data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In tetrahedron \\(ABCD\\), the dihedral angle between face \\(ABC\\) and face \\(BCD\\) is \\(60^\\circ\\). The orthogonal projection of vertex \\(A\\) onto face \\(BCD\\) is \\(H\\), which is the orthocenter of \\(\\triangle BCD\\). Point \\(G\\) is the centroid of \\(\\triang... | math | {
"style": "rule",
"ground_truth": "\\frac{4 \\sqrt{21}}{9}"
} | {
"split": "train",
"index": 1300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the probability of germination for each seed be 0.9, and 1000 seeds have been planted. For each seed that does not germinate, 2 more seeds need to be replanted. Let X be the number of replanted seeds. Calculate the expected value of X. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 1301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The table shows the vertical drops of six roller coasters in Fibonacci Fun Park: \n\\begin{tabular}{|l|c|}\n\\hline\nSpeed Demon & 150 feet \\\\ \n\\hline\nLooper & 230 feet \\\\ \n\\hline\nDare Devil & 160 feet \\\\ \n\\hline\nGiant Drop & 190 feet \\\\ \n\\hline\nSky Scream &... | math | {
"style": "rule",
"ground_truth": "1.67"
} | {
"split": "train",
"index": 1302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the volume of the parallelepiped formed by vectors $\\mathbf{a}$, $\\mathbf{b}$, and $\\mathbf{c}$ is 4, find the volume of the parallelepiped formed by the vectors $\\mathbf{2a} + \\mathbf{b}$, $\\mathbf{b} + 4\\mathbf{c}$, and $\\mathbf{c} - 5\\mathbf{a}$. Let's th... | math | {
"style": "rule",
"ground_truth": "232"
} | {
"split": "train",
"index": 1303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer greater than 1 that leaves a remainder of 1 when divided by each of 2, 3, 4, 5, 6, 7, 8 and 9? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2521"
} | {
"split": "train",
"index": 1304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a_0 = \\sin^2 \\left( \\frac{\\pi}{45} \\right)$ and\n\\[a_{n + 1} = 4a_n (1 - a_n)\\]for $n \\ge 0,$ find the smallest positive integer $n$ such that $a_n = a_0.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 1305,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $k$ be a real number such that $k > 1$ and\n\\[\\sum_{n=1}^{\\infty} \\frac{5n-1}{k^n} = \\frac{13}{4}.\\]Find $k.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 1306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures $8$ inches high and $10$ inches wide. What is the area of the border, in square inches? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 1307,
"solution": "1. **Calculate the area of the photograph**: \n The photograph measures $8$ inches in height and $10$ inches in width. Therefore, the area of the photograph is calculated as:\n \\[\n \\text{Area of photograph} = 8 \\times 10 = 80 \\text{ square inches}\n \\]... |
deepscaler | [
{
"role": "user",
"content": "What is the constant term of the expansion of $\\left(5x + \\frac{2}{5x}\\right)^8$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1120"
} | {
"split": "train",
"index": 1308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The real numbers $a,$ $b,$ $c,$ and $d$ satisfy\n\\[a^2 + b^2 + c^2 + 1 = d + \\sqrt{a + b + c - d}.\\]Find $d.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"split": "train",
"index": 1309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x \\cdot \\log_{27} 64 = 1$, then $4^x + 4^{-x} =$ \\_\\_\\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{10}{3}"
} | {
"split": "train",
"index": 1310,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\\begin{cases} x=3+t\\cos \\alpha \\\\ y=1+t\\sin \\alpha\\end{cases}$ (where $t$ is the parameter), in the polar coordinate system (with the same unit length as the Cartesian coordinate system $x... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {2}"
} | {
"split": "train",
"index": 1311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the volume of a regular triangular pyramid $P-ABC$ is $\\frac{1}{12}$, the center of its circumscribed sphere is $O$, and it satisfies $\\vec{OA} + \\vec{OB} + \\vec{OC} = \\vec{0}$, find the radius of the circumscribed sphere of the regular triangular pyramid $P-ABC... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 1312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five volunteers participate in community service for two days, Saturday and Sunday. Each day, two people are selected to serve. Find the number of ways to select exactly one person to serve for both days. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 1313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Analogous to the exponentiation of rational numbers, we define the division operation of several identical rational numbers (all not equal to $0$) as \"division exponentiation,\" denoted as $a^{ⓝ}$, read as \"$a$ circle $n$ times.\" For example, $2\\div 2\\div 2$ is denoted as ... | math | {
"style": "rule",
"ground_truth": "-73"
} | {
"split": "train",
"index": 1314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of length $19$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 1315,
"solution": "1. **Understanding the Triangle Inequality Theorem**: The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For a triangle with sides $a$, $b$, and $c$, this can be expres... |
deepscaler | [
{
"role": "user",
"content": "Given $\\delta(x) = 3x + 8$ and $\\phi(x) = 8x + 7$, what is $x$ if $\\delta(\\phi(x)) = 7$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{11}{12}"
} | {
"split": "train",
"index": 1316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The greatest common divisor of two integers is $(x+2)$ and their least common multiple is $x(x+2)$, where $x$ is a positive integer. If one of the integers is 24, what is the smallest possible value of the other one? Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 1317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A child builds towers using identically shaped cubes of different colors. How many different towers with a height 8 cubes can the child build with 2 red cubes, 3 blue cubes, and 4 green cubes? (One cube will be left out.) Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "1260"
} | {
"split": "train",
"index": 1318,
"solution": "1. **Identify the Total Number of Cubes and Colors**: The child has a total of $2$ red cubes, $3$ blue cubes, and $4$ green cubes, making a total of $9$ cubes. However, the tower must be built with only $8$ cubes, so one cube will be left out.\n\n2. **Calculate the ... |
deepscaler | [
{
"role": "user",
"content": "Determine the number of six-letter words where the first and last two letters are the same (e.g., \"aabbaa\"). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "456976"
} | {
"split": "train",
"index": 1319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A shooter has a probability of hitting the target of $0.8$ each time. Now, using the method of random simulation to estimate the probability that the shooter hits the target at least $3$ times out of $4$ shots: first, use a calculator to generate random integers between $0$ and... | math | {
"style": "rule",
"ground_truth": "0.75"
} | {
"split": "train",
"index": 1320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $F$ is taken on the extension of side $AD$ of parallelogram $ABCD$. $BF$ intersects diagonal $AC$ at $E$ and side $DC$ at $G$. If $EF = 40$ and $GF = 30$, find the length of $BE$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 1321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point \\( M \\) is chosen on the diameter \\( AB \\). Points \\( C \\) and \\( D \\), lying on the circumference on one side of \\( AB \\), are chosen such that \\(\\angle AMC=\\angle BMD=30^{\\circ}\\). Find the diameter of the circle given that \\( CD=12 \\). Let's think st... | math | {
"style": "rule",
"ground_truth": "8\\sqrt{3}"
} | {
"split": "train",
"index": 1322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the shortest distance between the circles defined by $x^2-10x +y^2-4y-7=0$ and $x^2+14x +y^2+6y+49=0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 1323,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $y(y-3x)$ for $x=3$ and $y=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 1324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Chloe wants to buy a hoodie that costs $\\$32.75$. She empties her wallet and finds she only has three $\\$10$ bills, eight quarters, and a pile of dimes. What is the minimum number of dimes that must be in her pile so she can pay for the hoodie? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 1325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $r$ be a real number. Two of the roots of $f(x)$ are $r + 1$ and $r + 7.$ Two of the roots of $g(x)$ are $r + 3$ and $r + 9,$ and\n\\[f(x) - g(x) = r\\]for all real numbers $x.$ Find $r.$ Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 1326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different 4-edge trips are there from $A$ to $B$ in a cube, where the trip can visit one vertex twice (excluding start and end vertices)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 1327,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is the center $O$ of a sphere with radius 10. The sphere intersects the face $A A_{1} D_{1} D$ along a circle with radius 1, the face $A_{1} B_{1} C_{1} D_{1}$ along a circle with radius 1, and the face $C D D_{1} C_{1}$ along a... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 1328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum number of angles less than $150^{\\circ}$ that a non-self-intersecting 2017-sided polygon can have if all its angles are strictly less than $180^{\\circ}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 1329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{b_n\\}$ with 8 terms satisfying $b_1=2014$, $b_8=2015$, and $b_{n+1}-b_n \\in \\{-1, \\frac{1}{3}, 1\\}$ (where $n=1,2,\\ldots,7$), determine the total number of such sequences $\\{b_n\\}$. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "252"
} | {
"split": "train",
"index": 1330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_{n}(n\\geqslant 2, n\\in N^{*})$ be the coefficient of the linear term of $x$ in the expansion of ${({3-\\sqrt{x}})^n}$. Find the value of $\\frac{3^2}{a_2}+\\frac{3^3}{a_3}+…+\\frac{{{3^{18}}}}{{{a_{18}}}}$. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 1331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A barcode is composed of alternate strips of black and white, where the leftmost and rightmost strips are always black. Each strip (of either color) has a width of 1 or 2. The total width of the barcode is 12. The barcodes are always read from left to right. How many distinct b... | math | {
"style": "rule",
"ground_truth": "116"
} | {
"split": "train",
"index": 1332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line $L_{1}$: $ax+(1-a)y=3$ and $L_{2}$: $(a-1)x+(2a+3)y=2$ are perpendicular to each other, find the values of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 1333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest digit $N$ for which $2345N$ is divisible by 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 1334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 10 boys, each with different weights and heights. For any two boys $\\mathbf{A}$ and $\\mathbf{B}$, if $\\mathbf{A}$ is heavier than $\\mathbf{B}$, or if $\\mathbf{A}$ is taller than $\\mathbf{B}$, we say that \" $\\mathrm{A}$ is not inferior to B\". If a boy is not i... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 1335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly $231$ of the 1-cm cubes cannot be seen. Find the smallest possible value of $N.$ Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "384"
} | {
"split": "train",
"index": 1336,
"solution": "The $231$ cubes which are not visible must lie below exactly one layer of cubes. Thus, they form a rectangular solid which is one unit shorter in each dimension. If the original block has dimensions $l \\times m \\times n$, we must have $(l - 1)\\times(m-1) \\times(... |
deepscaler | [
{
"role": "user",
"content": "Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is $\\tfrac{1}{2}$,... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 1337,
"solution": "1. **Identify the Event of Larry Winning**: Larry can win on his first, third, fifth, etc., turns. This is because he throws first and then alternates with Julius. If Larry wins on the $n$-th turn, it must be that $n$ is odd.\n\n2. **Define the Events**: Let $W$ rep... |
deepscaler | [
{
"role": "user",
"content": "A package of seeds was passed around a table. The first person took 1 seed, the second person took 2 seeds, the third took 3 seeds, and so forth, with each subsequent person taking one more seed than the previous one. It is known that during the second round a total of 100 more... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 1338,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of an infinite geometric series is $27$ times the series that results if the first three terms of the original series are removed. What is the value of the series' common ratio? 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": 1339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, with $b=1$, and $2\\cos C - 2a - c = 0$.\n\n(Ⅰ) Find the magnitude of angle $B$; \n(Ⅱ) Find the distance from the circumcenter of $\\triangle ABC$ to side $AC$. Let's think ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{6}"
} | {
"split": "train",
"index": 1340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\sum_{n = 0}^{\\infty}\\sin^{2n}\\theta = 4$, what is the value of $\\sin{2\\theta}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 1341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function f(x) = sinωx + cosωx, if there exists a real number x₁ such that for any real number x, f(x₁) ≤ f(x) ≤ f(x₁ + 2018) holds true, find the minimum positive value of ω. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2018}"
} | {
"split": "train",
"index": 1342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer cannot be divisible by 2 or 3, and there do not exist non-negative integers \\(a\\) and \\(b\\) such that \\(|2^a - 3^b| = n\\). Find the smallest value of \\(n\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 1343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a 4 by 4 grid, each of the 16 small squares measures 3 cm by 3 cm and is shaded. Four unshaded circles are then placed on top of the grid, one in each quadrant. The area of the visible shaded region can be written in the form $A-B\\pi$ square cm. What is the value of $A+B$? ... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 1344,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What two digits need to be added to the right of the number 2013 to make the resulting six-digit number divisible by 101? Find all possible answers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "94"
} | {
"split": "train",
"index": 1345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider five-dimensional Cartesian space $\\mathbb{R}^{5}=\\left\\{\\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\\right) \\mid x_{i} \\in \\mathbb{R}\\right\\}$ and consider the hyperplanes with the following equations: - $x_{i}=x_{j}$ for every $1 \\leq i<j \\leq 5$; - $x_{1}+x_{2... | math | {
"style": "rule",
"ground_truth": "480"
} | {
"split": "train",
"index": 1346,
"solution": "Note that given a set of plane equations $P_{i}\\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\\right)=0$, for $i=1,2, \\ldots, n$, each region that the planes separate the space into correspond to a $n$-tuple of -1 and 1 , representing the sign of $P_{1}, P_{2}, \\ldots P... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left(x\\right)=x-{e}^{-x}$, if the line $y=mx+n$ is a tangent line to the curve $y=f\\left(x\\right)$, find the minimum value of $m+n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1-\\dfrac{1}{e}"
} | {
"split": "train",
"index": 1347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For which values of the parameter \\(a\\) does the equation \\(x^{4} - 40 x^{2} + 144 = a(x^{2} + 4x - 12)\\) have exactly three distinct solutions? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 1348,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate\n\n$$\n\\frac{1 \\times 2 \\times 4+2 \\times 4 \\times 8+3 \\times 6 \\times 12+4 \\times 8 \\times 16}{1 \\times 3 \\times 9+2 \\times 6 \\times 18+3 \\times 9 \\times 27+4 \\times 12 \\times 36}\n$$\n\nOnly a numerical answer is expected here. The answer must be gi... | math | {
"style": "rule",
"ground_truth": "8/27"
} | {
"split": "train",
"index": 1349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For all \\( m \\) and \\( n \\) satisfying \\( 1 \\leq n \\leq m \\leq 5 \\), the polar equation\n$$\n\\rho = \\frac{1}{1 - C_{m}^{n} \\cos \\theta}\n$$\nrepresents how many different hyperbolas? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 1350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Carefully observe the following three rows of related numbers:<br/>First row: $-2$, $4$, $-8$, $16$, $-32$, $\\ldots$;<br/>Second row: $0$, $6$, $-6$, $18$, $-30$, $\\ldots$;<br/>Third row: $-1$, $2$, $-4$, $8$, $-16$, $\\ldots$;<br/>Answer the following questions:<br/>$(1)$ Th... | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 1351,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are real numbers $a, b, c,$ and $d$ such that $-20$ is a root of $x^3 + ax + b$ and $-21$ is a root of $x^3 + cx^2 + d.$ These two polynomials share a complex root $m + \\sqrt{n} \\cdot i,$ where $m$ and $n$ are positive integers and $i = \\sqrt{-1}.$ Find $m+n.$ Let's th... | math | {
"style": "rule",
"ground_truth": "330"
} | {
"split": "train",
"index": 1352,
"solution": "We plug -20 into the equation obtaining $(-20)^3-20a+b$, likewise, plugging -21 into the second equation gets $(-21)^3+441c+d$.\n Both equations must have 3 solutions exactly, so the other two solutions must be $m + \\sqrt{n} \\cdot i$ and $m - \\sqrt{n} \\cdot i$.\... |
deepscaler | [
{
"role": "user",
"content": "If $a,b>0$ and the triangle in the first quadrant bounded by the coordinate axes and the graph of $ax+by=6$ has area 6, then $ab=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 1353,
"solution": "1. **Identify the intercepts**: \n The equation of the line is given by $ax + by = 6$. To find the $x$-intercept, set $y = 0$:\n \\[\n ax = 6 \\implies x = \\frac{6}{a}.\n \\]\n Similarly, to find the $y$-intercept, set $x = 0$:\n \\[\n by = 6 \\implie... |
deepscaler | [
{
"role": "user",
"content": "Find all \\( t \\) such that \\( x-t \\) is a factor of \\( 10x^2 + 21x - 10 \\). 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": 1354,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given triangle $\\triangle ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, and $b \\sin A = \\sqrt{3} a \\cos B$.\n\n1. Find the measure of angle $B$.\n2. If $b = 3$ and $\\sin C = 2 \\sin A$, find the values of $a$ and $c$. Let's think step by... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 1355,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a function $f(x)$ that satisfies $f(x+3)=-f(x)$, when $x\\in \\left[-3,0\\right)$, $f(x)=2^{x}+\\sin \\frac{πx}{3}$, determine the value of $f(2023)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4} + \\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 1356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the curve $y=\\sin x$ ($0 \\leq x \\leq \\pi$) and the line $y= \\frac {1}{2}$, calculate the area of the enclosed shape formed by these two functions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}- \\frac {\\pi}{3}"
} | {
"split": "train",
"index": 1357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a round-robin chess tournament, only grandmasters and masters participated. The number of masters was three times the number of grandmasters, and the total points scored by the masters was 1.2 times the total points scored by the grandmasters.\n\nHow many people participated... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 1358,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(A B C, A B=6, B C=7\\) and \\(C A=8\\). Let \\(D, E, F\\) be the midpoints of sides \\(B C\\), \\(A C, A B\\), respectively. Also let \\(O_{A}, O_{B}, O_{C}\\) be the circumcenters of triangles \\(A F D, B D E\\), and \\(C E F\\), respectively. Find the area of t... | math | {
"style": "rule",
"ground_truth": "\\frac{21 \\sqrt{15}}{16}"
} | {
"split": "train",
"index": 1359,
"solution": "Let \\(A B=z, B C=x, C A=y\\). Let \\(X, Y, Z, O, N\\) be the circumcenter of \\(A E F, B F D, C D E, A B C, D E F\\) respectively. Note that \\(N\\) is the nine-point center of \\(A B C\\), and \\(X, Y, Z\\) are the midpoints of \\(O A, O B, O C\\) respectively, an... |
deepscaler | [
{
"role": "user",
"content": "[asy]\ndraw((-7,0)--(7,0),black+linewidth(.75));\ndraw((-3*sqrt(3),0)--(-2*sqrt(3),3)--(-sqrt(3),0)--(0,3)--(sqrt(3),0)--(2*sqrt(3),3)--(3*sqrt(3),0),black+linewidth(.75));\ndraw((-2*sqrt(3),0)--(-1*sqrt(3),3)--(0,0)--(sqrt(3),3)--(2*sqrt(3),0),black+linewidth(.75));\n[/asy]\nF... | math | {
"style": "rule",
"ground_truth": "12\\sqrt{3}"
} | {
"split": "train",
"index": 1360,
"solution": "1. **Calculate the area of one large equilateral triangle**: \n The formula for the area of an equilateral triangle with side length $s$ is given by:\n \\[\n \\text{Area} = \\frac{\\sqrt{3}}{4} s^2\n \\]\n Substituting $s = 2\\sqrt{3}$, we get:\n \\[\n ... |
deepscaler | [
{
"role": "user",
"content": "Three of the four vertices of a square are $(2, 8)$, $(13, 8)$, and $(13, -3)$. What is the area of the intersection of this square region and the region inside the graph of the equation $(x - 2)^2 + (y + 3)^2 = 16$? Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"split": "train",
"index": 1361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $a,b$ such that $b^{619}$ divides $a^{1000}+1$ and $a^{619}$ divides $b^{1000}+1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(1, 1)"
} | {
"split": "train",
"index": 1362,
"solution": "\nLet us find all positive integers \\( a, b \\) such that the following conditions are satisfied:\n1. \\( b^{619} \\mid a^{1000} + 1 \\)\n2. \\( a^{619} \\mid b^{1000} + 1 \\)\n\nTo find these solutions, we start by analyzing the given divisibility conditions. \n\n... |
deepscaler | [
{
"role": "user",
"content": "Suppose $x$ and $y$ are inversely proportional and positive. If $x$ increases by $p\\%$, then $y$ decreases by Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{100p}{100+p}\\%$"
} | {
"split": "train",
"index": 1363,
"solution": "1. **Understanding Inverse Proportionality**: Given that $x$ and $y$ are inversely proportional, we can express this relationship as:\n \\[\n xy = k\n \\]\n for some constant $k$.\n\n2. **Change in $x$**: If $x$ increases by $p\\%$, the new value of $x$, den... |
deepscaler | [
{
"role": "user",
"content": "How many 5-digit numbers have at least one zero? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30,951"
} | {
"split": "train",
"index": 1364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jimmy finds that 8 bananas weigh the same as 6 oranges. If Jimmy has 40 oranges, and the weight of 4 oranges equals the weight of 5 apples, how many apples and bananas would Jimmy need to equal the weight of his 40 oranges? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 1365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of ways to arrange the letters of the word PROOF. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 1366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_{1}, a_{2}, \\ldots, a_{n}$ be a sequence of distinct positive integers such that $a_{1}+a_{2}+\\cdots+a_{n}=2021$ and $a_{1} a_{2} \\cdots a_{n}$ is maximized. If $M=a_{1} a_{2} \\cdots a_{n}$, compute the largest positive integer $k$ such that $2^{k} \\mid M$. Let's th... | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 1367,
"solution": "We claim that the optimal set is $\\{2,3, \\cdots, 64\\} \\backslash\\{58\\}$. We first show that any optimal set is either of the form $\\{b, b+1, b+2, \\ldots, d\\}$ or $\\{b, b+1, \\ldots, d\\} \\backslash\\{c\\}$, for some $b<c<d$. Without loss of generality, as... |
deepscaler | [
{
"role": "user",
"content": "A truck travels $\\frac{b}{6}$ feet every $t$ seconds. There are $3$ feet in a yard. How many yards does the truck travel in $3$ minutes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10b}{t}"
} | {
"split": "train",
"index": 1368,
"solution": "1. **Identify the rate of travel**: The truck travels $\\frac{b}{6}$ feet every $t$ seconds.\n\n2. **Convert time from minutes to seconds**: Since there are $60$ seconds in a minute, $3$ minutes is $3 \\times 60 = 180$ seconds.\n\n3. **Set up a proportion to find th... |
deepscaler | [
{
"role": "user",
"content": "Five students, A, B, C, D, and E, participated in a labor skills competition. A and B asked about the results. The respondent told A, \"Unfortunately, neither you nor B got first place.\" To B, the respondent said, \"You certainly are not the worst.\" Determine the number of di... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 1369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Call a $7$-digit telephone number $d_1d_2d_3-d_4d_5d_6d_7$ memorable if the prefix sequence $d_1d_2d_3$ is exactly the same as either of the sequences $d_4d_5d_6$ or $d_5d_6d_7$ (possibly both). Assuming that each $d_i$ can be any of the ten decimal digits $0, 1, 2, \\ldots, 9$... | math | {
"style": "rule",
"ground_truth": "19990"
} | {
"split": "train",
"index": 1370,
"solution": "To solve this problem, we need to count the number of memorable telephone numbers based on the given conditions. A memorable telephone number is defined as $d_1d_2d_3-d_4d_5d_6d_7$ where the sequence $d_1d_2d_3$ matches either $d_4d_5d_6$ or $d_5d_6d_7$.\n\n1. **Cou... |
deepscaler | [
{
"role": "user",
"content": "How many pairs of positive integers $(x,y)$ satisfy $x^2-y^2=51$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 1371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify\n\\[\\frac{\\tan 30^\\circ + \\tan 40^\\circ + \\tan 50^\\circ + \\tan 60^\\circ}{\\cos 20^\\circ}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8 \\sqrt{3}}{3}"
} | {
"split": "train",
"index": 1372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\tan \\alpha +\\tan \\beta -\\tan \\alpha \\tan \\beta +1=0$, and $\\alpha ,\\beta \\in \\left(\\frac{\\pi }{2},\\pi \\right)$, calculate $\\alpha +\\beta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7\\pi}{4}"
} | {
"split": "train",
"index": 1373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. Let S be the area of triangle ABC. If 3a² = 2b² + c², find the maximum value of $\\frac{S}{b^{2}+2c^{2}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{14}}{24}"
} | {
"split": "train",
"index": 1374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what real number \\\\(m\\\\) is the complex number \\\\(z=m^{2}+m-2+(m^{2}-1)i\\\\) \n\\\\((1)\\\\) a real number; \\\\((2)\\\\) an imaginary number; \\\\((3)\\\\) a pure imaginary number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 1375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$, $y$, and $z$ be real numbers greater than $1$, and let $z$ be the geometric mean of $x$ and $y$. The minimum value of $\\frac{\\log z}{4\\log x} + \\frac{\\log z}{\\log y}$ is \\_\\_\\_\\_\\_\\_. 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": 1376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a 3x3 matrix where each row and each column forms an arithmetic sequence, and the middle element $a_{22} = 5$, find the sum of all nine elements. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 1377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the definition that an arithmetic sequence is a sequence in which the difference between any term and its preceding term is a constant. For example, the sequence $2, 4, 6, 8, \\ldots$ is an arithmetic sequence. A second-order arithmetic sequence is defined as a sequence i... | math | {
"style": "rule",
"ground_truth": "4050157"
} | {
"split": "train",
"index": 1378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by $30$. Find the sum of the four terms. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "129"
} | {
"split": "train",
"index": 1379,
"solution": "The sequence is of the form $a-d,$ $a,$ $a+d,$ $\\frac{(a+d)^2}{a}$. Since the first and last terms differ by 30, we have \\[\\frac{(a+d)^2}{a}-a+d=30\\] \\[d^2+3ad=30a\\] \\[d^2+3ad-30a=0\\] \\[d=\\frac{-3a + \\sqrt{9a^2+120a}}{2}.\\] Let $9a^2+120a=x^2$, where $x$... |
deepscaler | [
{
"role": "user",
"content": "A regular hexagon is inscribed in a circle and another regular hexagon is circumscribed about the same circle. What is the ratio of the area of the larger hexagon to the area of the smaller hexagon? Express your answer as a common fraction. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 1380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A random sample of $10$ households was taken from a residential area, and the monthly income data ${x_i}$ (in units of thousand yuan) of the $i$-th household was obtained. The following statistical values were calculated: $\\sum_{i=1}^{10} x_i = 80$, $\\sum_{i=1}^{10} y_i = 20$... | math | {
"style": "rule",
"ground_truth": "1.7"
} | {
"split": "train",
"index": 1381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two vectors in space, $\\overrightarrow{a} = (x - 1, 1, -x)$ and $\\overrightarrow{b} = (-x, 3, -1)$. If $\\overrightarrow{a}$ is perpendicular to $\\overrightarrow{b}$, find the value of $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 1382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), given that \\( \\frac{\\cos A}{\\sin B} + \\frac{\\cos B}{\\sin A} = 2 \\) and the perimeter of \\( \\triangle ABC \\) is 12, find the maximum possible area of the triangle. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "36(3 - 2\\sqrt{2})"
} | {
"split": "train",
"index": 1383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the $87^{\\mathrm{th}}$ odd positive integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "173"
} | {
"split": "train",
"index": 1384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Joanie takes a $\\$6,\\!000$ loan to pay for her car. The annual interest rate on the loan is $12\\%$. She makes no payments for 4 years, but has to pay back all the money she owes at the end of 4 years. How much more money will she owe if the interest compounds quarterly tha... | math | {
"style": "rule",
"ground_truth": "\\$187.12"
} | {
"split": "train",
"index": 1385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all nonnegative integer solutions $(x,y,z,w)$ of the equation\\[2^x\\cdot3^y-5^z\\cdot7^w=1.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(1, 1, 1, 0), (2, 2, 1, 1), (1, 0, 0, 0), (3, 0, 0, 1)"
} | {
"split": "train",
"index": 1386,
"solution": "\nWe are tasked with finding all nonnegative integer solutions \\((x, y, z, w)\\) to the equation:\n\\[\n2^x \\cdot 3^y - 5^z \\cdot 7^w = 1.\n\\]\n\nFirst, we note that \\(x \\geq 1\\) because if \\(x = 0\\), the left-hand side would be a fraction, which cannot equ... |
deepscaler | [
{
"role": "user",
"content": "On the trip home from the meeting where this AMC $ 10$ was constructed, the Contest Chair noted that his airport parking receipt had digits of the form $ bbcac$, where $ 0 \\le a < b < c \\le 9$, and $ b$ was the average of $ a$ and $ c$. How many different five-digit numbers s... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 1387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given square $ABCD$ with side $8$ feet. A circle is drawn through vertices $A$ and $D$ and tangent to side $BC$. The radius of the circle, in feet, is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 1388,
"solution": "1. **Identify the Geometry and Setup**: We are given a square $ABCD$ with side length $8$ feet. A circle passes through vertices $A$ and $D$ and is tangent to side $BC$. We need to find the radius of this circle.\n\n2. **Define Points and Variables**: Let $O$ be the... |
deepscaler | [
{
"role": "user",
"content": "The germination rate of cotton seeds is $0.9$, and the probability of developing into strong seedlings is $0.6$,\n\n$(1)$ If two seeds are sown per hole, the probability of missing seedlings in this hole is _______; the probability of having no strong seedlings in this hole is ... | math | {
"style": "rule",
"ground_truth": "0.936"
} | {
"split": "train",
"index": 1389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area enclosed by the region defined by the equation $x^2+y^2+12x+16y=0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100\\pi"
} | {
"split": "train",
"index": 1390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let point G be the centroid of triangle ABC. If $\\angle A=120^\\circ$ and $\\overrightarrow {AB} \\cdot \\overrightarrow {AC}=-1$, find the minimum value of $|\\overrightarrow {AG}|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{3}"
} | {
"split": "train",
"index": 1391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five rays $\\overrightarrow{OA}$ , $\\overrightarrow{OB}$ , $\\overrightarrow{OC}$ , $\\overrightarrow{OD}$ , and $\\overrightarrow{OE}$ radiate in a clockwise order from $O$ forming four non-overlapping angles such that $\\angle EOD = 2\\angle COB$ , $\\angle COB = 2... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 1392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many five-digit positive integers have the product of their digits equal to 1000? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 1393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A token starts at the point $(0,0)$ of an $xy$-coordinate grid and then makes a sequence of six moves. Each move is 1 unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other move... | math | {
"style": "rule",
"ground_truth": "391"
} | {
"split": "train",
"index": 1394,
"solution": "Perform the coordinate transformation $(x, y)\\rightarrow (x+y, x-y)$. Then we can see that a movement up, right, left, or down in the old coordinates adds the vectors $\\langle 1, -1 \\rangle$, $\\langle 1, 1 \\rangle$, $\\langle -1, -1 \\rangle$, $\\langle -1, 1 \... |
deepscaler | [
{
"role": "user",
"content": "Consider the following sequence $$\\left(a_{n}\\right)_{n=1}^{\\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1, \\ldots)$$ Find all pairs $(\\alpha, \\beta)$ of positive real numbers such that $\\lim _{n \\rightarrow \\infty} \\frac{\\sum_{k=1}^{n} a_{k}}{n^{\\alpha}}=\\beta$. Let's t... | math | {
"style": "rule",
"ground_truth": "(\\alpha, \\beta)=\\left(\\frac{3}{2}, \\frac{\\sqrt{2}}{3}\\right)"
} | {
"split": "train",
"index": 1395,
"solution": "Let $N_{n}=\\binom{n+1}{2}$ (then $a_{N_{n}}$ is the first appearance of number $n$ in the sequence) and consider limit of the subsequence $$b_{N_{n}}:=\\frac{\\sum_{k=1}^{N_{n}} a_{k}}{N_{n}^{\\alpha}}=\\frac{\\sum_{k=1}^{n} 1+\\cdots+k}{\\binom{n+1}{2}^{\\alpha}}=... |
deepscaler | [
{
"role": "user",
"content": "If $x$ varies as the cube of $y$, and $y$ varies as the fifth root of $z$, then $x$ varies as the nth power of $z$, where n is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 1396,
"solution": "1. **Understanding the relationship between $x$ and $y$:**\n Given that $x$ varies as the cube of $y$, we can express this relationship mathematically as:\n \\[\n x = k \\cdot y^3\n \\]\n where $k$ is a constant of proportionality.\n\n2. **Understanding th... |
deepscaler | [
{
"role": "user",
"content": "If income of $5$ yuan is denoted as $+5$ yuan, then expenses of $5$ yuan are denoted as what? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 1397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integers with three not necessarily distinct digits, $abc$, with $a \\neq 0$ and $c \\neq 0$ such that both $abc$ and $cba$ are multiples of $4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 1398,
"solution": "A number is divisible by four if its last two digits are divisible by 4. Thus, we require that $10b + a$ and $10b + c$ are both divisible by $4$. If $b$ is odd, then $a$ and $c$ must both be $2 \\pmod 4$ meaning that $a$ and $c$ are $2$ or $6$. If $b$ is even, then ... |
deepscaler | [
{
"role": "user",
"content": "A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three pric... | math | {
"style": "rule",
"ground_truth": "420"
} | {
"split": "train",
"index": 1399,
"solution": "[Clarification: You are supposed to find the number of all possible tuples of prices, $(A, B, C)$, that could have been on that day.]\n Since we have three numbers, consider the number of ways we can put these three numbers together in a string of 7 digits. For exam... |
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