data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "let $x,y,z$ be positive reals , such that $x+y+z=1399$ find the\n $$ \\max( [x]y + [y]z + [z]x ) $$ \n( $[a]$ is the biggest integer not exceeding $a$ ) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "652400"
} | {
"split": "train",
"index": 21249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many whole numbers between $200$ and $500$ contain the digit $3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "138"
} | {
"split": "train",
"index": 21250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the array: $(1,1,1)$, $(2,2,4)$, $(3,4,12)$, $(4,8,32)$, $\\ldots$, $(a_{n}, b_{n}, c_{n})$, find the value of $c_{7}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "448"
} | {
"split": "train",
"index": 21251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A$ and $B$ have the same $y$-coordinate of 13, but different $x$-coordinates. What is the sum of the slope and the $y$-intercept of the line containing both points? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 21252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of $x + x(x^x)$ when $x = 2$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 21253,
"solution": "1. **Substitute $x = 2$ into the expression**: We start by substituting $x = 2$ into the expression $x + x(x^x)$.\n \\[\n x + x(x^x) = 2 + 2(2^2)\n \\]\n\n2. **Evaluate the exponentiation**: Next, we calculate $2^2$.\n \\[\n 2^2 = 4\n \\]\n\n3. **Substi... |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the greatest common factor of 3 and 6 and the least common multiple of 3 and 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 21254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The figure below shows line $\\ell$ with a regular, infinite, recurring pattern of squares and line segments.\n\nHow many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transfo... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 21255,
"solution": "To determine which transformations will map the figure onto itself, we analyze each type of transformation given in the problem:\n\n1. **Some rotation around a point on line $\\ell$:**\n - Consider a rotation of $180^\\circ$ around a point exactly halfway between... |
deepscaler | [
{
"role": "user",
"content": "Given that construction teams A and B each have a certain number of workers. If team A lends 90 workers to team B, then team B's total number of workers will be twice that of team A. If team B lends a certain number of workers to team A, then team A's total number of workers wi... | math | {
"style": "rule",
"ground_truth": "153"
} | {
"split": "train",
"index": 21256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lines in the $xy$-plane are drawn through the point $(3,4)$ and the trisection points of the line segment joining the points $(-4,5)$ and $(5,-1)$. One of these lines has the equation Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x-4y+13=0"
} | {
"split": "train",
"index": 21257,
"solution": "1. **Finding the Trisection Points:**\n The trisection points of the line segment joining $(-4, 5)$ and $(5, -1)$ are calculated by dividing the segment into three equal parts. We start by finding the differences in the x-coordinates and y-coordinates:\n - Diff... |
deepscaler | [
{
"role": "user",
"content": "Let $b(n)$ be the number of digits in the base -4 representation of $n$. Evaluate $\\sum_{i=1}^{2013} b(i)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12345"
} | {
"split": "train",
"index": 21258,
"solution": "We have the following: - $b(n)=1$ for $n$ between 1 and 3 . - $b(n)=3$ for $n$ between $4^{2}-3 \\cdot 4=4$ and $3 \\cdot 4^{2}+3=51$. (Since $a \\cdot 4^{2}-b \\cdot 4+c$ takes on $3 \\cdot 4 \\cdot 4$ distinct values over $1 \\leq a \\leq 3,0 \\leq b \\leq 3,0 \\... |
deepscaler | [
{
"role": "user",
"content": "As shown in the diagram, in the square \\(ABCD\\), \\(AB = 2\\). Draw an arc with center \\(C\\) and radius equal to \\(CD\\), and another arc with center \\(B\\) and radius equal to \\(BA\\). The two arcs intersect at \\(E\\). What is the area of the sector \\(BAE\\)? Let's th... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 21259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the number $x$ . Using multiply and division operations of any 2 given or already given numbers we can obtain powers with natural exponent of the number $x$ (for example, $x\\cdot x=x^{2}$ , $x^{2}\\cdot x^{2}=x^{4}$ , $x^{4}: x=x^{3}$ , etc). Determine the minimal n... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 21260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of rectangle $ABCD$ with vertices $A$(0, 0), $B$(0, 4), $C$($x$, 4) and $D$($x$, 0) is 28 square units. If $x > 0$, what is the value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 21261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jun Jun is looking at an incorrect single-digit multiplication equation \\( A \\times B = \\overline{CD} \\), where the digits represented by \\( A \\), \\( B \\), \\( C \\), and \\( D \\) are all different from each other. Clever Jun Jun finds that if only one digit is changed... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 21262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the set \\( \\{1, 2, 3, \\ldots, 8\\} \\) and each of its non-empty subsets, define a unique alternating sum as follows: arrange the numbers in the subset in decreasing order and alternately add and subtract successive numbers. For instance, the alternating sum for \\( \\{1... | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"split": "train",
"index": 21263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The cost $C$ of sending a parcel post package weighing $P$ pounds, $P$ an integer, is $10$ cents for the first pound and $3$ cents for each additional pound. The formula for the cost is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "C=10+3(P-1)"
} | {
"split": "train",
"index": 21264,
"solution": "To find the correct formula for the cost $C$ of sending a parcel post package weighing $P$ pounds, we analyze the given information and the structure of the cost:\n\n1. **Cost for the first pound**: The cost for the first pound is a fixed amount of $10$ cents.\n\n2... |
deepscaler | [
{
"role": "user",
"content": "If the system of equations\n\n\\begin{align*}\n2x-y&=a,\\\\\n3y-6x &=b.\n\\end{align*}has a solution, find $\\frac{a}{b},$ assuming $b \\neq 0.$ 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": 21265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), \\( M \\) is the midpoint of side \\( AC \\), \\( D \\) is a point on side \\( BC \\) such that \\( AD \\) is the angle bisector of \\( \\angle BAC \\), and \\( P \\) is the point of intersection of \\( AD \\) and \\( BM \\). Given that \\( A... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 21266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the triangle $ABC$ have area $1$ . The interior bisectors of the angles $\\angle BAC,\\angle ABC, \\angle BCA$ intersect the sides $(BC), (AC), (AB) $ and the circumscribed circle of the respective triangle $ABC$ at the points $L$ and $G, N$ and $F, Q$ and $... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 21267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a large triangle divided into squares and triangles. Let \\( S \\) be the number of squares of any size in the diagram and \\( T \\) be the number of triangles of any size in the diagram. What is the value of \\( S \\times T \\)? \n\nA) 30\n\nB) 35\n\nC) 48\n\... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 21268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a square piece of grid paper containing an integer number of cells, a hole in the shape of a square, also consisting of an integer number of cells, was cut out. How many cells did the large square contain if, after cutting out the hole, 209 cells remained? Let's think step b... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 21269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the display of a certain instrument, each indicator light can show different signals by lighting up in red, yellow, or blue. It is known that there are 6 indicator lights in a row, and each time 3 of them light up, with only 2 being adjacent, then the total number of differe... | math | {
"style": "rule",
"ground_truth": "324"
} | {
"split": "train",
"index": 21270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The six-digit number $20210A$ is prime for only one digit $A.$ What is $A?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 21271,
"solution": "1. **Check divisibility by 5**: The number is $\\underline{2}\\,\\underline{0}\\,\\underline{2}\\,\\underline{1}\\,\\underline{0}\\,\\underline{A}$. If $A = 5$, the number ends in $5$ and is divisible by $5$. Therefore, $A = 5$ cannot be the correct answer since th... |
deepscaler | [
{
"role": "user",
"content": "In three-digit numbers, if the digit in the tens place is smaller than the digits in both the hundreds and ones places, the number is called a \"concave number,\" such as 304, 968, etc. How many distinct three-digit concave numbers are there without any repeated digits? Let's t... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 21272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest prime divisor of $36^2 + 49^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3697"
} | {
"split": "train",
"index": 21273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Richard is building a rectangular backyard from 360 feet of fencing. The fencing must cover three sides of the backyard (the fourth side is bordered by Richard's house). What is the maximum area of this backyard? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "16200"
} | {
"split": "train",
"index": 21274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let a sequence $\\left\\{a_{n}\\right\\}_{n=0}^{\\infty}$ be defined by $a_{0}=\\sqrt{2}, a_{1}=2$, and $a_{n+1}=a_{n} a_{n-1}^{2}$ for $n \\geq 1$. The sequence of remainders when $a_{0}, a_{1}, a_{2}, \\cdots$ are divided by 2014 is eventually periodic with some minimal perio... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 21275,
"solution": "Let $a_{n}=2^{b_{n}}$, so notice $b_{1}=1, b_{2}=2$, and $b_{n+1}=b_{n}+2 b_{n-1}$ for $n \\geq 1$, so by inspection $b_{n}=2^{n-1}$ for all $n$; thus $a_{n}=2^{2^{n-1}} .2014=2 \\cdot 19 \\cdot 53$ so we just want to find the lcm of the eventual periods of $2^{n} ... |
deepscaler | [
{
"role": "user",
"content": "Let $a, b$, and $c$ be real numbers. Consider the system of simultaneous equations in variables $x$ and $y:$ $a x+b y =c-1$ and $(a+5) x+(b+3) y =c+1$. Determine the value(s) of $c$ in terms of $a$ such that the system always has a solution for any $a$ and $b$. Let's think step... | math | {
"style": "rule",
"ground_truth": "2a/5 + 1 \\text{ or } \\frac{2a+5}{5}"
} | {
"split": "train",
"index": 21276,
"solution": "We have to only consider when the determinant of $\\begin{pmatrix}a & b \\\\ a+5 & b+3\\end{pmatrix}$ is zero. That is, when $b=3 a / 5$. Plugging in $b=3 a / 5$, we find that $(a+5)(c-1)=a(c+1)$ or that $c=2 a / 5+1$."
} |
deepscaler | [
{
"role": "user",
"content": "For any real number $x$, the symbol $\\lfloor x \\rfloor$ represents the integer part of $x$, which is the greatest integer not exceeding $x$. This function, $\\lfloor x \\rfloor$, is called the \"floor function\". Calculate the sum $\\lfloor \\log_3 1 \\rfloor + \\lfloor \\log... | math | {
"style": "rule",
"ground_truth": "857"
} | {
"split": "train",
"index": 21277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A factory produces a certain type of component, and the inspector randomly selects 16 of these components from the production line each day to measure their dimensions (in cm). The dimensions of the 16 components selected in one day are as follows:\n\n10.12, 9.97, 10.01, 9.95, ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 21278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ana's monthly salary was $2000$ in May. In June she received a 20% raise. In July she received a 20% pay cut. After the two changes in June and July, Ana's monthly salary was Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1920"
} | {
"split": "train",
"index": 21279,
"solution": "1. **Calculate the salary after the raise in June:**\n Ana's initial salary in May is $2000. A 20\\% raise means her salary is increased by $2000 \\times 20\\% = $400. Therefore, her new salary in June becomes:\n \\[\n 2000 + 400 = 2400\n \\]\n Alternativ... |
deepscaler | [
{
"role": "user",
"content": "Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 22x^2 + 80x - 67$. It is given that there exist real numbers $A$, $B$, and $C$ such that \\begin{equation*} \\frac{1}{s^3 - 22s^2 + 80s - 67} = \\frac{A}{s-p} + \\frac{B}{s-q} + \\frac{C}{s-r} \\end{equation*}... | math | {
"style": "rule",
"ground_truth": "244"
} | {
"split": "train",
"index": 21280,
"solution": "We start by expressing the given partial fraction decomposition:\n\\[\n\\frac{1}{s^3 - 22s^2 + 80s - 67} = \\frac{A}{s-p} + \\frac{B}{s-q} + \\frac{C}{s-r}\n\\]\nfor all \\( s \\not\\in \\{p, q, r\\} \\), where \\( p, q, r \\) are the roots of the polynomial \\( x^... |
deepscaler | [
{
"role": "user",
"content": "Find the number of solutions to\n\\[\\sin x = \\left( \\frac{1}{3} \\right)^x\\] on the interval $(0,150 \\pi).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 21281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the center of the circle with equation $x^2 - 6x + y^2 + 2y = 9$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(3, -1)"
} | {
"split": "train",
"index": 21282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There exist two distinct unit vectors $\\mathbf{v}$ such that the angle between $\\mathbf{v}$ and $\\begin{pmatrix} 2 \\\\ 2 \\\\ -1 \\end{pmatrix}$ is $45^\\circ,$ and the angle between $\\mathbf{v}$ and $\\begin{pmatrix} 0 \\\\ 1 \\\\ -1 \\end{pmatrix}$ is $60^\\circ.$ Let $... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 21283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ordered triples $(x,y,z)$ of positive integers satisfy $\\text{lcm}(x,y) = 72, \\text{lcm}(x,z) = 600 \\text{ and lcm}(y,z)=900$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 21284,
"solution": "1. **Understanding the Problem:**\n We need to find the number of ordered triples $(x, y, z)$ of positive integers such that:\n - $\\text{lcm}(x, y) = 72$\n - $\\text{lcm}(x, z) = 600$\n - $\\text{lcm}(y, z) = 900$\n\n2. **Prime Factorization:**\n - $72 =... |
deepscaler | [
{
"role": "user",
"content": "Given that the terminal side of angle $θ$ lies on the ray $y=2x(x≤0)$, find the value of $\\sin θ + \\cos θ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 21285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify: $-{-\\left[-|-1|^2\\right]^3}^4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 21286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five cards are lying on a table as shown.\n\\[\\begin{matrix} & \\qquad & \\boxed{\\tt{P}} & \\qquad & \\boxed{\\tt{Q}} \\\\ \\\\ \\boxed{\\tt{3}} & \\qquad & \\boxed{\\tt{4}} & \\qquad & \\boxed{\\tt{6}} \\end{matrix}\\]\nEach card has a letter on one side and a whole number ... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 21287,
"solution": "To solve this problem, we need to test Jane's statement: \"If a vowel is on one side of any card, then an even number is on the other side.\" We need to find a counterexample to disprove this statement.\n\n1. **Understanding the statement**: The statement can be in... |
deepscaler | [
{
"role": "user",
"content": "Suppose that $x_1+1=x_2+2=x_3+3=\\cdots=x_{2008}+2008=x_1+x_2+x_3+\\cdots+x_{2008}+2009$. Find the value of $\\left\\lfloor|S|\\right\\rfloor$, where $S=\\sum_{n=1}^{2008}x_n$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1005"
} | {
"split": "train",
"index": 21288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Aunt Anna is $42$ years old. Caitlin is $5$ years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 21289,
"solution": "1. **Determine Brianna's Age:**\n Given that Aunt Anna is $42$ years old and Brianna is half as old as Aunt Anna, we calculate Brianna's age as follows:\n \\[\n \\text{Brianna's age} = \\frac{1}{2} \\times \\text{Aunt Anna's age} = \\frac{1}{2} \\times 42 = 2... |
deepscaler | [
{
"role": "user",
"content": "Oil, as an important strategic reserve commodity, has always been of concern to countries. According to reports from relevant departments, it is estimated that the demand for oil in China in 2022 will be 735,000,000 tons. Express 735,000,000 in scientific notation as ____. Let'... | math | {
"style": "rule",
"ground_truth": "7.35 \\times 10^{8}"
} | {
"split": "train",
"index": 21290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle PQR$, we have $PQ = QR = 34$ and $PR = 32$. Point $M$ is the midpoint of $\\overline{QR}$. Find $PM$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{89}"
} | {
"split": "train",
"index": 21291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The volume of the parallelepiped determined by vectors $\\mathbf{a}$, $\\mathbf{b}$, and $\\mathbf{c}$ is 6. Find the volume of the parallelepiped determined by $\\mathbf{a} + 2\\mathbf{b},$ $\\mathbf{b} + \\mathbf{c},$ and $2\\mathbf{c} - 5\\mathbf{a}.$ Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 21292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $$1+6+11+16+21+26+\\cdots+91+96+101 \\equiv n\\pmod{15},$$where $0\\le n<15$, what is the value of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 21293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square with a side length of 100 cm is drawn on a board. Alexei intersected it with two lines parallel to one pair of its sides. After that, Danil intersected the square with two lines parallel to the other pair of sides. As a result, the square was divided into 9 rectangles,... | math | {
"style": "rule",
"ground_truth": "2400"
} | {
"split": "train",
"index": 21294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the teacher has 30 cookies to distribute among five students so that each student receives at least three cookies, find the number of ways the cookies can be distributed. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3876"
} | {
"split": "train",
"index": 21295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A basketball player scored a mix of free throws, 2-pointers, and 3-pointers during a game, totaling 7 successful shots. Find the different numbers that could represent the total points scored by the player, assuming free throws are worth 1 point each. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 21296,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Completely factor the following expression: \\[(6a^3+92a^2-7)-(-7a^3+a^2-7)\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13a^2(a+7)"
} | {
"split": "train",
"index": 21297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function \\( f \\) satisfies the equation \\((n - 2019) f(n) - f(2019 - n) = 2019\\) for every integer \\( n \\).\nWhat is the value of \\( f(2019) \\)?\nA) 0\nB) 1\nC) \\(2018 \\times 2019\\)\nD) \\(2019^2\\)\nE) \\(2019 \\times 2020\\) Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "2019 \\times 2018"
} | {
"split": "train",
"index": 21298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many irreducible fractions with a numerator of 2015 are there that are less than \\( \\frac{1}{2015} \\) and greater than \\( \\frac{1}{2016} \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1440"
} | {
"split": "train",
"index": 21299,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A nine-digit number is formed by repeating a three-digit number three times. For example, 123,123,123 or 456,456,456 are numbers of this form. What is the greatest common divisor of all nine-digit numbers of this form? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "1001001"
} | {
"split": "train",
"index": 21300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number $N$ has three digits when expressed in base $7$. When $N$ is expressed in base $9$ the digits are reversed. Then the middle digit is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 21301,
"solution": "1. **Expressing $N$ in different bases**: Let $N$ be represented as $\\overline{abc}_7$ in base $7$ and as $\\overline{cba}_9$ in base $9$. This means:\n - In base $7$: $N = 49a + 7b + c$\n - In base $9$: $N = 81c + 9b + a$\n\n2. **Setting up the equation**: Si... |
deepscaler | [
{
"role": "user",
"content": "Two different natural numbers are selected from the set $\\ \\allowbreak \\{1, 2, 3, \\ldots, 6\\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{15}"
} | {
"split": "train",
"index": 21302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mrs. Carter's algebra class consists of 48 students. Due to a schedule conflict, 40 students took the Chapter 5 test, averaging 75%, while the remaining 8 students took it the following day, achieving an average score of 82%. What is the new overall mean score of the class on t... | math | {
"style": "rule",
"ground_truth": "76.17\\%"
} | {
"split": "train",
"index": 21303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A TV station is broadcasting 5 advertisements in a row, including 3 different commercial advertisements and 2 different public service advertisements. The last advertisement cannot be a commercial one, and the two public service advertisements cannot be broadcast consecutively.... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 21304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum possible number of three term arithmetic progressions in a monotone sequence of $n$ distinct reals. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(n) = \\left\\lfloor \\frac{(n-1)^2}{2} \\right\\rfloor"
} | {
"split": "train",
"index": 21305,
"solution": "Consider the first few cases for $n$ with the entire $n$ numbers forming an arithmetic sequence \\[(1, 2, 3, \\ldots, n)\\] If $n = 3$ , there will be one ascending triplet (123). Let's only consider the ascending order for now.\nIf $n = 4$ , the first 3 numbers gi... |
deepscaler | [
{
"role": "user",
"content": "Consider the number $99,\\!999,\\!999,\\!999$ squared. Following a pattern observed in previous problems, determine how many zeros are in the decimal expansion of this number squared. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 21306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum number of points that can be placed on a segment of length 1 such that on any subsegment of length \\( d \\) contained in this segment, there are no more than \\( 1 + 1000 d^2 \\) points? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 21307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a cube $ABCDEFGH$ with an edge length of $6$, let $M$ and $N$ be points on $BB_1$ and $B_1C_1$ respectively, such that $B_1M = B_1N = 2$. Let $S$ and $P$ be the midpoints of segments $AD$ and $MN$ respectively. Find the distance between the skew lines $SP$ and $AC_1$. Let... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{114}}{38}"
} | {
"split": "train",
"index": 21308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mindy is attempting to solve the quadratic equation by completing the square: $$100x^2+80x-144 = 0.$$ She rewrites the given quadratic equation in the form $$(dx + e)^2 = f,$$ where \\(d\\), \\(e\\), and \\(f\\) are integers and \\(d > 0\\). What are the values of \\(d + e + f\... | math | {
"style": "rule",
"ground_truth": "174"
} | {
"split": "train",
"index": 21309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ with the property that in the set $\\{70, 71, 72,... 70 + n\\}$ you can choose two different numbers whose product is the square of an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 21310,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $150$-gon $Q_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $150$ vertices equals $3000$. The midpoints of the sides of $Q_1$ form a second $150$-gon, $Q_2$. Finally, the midpoints of the sides of $Q_2$ form a third $150$-gon, $Q_3$. Find the sum of... | math | {
"style": "rule",
"ground_truth": "3000"
} | {
"split": "train",
"index": 21311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit positive integers $x$ satisfy $3874x + 481 \\equiv 1205 \\pmod{31}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "290"
} | {
"split": "train",
"index": 21312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer that ends in 9 and is divisible by 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 21313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the six digits 1, 2, 3, 5, 5 and 8 are randomly arranged into a six-digit positive integer, what is the probability that the integer is divisible by 15? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 21314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three $1 \\times 1 \\times 1$ cubes are joined face to face in a single row and placed on a table. The cubes have a total of 11 exposed $1 \\times 1$ faces. If sixty $1 \\times 1 \\times 1$ cubes are joined face to face in a single row and placed on a table, how many $1 \\times... | math | {
"style": "rule",
"ground_truth": "182"
} | {
"split": "train",
"index": 21315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that there are 21 students in Dr. Smith's physics class, the average score before including Simon's project score was 86. After including Simon's project score, the average for the class rose to 88. Calculate Simon's score on the project. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 21316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( A \\cup B = \\left\\{a_{1}, a_{2}, a_{3}\\right\\} \\) and \\( A \\neq B \\), where \\((A, B)\\) and \\((B, A)\\) are considered different pairs, find the number of such pairs \\((A, B)\\). Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 21317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p(x)$ be a monic polynomial of degree 4, such that $p(1) = 17,$ $p(2) = 34,$ and $p(3) = 51.$ Find $p(0) + p(4).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "92"
} | {
"split": "train",
"index": 21318,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the final 5 contestants of \"The Voice\" season 4 must sign with one of the three companies A, B, and C, with each company signing at least 1 person and at most 2 people, calculate the total number of possible different signing schemes. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 21319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the diameter in centimeters of a circle whose area is $100\\pi \\text{cm}^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 21320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the number of terms in the simplified expression of \\[(x+y+z)^{2020} + (x-y-z)^{2020},\\] by expanding it and combining like terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1,022,121"
} | {
"split": "train",
"index": 21321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $x \\gt 0$, $y \\gt 0$, when $x=$______, the maximum value of $\\sqrt{xy}(1-x-2y)$ is _______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{16}"
} | {
"split": "train",
"index": 21322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the square root of $\\dfrac{9!}{126}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12.648"
} | {
"split": "train",
"index": 21323,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the tens digit and the ones digit of the integer form of $(2+3)^{23}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 21324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A transformation of the first quadrant of the coordinate plane maps each point $(x,y)$ to the point $(\\sqrt{x},\\sqrt{y}).$ The vertices of quadrilateral $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$ Let $k_{}$ be the area of the region enclosed by t... | math | {
"style": "rule",
"ground_truth": "314"
} | {
"split": "train",
"index": 21325,
"solution": "\\begin{eqnarray*}A' = & (\\sqrt {900}, \\sqrt {300})\\\\ B' = & (\\sqrt {1800}, \\sqrt {600})\\\\ C' = & (\\sqrt {600}, \\sqrt {1800})\\\\ D' = & (\\sqrt {300}, \\sqrt {900}) \\end{eqnarray*}\n First we see that lines passing through $AB$ and $CD$ have equations $... |
deepscaler | [
{
"role": "user",
"content": "The degree measures of the angles of nondegenerate hexagon $ABCDEF$ are integers that form a non-constant arithmetic sequence in some order, and $\\angle A$ is the smallest angle of the (not necessarily convex) hexagon. Compute the sum of all possible degree measures of $\... | math | {
"style": "rule",
"ground_truth": "1500"
} | {
"split": "train",
"index": 21326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of ways to select a sequence of 8 sets $A_{1}, A_{2}, \\ldots, A_{8}$, such that each is a subset (possibly empty) of \\{1,2\\}, and $A_{m}$ contains $A_{n}$ if $m$ divides $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2025"
} | {
"split": "train",
"index": 21327,
"solution": "Consider an arbitrary $x \\in\\{1,2\\}$, and let us consider the number of ways for $x$ to be in some of the sets so that the constraints are satisfied. We divide into a few cases: - Case: $x \\notin A_{1}$. Then $x$ cannot be in any of the sets. So there is one po... |
deepscaler | [
{
"role": "user",
"content": "Point $A$ lies on the line $y = \\frac{12}{5} x - 3$, and point $B$ lies on the parabola $y = x^2$. What is the minimum length of the segment $AB$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.6"
} | {
"split": "train",
"index": 21328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $|\\overrightarrow{a}|=4, |\\overrightarrow{b}|=8$, and the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is $\\frac{2\\pi}{3}$.\n\n(I) Find $|\\overrightarrow{a}+\\overrightarrow{b}|$;\n\n(II) Find the value of $k$ such that $(\\overrightarrow{a}+2\\... | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 21329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the polynomial expression \\( x^4 - 61x^2 + 60 \\), for how many integers \\( x \\) is the expression negative. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 21330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A perfect power is an integer $n$ that can be represented as $a^{k}$ for some positive integers $a \\geq 1$ and $k \\geq 2$. Find the sum of all prime numbers $0<p<50$ such that $p$ is 1 less than a perfect power. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 21331,
"solution": "First, it is known that $a^{k}-1=(a-1)\\left(a^{k-1}+a^{k-2}+\\ldots\\right)$. This means either $a-1$ or $a^{k-1}+a^{k-2}+\\ldots+1$ must be 1 in order for $a^{k}-1$ to be prime. But this only occurs when $a$ is 2 . Thus, the only possible primes are of the form $... |
deepscaler | [
{
"role": "user",
"content": "In a set of 10 programs, there are 6 singing programs and 4 dance programs. The requirement is that there must be at least one singing program between any two dance programs. Determine the number of different ways to arrange these programs. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "604800"
} | {
"split": "train",
"index": 21332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $0 \\leqslant x \\leqslant 2$, find the minimum and maximum values of the function $f(x) = 4^{x - \\frac{1}{2}} - 3 \\cdot 2^x + 5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 21333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p_{}$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of $5$ heads before one encounters a run of $2$ tails. Given that $p_{}$ can be written in the form $m/n$ where $m_{}$ and $n_{}$ are relatively prime positive intege... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 21334,
"solution": "Let $p_H, p_T$ respectively denote the probabilities that a string beginning with H's and T's are successful. Thus,\n$p_T = \\frac 12p_H.$\n A successful string can either start with 1 to 4 H's, start with a T and then continue with a string starting with H (as the... |
deepscaler | [
{
"role": "user",
"content": "Given that the function $F(x) = f(x) + x^2$ is an odd function, and $f(2) = 1$, find $f(-2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"split": "train",
"index": 21335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of values of $x$ satisfying the equation\n\\[\\frac {2x^2 - 10x}{x^2 - 5x} = x - 3\\]is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 21336,
"solution": "1. **Identify the domain of the equation**: The given equation is\n \\[\n \\frac{2x^2 - 10x}{x^2 - 5x} = x - 3.\n \\]\n We first note that the denominator $x^2 - 5x$ must not be zero to avoid division by zero. Factoring out $x$, we get:\n \\[\n x(x - 5)... |
deepscaler | [
{
"role": "user",
"content": "Farmer James has some strange animals. His hens have 2 heads and 8 legs, his peacocks have 3 heads and 9 legs, and his zombie hens have 6 heads and 12 legs. Farmer James counts 800 heads and 2018 legs on his farm. What is the number of animals that Farmer James has on his farm?... | math | {
"style": "rule",
"ground_truth": "203"
} | {
"split": "train",
"index": 21337,
"solution": "Note that each animal has 6 more legs than heads. Thus, if there are $n$ animals, then there are $6 n$ more legs than heads in total. There are $2018-800=1218$ more legs than heads in total, so there are $\\frac{1218}{6}=203$ animals."
} |
deepscaler | [
{
"role": "user",
"content": "Using the four arithmetic operators and parentheses, find a way to combine the numbers 10, 10, 4, and 2 such that the result is 24. What is the arithmetic expression? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2 + 4 \\div 10) \\times 10"
} | {
"split": "train",
"index": 21338,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a busy afternoon, David decides to drink a cup of water every 20 minutes to stay hydrated. Assuming he maintains this pace, how many cups of water does David drink in 3 hours and 45 minutes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11.25"
} | {
"split": "train",
"index": 21339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit positive integers are multiples of 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1286"
} | {
"split": "train",
"index": 21340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right triangle has integer side lengths. One of its legs is 1575 units shorter than its hypotenuse, and the other leg is less than 1991 units. Find the length of the hypotenuse of this right triangle. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "1799"
} | {
"split": "train",
"index": 21341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the village of Matitika, five friends live along a straight road in the following order: Alya, Bella, Valya, Galya, and Dilya. Each of them calculated the sum of distances (in meters) from her house to the houses of the others. Bella reported the number 700, Valya reported 6... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 21342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the increment of the argument and the function \\( y = x^2 \\) if the argument \\( x \\) changes from 2 to 2.5. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2.25"
} | {
"split": "train",
"index": 21343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of triples $(f, g, h)$ of permutations on $\\{1,2,3,4,5\\}$ such that $$ \\begin{aligned} & f(g(h(x)))=h(g(f(x)))=g(x), \\\\ & g(h(f(x)))=f(h(g(x)))=h(x), \\text { and } \\\\ & h(f(g(x)))=g(f(h(x)))=f(x) \\end{aligned} $$ for all $x \\in\\{1,2,3,4,5\\}$. Let'... | math | {
"style": "rule",
"ground_truth": "146"
} | {
"split": "train",
"index": 21344,
"solution": "Let $f g$ represent the composition of permutations $f$ and $g$, where $(f g)(x)=f(g(x))$ for all $x \\in\\{1,2,3,4,5\\}$. Evaluating fghfh in two ways, we get $$ f=g f h=(f g h) f h=f g h f h=f(g h f) h=f h h, $$ so $h h=1$. Similarly, we get $f, g$, and $h$ are a... |
deepscaler | [
{
"role": "user",
"content": "Given Allison's birthday cake is in the form of a $5 \\times 5 \\times 3$ inch rectangular prism with icing on the top, front, and back sides but not on the sides or bottom, calculate the number of $1 \\times 1 \\times 1$ inch smaller prisms that will have icing on exactly two ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 21345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains 5 red, 6 green, 7 yellow, and 8 blue jelly beans. A jelly bean is selected at random. What is the probability that it is blue? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{13}"
} | {
"split": "train",
"index": 21346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=3x-2$, and let $g(x)=f(f(f(f(x))))$. If the domain of $g$ is $0\\leq x\\leq 2$, compute the range of $g$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-80\\leq g(x)\\leq 82"
} | {
"split": "train",
"index": 21347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{72}"
} | {
"split": "train",
"index": 21348,
"solution": "To solve this problem, we need to calculate the probability of two scenarios: \n1. Exactly three dice show the same number.\n2. All four dice show the same number.\n\n#### Case 1: All four dice show the same number\n- The probability that the first die shows any nu... |
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