. If the candidate we... | math | {
"style": "rule",
"ground_truth": "580"
} | {
"split": "train",
"index": 28261,
"solution": "On the first day, the candidate moves $[4(0) + 1]^2/2\\ \\text{east},\\, [4(0) + 2]^2/2\\ \\text{north},\\, [4(0) + 3]^2/2\\ \\text{west},\\, [4(0) + 4]^2/2\\ \\text{south}$, and so on. The E/W displacement is thus $1^2 - 3^2 + 5^2 \\ldots +37^2 - 39^2 = \\left|\\s... |
deepscaler | [
{
"role": "user",
"content": "Let $x, y, z$ be real numbers such that\n\\[\nx + y + z = 5,\n\\]\n\\[\nx^2 + y^2 + z^2 = 11.\n\\]\nFind the smallest and largest possible values of $x$, and compute their sum. 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": 28262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A man named Juan has three rectangular solids, each having volume 128. Two of the faces of one solid have areas 4 and 32. Two faces of another solid have areas 64 and 16. Finally, two faces of the last solid have areas 8 and 32. What is the minimum possible exposed surface area... | math | {
"style": "rule",
"ground_truth": "688"
} | {
"split": "train",
"index": 28263,
"solution": "Suppose that $x, y, z$ are the sides of the following solids. Then Volume $=xyz=128$. For the first solid, without loss of generality (with respect to assigning lengths to $x, y, z$), $xy=4$ and $yz=32$. Then $xy^{2}z=128$. Then $y=1$. Solving the remaining equatio... |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $7, n,$ and $n+2$ cents, $120$ cents is the greatest postage that cannot be formed. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 28264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among 150 schoolchildren, only boys collect stamps. 67 people collect USSR stamps, 48 people collect African stamps, and 32 people collect American stamps. 11 people collect only USSR stamps, 7 people collect only African stamps, 4 people collect only American stamps, and only ... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 28265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence ${a_{n}}$ with the sum of its first $n$ terms denoted as $S_{n}$, if $S_{5}$, $S_{4}$, and $S_{6}$ form an arithmetic sequence, then determine the common ratio of the sequence ${a_{n}}$, denoted as $q$. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 28266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mrs. Everett recorded the performance of her students in a chemistry test. However, due to a data entry error, 5 students who scored 60% were mistakenly recorded as scoring 70%. Below is the corrected table after readjusting these students. Using the data, calculate the average... | math | {
"style": "rule",
"ground_truth": "75.33"
} | {
"split": "train",
"index": 28267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the shaded region. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\dfrac{1}{2}"
} | {
"split": "train",
"index": 28268,
"solution": "1. **Identify the equations of the lines**: \n - For the first line, using the points (0,4) and (8,1), the slope $m$ is calculated as:\n \\[\n m = \\frac{1-4}{8-0} = -\\frac{3}{8}\n \\]\n Thus, the equation of the line is:\n \\[\n y_1 = -\... |
deepscaler | [
{
"role": "user",
"content": "A product originally priced at \\$120 receives a discount of 8%. Calculate the percentage increase needed to return the reduced price to its original amount. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8.7\\%"
} | {
"split": "train",
"index": 28269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\\overline{BD} \\perp \\overline{BC}$. The line $\\ell$ through $D$ parallel to line $BC$ intersects sides $\\overline{AB}$ and $\\overline{AC}$ at points $E$ and $F... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"split": "train",
"index": 28270,
"solution": "Since $\\triangle AFG$ is isosceles, $AF = FG$, and since $\\triangle AEF$ is equilateral, $AF = EF$. Thus, $EF = FG$, and since these triangles share an altitude, they must have the same area.\n Drop perpendiculars from $E$ and $F$ to line $BC$; call the meeting p... |
deepscaler | [
{
"role": "user",
"content": "Find all functions $ f: (0, \\infty) \\mapsto (0, \\infty)$ (so $ f$ is a function from the positive real numbers) such that\n\\[ \\frac {\\left( f(w) \\right)^2 \\plus{} \\left( f(x) \\right)^2}{f(y^2) \\plus{} f(z^2) } \\equal{} \\frac {w^2 \\plus{} x^2}{y^2 \\plus{} z^2}\n\\... | math | {
"style": "rule",
"ground_truth": "f(x) = x \\text{ or } f(x) = \\frac{1}{x}"
} | {
"split": "train",
"index": 28271,
"solution": "\nTo find all functions \\( f: (0, \\infty) \\to (0, \\infty) \\) satisfying the given functional equation:\n\n\\[\n\\frac {\\left( f(w) \\right)^2 + \\left( f(x) \\right)^2}{f(y^2) + f(z^2) } = \\frac {w^2 + x^2}{y^2 + z^2}\n\\]\n\nfor all positive real numbers \\... |
deepscaler | [
{
"role": "user",
"content": "Find the number of different patterns that can be created by shading exactly three of the nine small triangles, no two of which can share a side, considering patterns that can be matched by rotations or by reflections as the same. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 28272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the sum of all positive real numbers \\(x \\leq 5\\) satisfying \\(x=\\frac{\\left\\lceil x^{2}\\right\\rceil+\\lceil x\\rceil \\cdot\\lfloor x\\rfloor}{\\lceil x\\rceil+\\lfloor x\\rfloor}\\). Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 28273,
"solution": "Note that all integer \\(x\\) work. If \\(x\\) is not an integer then suppose \\(n<x<n+1\\). Then \\(x=n+\\frac{k}{2n+1}\\), where \\(n\\) is an integer and \\(1 \\leq k \\leq 2n\\) is also an integer, since the denominator of the fraction on the right hand side is... |
deepscaler | [
{
"role": "user",
"content": "Point $Q$ lies on the diagonal $AC$ of square $EFGH$ with $EQ > GQ$. Let $R_{1}$ and $R_{2}$ be the circumcenters of triangles $EFQ$ and $GHQ$ respectively. Given that $EF = 8$ and $\\angle R_{1}QR_{2} = 90^{\\circ}$, find the length $EQ$ in the form $\\sqrt{c} + \\sqrt{d}$, wh... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 28274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=e^{x}-mx^{3}$ ($m$ is a nonzero constant). \n$(1)$ If the function $f(x)$ is increasing on $(0,+\\infty)$, find the range of real numbers for $m$. \n$(2)$ If $f_{n+1}(x)$ ($n\\in \\mathbb{N}$) represents the derivative of $f_{n}(x)$, where $f_{0}(x)=f... | math | {
"style": "rule",
"ground_truth": "n = 8"
} | {
"split": "train",
"index": 28275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x) = 3x+2$ and $g(x) = (x-1)^2$, what is $f(g(-2))$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 28276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A tourist is learning an incorrect way to sort a permutation $(p_{1}, \\ldots, p_{n})$ of the integers $(1, \\ldots, n)$. We define a fix on two adjacent elements $p_{i}$ and $p_{i+1}$, to be an operation which swaps the two elements if $p_{i}>p_{i+1}$, and does nothing otherwi... | math | {
"style": "rule",
"ground_truth": "1009! \\cdot 1010!"
} | {
"split": "train",
"index": 28277,
"solution": "Note that the given algorithm is very similar to the well-known Bubble Sort algorithm for sorting an array. The exception is that in the $i$-th round through the array, the first $i-1$ pairs are not checked. We claim a necessary and sufficient condition for the arr... |
deepscaler | [
{
"role": "user",
"content": "Let $a_0=29$ , $b_0=1$ and $$ a_{n+1} = a_n+a_{n-1}\\cdot b_n^{2019}, \\qquad b_{n+1}=b_nb_{n-1} $$ for $n\\geq 1$ . Determine the smallest positive integer $k$ for which $29$ divides $\\gcd(a_k, b_k-1)$ whenever $a_1,b_1$ are positive integers and $29$ does n... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 28278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The digits 2, 4, 6, and 8 are each used once to create two 2-digit numbers. What is the smallest possible difference between the two 2-digit numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 28279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the largest regular hexagon that can fit inside of a rectangle with side lengths 20 and 22 can be expressed as $a \\sqrt{b}-c$, for positive integers $a, b$, and $c$, where $b$ is squarefree. Compute $100 a+10 b+c$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "134610"
} | {
"split": "train",
"index": 28280,
"solution": "Let $s$ be the sidelength of the hexagon. We can view this problem as finding the maximal rectangle of with sides $s$ and $s \\sqrt{3}$ that can fit inside this rectangle. Let $A B C D$ be a rectangle with $A B=20$ and $B C=22$ and let $X Y Z W$ be an inscribed rec... |
deepscaler | [
{
"role": "user",
"content": "\nThe standard enthalpy of formation (ΔH_f°) of a substance is equal to the heat effect of the formation reaction of 1 mole of the substance from simple substances in their standard states (at 1 atm pressure and a given temperature). Therefore, it is necessary to find the heat ... | math | {
"style": "rule",
"ground_truth": "-82.9"
} | {
"split": "train",
"index": 28281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( ABC \\), \\( AB = 33 \\), \\( AC = 21 \\), and \\( BC = m \\), where \\( m \\) is a positive integer. If point \\( D \\) can be found on \\( AB \\) and point \\( E \\) can be found on \\( AC \\) such that \\( AD = DE = EC = n \\), where \\( n \\) is a positive i... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 28282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain football invitational tournament, 16 cities participate, with each city sending two teams, Team A and Team B. According to the competition rules, after several days of matches, it was found that aside from Team A from city $A$, the number of matches already played ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 28283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it; this grid's *score* is the sum of all numbers recorded this way. Deyuan shades each square in a blank $n\\times n$... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 28284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pyramid is formed on a $6\\times 8$ rectangular base. The four edges joining the apex to the corners of the rectangular base each have length $13$. What is the volume of the pyramid? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 28285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different positive three-digit integers can be formed using only the digits in the set $\\{2, 3, 5, 5, 7, 7, 7\\}$ if no digit may be used more times than it appears in the given set of available digits? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 28286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The matrix\n\\[\\mathbf{M} = \\begin{pmatrix} 0 & 2y & z \\\\ x & y & -z \\\\ x & -y & z \\end{pmatrix}\\]satisfies $\\mathbf{M}^T \\mathbf{M} = \\mathbf{I}.$ Find $x^2 + y^2 + z^2.$\n\nNote: For a matrix $\\mathbf{A},$ $\\mathbf{A}^T$ is the transpose of $\\mathbf{A},$ which ... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 28287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nikita usually leaves home at 8:00 AM, gets into Uncle Vanya's car, and his uncle drives him to school at a certain time. But on Friday, Nikita left home at 7:10 AM and ran in the opposite direction. Uncle Vanya waited for him and at 8:10 AM drove after him, caught up with Niki... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 28288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many triangles with positive area can be formed with vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $6$, inclusive? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6788"
} | {
"split": "train",
"index": 28289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expression $\\sqrt{\\frac{4}{3}} - \\sqrt{\\frac{3}{4}}$ is equal to: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{6}"
} | {
"split": "train",
"index": 28290,
"solution": "1. **Simplify $\\sqrt{\\frac{4}{3}}$:**\n \\[\n \\sqrt{\\frac{4}{3}} = \\frac{\\sqrt{4}}{\\sqrt{3}} = \\frac{2}{\\sqrt{3}}\n \\]\n Rationalizing the denominator:\n \\[\n \\frac{2}{\\sqrt{3}} \\cdot \\frac{\\sqrt{3}}{\\sqrt{3}} = \\frac{2\\sqrt{3}}{3}\n ... |
deepscaler | [
{
"role": "user",
"content": "Given that the rhombus has diagonals of length $8$ and $30$, calculate the radius of the circle inscribed in the rhombus. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{30}{\\sqrt{241}}"
} | {
"split": "train",
"index": 28291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two wholesalers, A and B, sell the same brand of teapots and teacups at the same price, with the teapot priced at 30 yuan each and the teacup at 5 yuan each. Both stores are offering a promotional sale: Store A has a 'buy one get one free' promotion (buy a teapot and get a teac... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 28292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product by at least 2? Express your answer as a common fraction. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"split": "train",
"index": 28293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ be a real number. Consider the following five statements:\n\n$0 < x^2 < 1$\n$x^2 > 1$\n$-1 < x < 0$\n$0 < x < 1$\n$0 < x - x^2 < 1$\n\nWhat is the maximum number of these statements that can be true for any value of $x$? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 28294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During what year after 1994 did sales increase the most number of dollars? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 28295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function \\( f(x) \\) be defined on \\([0,1]\\), satisfying \\( f(0) = f(1) \\) and for any \\( x, y \\in [0,1] \\) having \\( |f(x) - f(y)| < |x - y| \\). Find the smallest real number \\( m \\) such that for any function \\( f(x) \\) meeting the above conditions and f... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 28296,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equations $y = 2x^2$ and $y = 4x + c$, determine the value(s) of $c$ for which these two equations have precisely two identical solutions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 28297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The points $(-1,4)$ and $(2,-3)$ are adjacent vertices of a square. What is the area of the square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 28298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the modular inverse of $13$, modulo $2000$?\n\nExpress your answer as an integer from $0$ to $1999$, inclusive. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1077"
} | {
"split": "train",
"index": 28299,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the car's average miles-per-gallon for the entire trip given that the odometer readings are $34,500, 34,800, 35,250$, and the gas tank was filled with $8, 10, 15$ gallons of gasoline. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22.7"
} | {
"split": "train",
"index": 28300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( M(x, y, z) \\) represent the minimum of the three numbers \\( x, y, z \\). If the quadratic function \\( f(x) = ax^2 + bx + c \\) (where \\( a, b, c > 0 \\)) has a zero, determine the maximum value of \\( M \\left( \\frac{b+c}{a}, \\frac{c+a}{b}, \\frac{a+b}{c} \\right)... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"split": "train",
"index": 28301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\(x = \\frac{1}{3}\\) and \\(y = 3\\), find the value of \\(\\frac{1}{4}x^3y^8\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60.75"
} | {
"split": "train",
"index": 28302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\(x\\) is a real number, find the least possible value of \\((x+2)(x+3)(x+4)(x+5)+3033\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3032"
} | {
"split": "train",
"index": 28303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest four-digit number whose product of all digits equals 512. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1888"
} | {
"split": "train",
"index": 28304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 10$, $BC = 5$, and the height of the pyramid, $PA$, while not perpendicular to the plane of $ABCD$, ends at the center of rectangle $ABCD$ and is twice the length of $BC$. What is the volume of $PABCD$? Let's think step ... | math | {
"style": "rule",
"ground_truth": "\\frac{500}{3}"
} | {
"split": "train",
"index": 28305,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when the value of $m$ is divided by 1000 in the number of increasing sequences of positive integers $a_1 \\le a_2 \\le a_3 \\le \\cdots \\le a_6 \\le 1500$ such that $a_i-i$ is odd for $1\\le i \\le 6$. The total number of sequences can be expressed as ${m \\... | math | {
"style": "rule",
"ground_truth": "752"
} | {
"split": "train",
"index": 28306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD$, $P$ is a point on side $\\overline{BC}$ such that $BP = 20$ and $CP = 5.$ If $\\tan \\angle APD = 2,$ then find $AB.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 28307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle has side lengths of $x,75,100$ where $x<75$ and altitudes of lengths $y,28,60$ where $y<28$ . What is the value of $x+y$ ?\n\n*2019 CCA Math Bonanza Team Round #2* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 28308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The seven digits in Sam's phone number and the four digits in his house number have the same sum. The four digits in his house number are distinct, and his phone number is 271-3147. What is the largest possible value of Sam's house number? Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "9871"
} | {
"split": "train",
"index": 28309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence $s_{0}, s_{1}, s_{2}, s_{3}, \\ldots$ is defined by $s_{0}=s_{1}=1$ and, for every positive integer $n, s_{2 n}=s_{n}, s_{4 n+1}=s_{2 n+1}, s_{4 n-1}=s_{2 n-1}+s_{2 n-1}^{2} / s_{n-1}$. What is the value of $s_{1000}$? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 28310,
"solution": "720 Some experimentation with small values may suggest that $s_{n}=k$!, where $k$ is the number of ones in the binary representation of $n$, and this formula is in fact provable by a straightforward induction. Since $1000_{10}=1111101000_{2}$, with six ones, $s_{10... |
deepscaler | [
{
"role": "user",
"content": "Marla has a large white cube that has an edge of 8 feet. She also has enough green paint to cover 200 square feet. Marla uses all the paint to create a white circle centered on each face, surrounded by a green border. What is the area of one of the white circles, in square feet... | math | {
"style": "rule",
"ground_truth": "30.67"
} | {
"split": "train",
"index": 28311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The domain of the function $f(x) = \\arcsin(\\log_{m}(nx))$ is a closed interval of length $\\frac{1}{2013}$ , where $m$ and $n$ are positive integers and $m>1$. Find the the smallest possible value of $m+n.$ Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "5371"
} | {
"split": "train",
"index": 28312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $JKLM$, $P$ is a point on $LM$ so that $\\angle JPL=90^{\\circ}$. $UV$ is perpendicular to $LM$ with $LU=UP$, as shown. $PL$ intersects $UV$ at $Q$. Point $R$ is on $LM$ such that $RJ$ passes through $Q$. In $\\triangle PQL$, $PL=25$, $LQ=20$ and $QP=15$. Find $VD$... | math | {
"style": "rule",
"ground_truth": "\\dfrac{28}{3}"
} | {
"split": "train",
"index": 28313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\tan A=\\frac{12}{5}$ , $\\cos B=-\\frac{3}{5}$ and that $A$ and $B$ are in the same quadrant, find the value of $\\cos (A-B)$ . 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": 28314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a right triangle with sides of length $5$, $12$, and $13$, and a square with side length $x$ inscribed in it so that one vertex of the square coincides with the right-angle vertex of the triangle, and another square with side length $y$ inscribed in a different right tria... | math | {
"style": "rule",
"ground_truth": "\\frac{39}{51}"
} | {
"split": "train",
"index": 28315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive number that is prime and $10$ less than a perfect square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "71"
} | {
"split": "train",
"index": 28316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair 6-sided die is rolled. What is the probability that the number rolled is a divisor of 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac23"
} | {
"split": "train",
"index": 28317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right triangle has perimeter $32$ and area $20$. What is the length of its hypotenuse? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{59}{4}"
} | {
"split": "train",
"index": 28318,
"solution": "\nLet the legs of the triangle be $a$ and $b$, and the hypotenuse be $c$. We are given that the perimeter is 32 and the area is 20. Therefore, we have the following equations:\n\n1. \\(a + b + c = 32\\)\n2. \\(\\frac{1}{2}ab = 20 \\Rightarrow ab = 40\\)\n\nUsing th... |
deepscaler | [
{
"role": "user",
"content": "A set of $36$ square blocks is arranged into a $6 \\times 6$ square. How many different combinations of $4$ blocks can be selected from that set so that no two are in the same row or column? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5400"
} | {
"split": "train",
"index": 28319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the binomial coefficients in the expansion of $(5x- \\frac{1}{\\sqrt{x}})^n$ is 64, determine the constant term in its expansion. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "375"
} | {
"split": "train",
"index": 28320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given vectors $\\overrightarrow{m} = (2\\sin B, -\\sqrt{3})$ and $\\overrightarrow{n} = (\\cos 2B, 2\\cos^2 B - 1)$ and $\\overrightarrow{m} \\parallel \\overrightarrow{n}$:\n... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 28321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\tan\\alpha= \\dfrac {1}{3}$, find the values of the following expressions: \n1. $\\dfrac {\\sin \\alpha+\\cos\\alpha}{5\\cos\\alpha-\\sin\\alpha}$\n2. $\\dfrac {1}{2\\sin\\alpha\\cdot \\cos\\alpha+\\cos ^{2}\\alpha}$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\dfrac {2}{3}"
} | {
"split": "train",
"index": 28322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of two nonzero real numbers is $4$ times their product. What is the sum of the reciprocals of the two numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 28323,
"solution": "1. **Define the variables**: Let the two nonzero real numbers be $x$ and $y$.\n\n2. **Translate the given condition into an equation**: We are given that the sum of $x$ and $y$ is $4$ times their product. This can be written as:\n \\[\n x + y = 4xy\n \\]\n\n3... |
deepscaler | [
{
"role": "user",
"content": "A product of five primes is of the form $A B C, A B C$, where $A, B$, and $C$ represent digits. If one of the primes is 491, find the product $A B C, A B C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "982,982"
} | {
"split": "train",
"index": 28324,
"solution": "$491 \\cdot 1001 \\cdot 2=982,982$."
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x)=\\sqrt{x-3}$, what is the smallest real number $x$ in the domain of $f(f(x))$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 28325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function $g$ is defined by $g(z) = (3 - 2i) z^2 + \\beta z + \\delta$ for all complex numbers $z$, where $\\beta$ and $\\delta$ are complex numbers and $i^2 = -1$. Suppose that $g(1)$ and $g(-i)$ are both real. What is the smallest possible value of $|\\beta| + |\\delta|$? Le... | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"split": "train",
"index": 28326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c,$ $x,$ $y,$ $z$ be nonzero complex numbers such that\n\\[a = \\frac{b + c}{x - 2}, \\quad b = \\frac{a + c}{y - 2}, \\quad c = \\frac{a + b}{z - 2},\\]and $xy + xz + yz = 5$ and $x + y + z = 3,$ find $xyz.$ Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 28327,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_1=1$, $a_2=3$, if $|a_{n+1}-a_n|=2^n$ $(n\\in\\mathbb{N}^*)$, and the sequence $\\{a_{2n-1}\\}$ is increasing while $\\{a_{2n}\\}$ is decreasing, then $\\lim\\limits_{n\\to\\infty} \\frac{a_{2n-1}}{a_{2n}}=$ ______. Let's think step by... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 28328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Piravena must make a trip from $A$ to $B,$ then from $B$ to $C,$ then from $C$ to $A.$ Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with $C$ a distance of $3000\\text{ km}$ from $A$ and ... | math | {
"style": "rule",
"ground_truth": "\\$425"
} | {
"split": "train",
"index": 28329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest possible value of $n$ if Juliana chooses three different numbers from the set $\\{-6,-4,-2,0,1,3,5,7\\}$ and multiplies them together to obtain the integer $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 28330,
"solution": "Since $3 \\times 5 \\times 7=105$, then the greatest possible value of $n$ is at least 105. For the product of three numbers to be positive, either all three numbers are positive or one number is positive and two numbers are negative. If all three numbers are posit... |
deepscaler | [
{
"role": "user",
"content": "What is the smallest four-digit positive integer that is divisible by 47? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1034"
} | {
"split": "train",
"index": 28331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 28332,
"solution": "\nGiven 7 arbitrary points in the plane, we need to determine the maximum number of circles that can be drawn through every 4 possible concyclic points.\n\nTo solve this, we consider the combinatorial aspect of selecting 4 points out of 7. The number of ways to cho... |
deepscaler | [
{
"role": "user",
"content": "If $x^2 + \\frac{1}{x^2} = A,$ and $x - \\frac{1}{x} = B,$ where $A$ and $B$ are positive numbers, find the minimum possible numerical value for $\\frac{A}{B}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{2}"
} | {
"split": "train",
"index": 28333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f : \\mathbb{R} \\to \\mathbb{R}$ such that\n\\[f(x(x + f(y))) = (x + y)f(x),\\]\nfor all $x, y \\in\\mathbb{R}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = 0 \\text{ and } f(x) = x"
} | {
"split": "train",
"index": 28334,
"solution": "\nLet's consider the functional equation \\( f(x(x + f(y))) = (x + y)f(x) \\) for all \\( x, y \\in \\mathbb{R} \\).\n\n### Step 1: Test simple functions\nFirst, let's test the simplest potential solutions.\n\n1. **\\( f(x) = 0 \\):**\n - Substituting \\( f(x) = ... |
deepscaler | [
{
"role": "user",
"content": "A positive integer $n$ not exceeding $120$ is chosen such that if $n\\le 60$, then the probability of choosing $n$ is $p$, and if $n > 60$, then the probability of choosing $n$ is $2p$. The probability that a perfect square is chosen is?\nA) $\\frac{1}{180}$\nB) $\\frac{7}{180}... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{180}"
} | {
"split": "train",
"index": 28335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a $3 \\times 4$ grid, you need to place 4 crosses so that there is at least one cross in each row and each column. How many ways are there to do this? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 28336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of the following expressions: \n(1) $$\\sqrt[4]{(3-\\pi )^{4}}$$ + (0.008) $$^{- \\frac {1}{3}}$$ - (0.25) $$^{ \\frac {1}{2}}$$ × ($$\\frac {1}{ \\sqrt {2}}$$)$$^{-4}$$; \n(2) $\\log_{3}$$\\sqrt {27}$$ - $\\log_{3}$$\\sqrt {3}$$ - $\\lg{625}$ - $\\lg{4}$ ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 28337,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin(2x+\\varphi)$ where $(0 < \\varphi < \\pi)$ satisfies $f(x) \\leq |f(\\frac{\\pi}{6})|$, and $f(x_{1}) = f(x_{2}) = -\\frac{3}{5}$, calculate the value of $\\sin(x_{2}-x_{1})$. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 28338,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all natural numbers $n (n \\geq 2)$ such that there exists reals $a_1, a_2, \\dots, a_n$ which satisfy \\[ \\{ |a_i - a_j| \\mid 1\\leq i<j \\leq n\\} = \\left\\{1,2,\\dots,\\frac{n(n-1)}{2}\\right\\}. \\]\r\n\r\nLet $A=\\{1,2,3,4,5,6\\}, B=\\{7,8,9,\\dots,n\\}$. $A_i(i=1,... | math | {
"style": "rule",
"ground_truth": "2, 3, 4"
} | {
"split": "train",
"index": 28339,
"solution": "\n\nWe need to find all natural numbers \\( n \\) (where \\( n \\geq 2 \\)) such that there exist real numbers \\( a_1, a_2, \\dots, a_n \\) which satisfy the condition:\n\\[ \\{ |a_i - a_j| \\mid 1 \\leq i < j \\leq n \\} = \\left\\{ 1, 2, \\dots, \\frac{n(n-1)}{2... |
deepscaler | [
{
"role": "user",
"content": "Determine the value of the expression\n\\[\\log_3 (64 + \\log_3 (64 + \\log_3 (64 + \\cdots))),\\]\nassuming it is positive. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3.8"
} | {
"split": "train",
"index": 28340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of \\(a + b + c + d\\) if\n\n$$\n\\begin{gathered}\n6a + 2b = 3848 \\\\\n6c + 3d = 4410 \\\\\na + 3b + 2d = 3080\n\\end{gathered}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1986"
} | {
"split": "train",
"index": 28341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ such that\n$$f(xy) = f(x)f(y) + f(f(x + y))$$\nholds for all $x, y \\in \\mathbb{R}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = 0 \\text{ and } f(x) = x - 1."
} | {
"split": "train",
"index": 28342,
"solution": "\nWe are tasked with finding all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) satisfying the equation:\n\n\\[\nf(xy) = f(x)f(y) + f(f(x+y))\n\\]\n\nfor all \\( x, y \\in \\mathbb{R} \\).\n\n### Step 1: Consideration of Simple Cases\n\nFirst, let us con... |
deepscaler | [
{
"role": "user",
"content": "Given the positive integer \\( A = \\overline{a_{n} a_{n-1} \\cdots a_{1} a_{0}} \\), where \\( a_{n}, a_{n-1}, \\ldots, a_{0} \\) are all non-zero and not all equal (with \\( n \\) being a positive integer), consider the following cyclic permutations of \\( A \\):\n$$\n\\begin... | math | {
"style": "rule",
"ground_truth": "142857"
} | {
"split": "train",
"index": 28343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If I expand $25\\cdot 24\\cdot 23 \\cdots 3 \\cdot 2 \\cdot 1$, how many zeros are there at the end of the number I get? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 28344,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $y^{2}=2x$ and a point $P(m,2)$ on it, find the value of $m$ and the distance between point $P$ and the focus $F$ of the parabola. 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": 28345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, four different numbers are selected, denoted as $a$, $b$, $c$, $d$ respectively. If the parity of $a+b$ is the same as the parity of $c+d$, then the total number of ways to select $a$, $b$, $c$, $d$ is ______ (provide the... | math | {
"style": "rule",
"ground_truth": "912"
} | {
"split": "train",
"index": 28346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a rectangular grid constructed with toothpicks of equal length, with a height of 15 toothpicks and a width of 12 toothpicks, calculate the total number of toothpicks required to build the grid. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "387"
} | {
"split": "train",
"index": 28347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2\\cos x(\\sin x-\\cos x)$, where $x\\in R$.\n(I) Find the symmetry center of the graph of the function $f(x)$;\n(II) Find the minimum and maximum values of the function $f(x)$ on the interval $[\\frac{\\pi}{8}, \\frac{3\\pi}{4}]$. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 28348,
"solution": ""
} |
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