data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Four people are sitting around a round table, with identical coins placed in front of each person. Everyone flips their coin simultaneously. If the coin lands heads up, the person stands up; if it lands tails up, the person remains seated. Calculate the probability that no two ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"split": "train",
"index": 37449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be real numbers such that\n\\[xy - \\frac{x}{y^2} - \\frac{y}{x^2} = 3.\\]Find the sum of all possible values of $(x - 1)(y - 1).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 37450,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number \\( n \\) such that both \\( n^2 \\) and \\( (n+1)^2 \\) contain the digit 7. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 37451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ be the greatest common factor and let $B$ be the least common multiple of 8, 12 and 24. What is the value of $A + B$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 37452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all integer solutions to the inequality $|n| < |n-3| < 10$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-20"
} | {
"split": "train",
"index": 37453,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A palindrome between $10000$ and $100000$ is chosen at random. What is the probability that it is divisible by $11$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{41}{450}"
} | {
"split": "train",
"index": 37454,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\theta \\in (0, \\frac{\\pi}{2})$, and $\\sin\\theta = \\frac{4}{5}$, find the value of $\\cos\\theta$ and $\\sin(\\theta + \\frac{\\pi}{3})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4 + 3\\sqrt{3}}{10}"
} | {
"split": "train",
"index": 37455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $g(x)=\\ln x+\\frac{1}{2}x^{2}-(b-1)x$.\n\n (1) If the function $g(x)$ has a monotonically decreasing interval, find the range of values for the real number $b$;\n\n (2) Let $x_{1}$ and $x_{2}$ ($x_{1} < x_{2}$) be the two extreme points of the function... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{8}-2\\ln 2"
} | {
"split": "train",
"index": 37456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which digit of $.12345$, when changed to $9$, gives the largest number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 37457,
"solution": "To determine which digit change in $.12345$ to $9$ results in the largest number, we need to consider the positional value of each digit in the decimal.\n\n1. **Understanding Decimal Places**:\n - The decimal $.12345$ can be expanded as:\n \\[\n 0.12345 =... |
deepscaler | [
{
"role": "user",
"content": "Find the least prime factor of the number represented by \\(1 \\underbrace{0000 \\cdots 00}_{2010 \\text{-many}} 1\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 37458,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Mathematics College Entrance Examination scores distribution $\\xi$ closely follows the normal distribution $N(100, 5^2)$, and $P(\\xi < 110) = 0.96$. Find the value of $P(90 < \\xi < 100)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.46"
} | {
"split": "train",
"index": 37459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average of the numbers $1, 2, 3,\\dots, 148, 149,$ and $x$ is $50x$. What is $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11175}{7499}"
} | {
"split": "train",
"index": 37460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, four circles with centers $P$, $Q$, $R$, and $S$ each have a radius of 2. These circles are tangent to one another and to the sides of $\\triangle ABC$ as shown. The circles centered at $P$ and $Q$ are tangent to side $AB$, the circle at $R$ is tangent to side $... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 37461,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle with radius $R$ and a fixed point $A$ on the circumference, a point is randomly chosen on the circumference and connected to point $A$. The probability that the length of the chord formed is between $R$ and $\\sqrt{3}R$ is ______. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 37462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all solutions to the equation $(x-6)^2=25$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 37463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular dodecagon \\(Q_1 Q_2 \\ldots Q_{12}\\) is drawn in the coordinate plane with \\(Q_1\\) at \\((2,0)\\) and \\(Q_7\\) at \\((4,0)\\). If \\(Q_n\\) is the point \\((x_n, y_n)\\), compute the numerical value of the product:\n\\[\n(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \\... | math | {
"style": "rule",
"ground_truth": "531440"
} | {
"split": "train",
"index": 37464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of butter:flour:sugar in a recipe is 1:6:4. When using 8 cups of sugar in this recipe, how many total cups of these three ingredients will be used? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 37465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ (a_n)_{n \\equal{} 1}^\\infty$ is defined on real numbers with $ a_n \\not \\equal{} 0$ , $ a_na_{n \\plus{} 3} \\equal{} a_{n \\plus{} 2}a_{n \\plus{} 5}$ and $ a_1a_2 \\plus{} a_3a_4 \\plus{} a_5a_6 \\equal{} 6$ . So $ a_1a_2 \\plus{} a_3a_4 \\plus{} \\cdots \\plus{}... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 37466,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $b\\tan A = (2c-b)\\tan B$. \n$(1)$ Find angle $A$; \n$(2)$ If $\\overrightarrow{m}=(0,-1)$ and $\\overrightarrow{n}=(\\cos B, 2\\cos^2\\frac{C}{2})$, find the mi... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 37467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the domains of functions $f(x)$ and $g(x)$ are both $\\mathbb{R}$, and $f(x) + g(2-x) = 5$, $g(x) - f(x-4) = 7$, if the graph of $y=g(x)$ is symmetric about the line $x=2$ and $g(2) = 4$, find $\\sum _{k=1}^{22}f(k)$. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "-24"
} | {
"split": "train",
"index": 37468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute: $5^2-3(4)+3^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 37469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "143"
} | {
"split": "train",
"index": 37470,
"solution": "1. **Understanding the problem**: Claire calculated the sum of the interior angles of a convex polygon as $2017^\\circ$. However, she missed one angle. We need to find the measure of this forgotten angle.\n\n2. **Using the formula for the sum of interior angles**: ... |
deepscaler | [
{
"role": "user",
"content": "A and B began riding bicycles from point A to point C, passing through point B on the way. After a while, A asked B, \"How many kilometers have we ridden?\" B responded, \"We have ridden a distance equivalent to one-third of the distance from here to point B.\" After riding ano... | math | {
"style": "rule",
"ground_truth": "\\frac{40}{3}"
} | {
"split": "train",
"index": 37471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is four, Elías is five, and so following each person is friend of a person more than the previous person, until reaching Yvonne, th... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 37472,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the tens digit and the units digit in the decimal representation of $9^{2004}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 37473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows two 10 by 14 rectangles which are edge-to-edge and share a common vertex. It also shows the center \\( O \\) of one rectangle and the midpoint \\( M \\) of one edge of the other. What is the distance \\( OM \\)?\n\nA) 12\n\nB) 15\n\nC) 18\n\nD) 21\n\nE) 24 Let... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 37474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the arithmetic sequences $\\{a_n\\}$ and $\\{b_n\\}$ have the sum of the first $n$ terms denoted by $S_n$ and $T_n$ respectively. If for any natural number $n$ it holds that $\\dfrac{S_n}{T_n} = \\dfrac{2n-3}{4n-3}$, find the value of $\\dfrac{a_9}{b_5+b_7} + \\dfrac{a_3}{b... | math | {
"style": "rule",
"ground_truth": "\\dfrac{19}{41}"
} | {
"split": "train",
"index": 37475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many different ways can four couples sit around a circular table such that no couple sits next to each other? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1488"
} | {
"split": "train",
"index": 37476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the expression $(1296^{\\log_6 4096})^{\\frac{1}{4}}$, calculate its value. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4096"
} | {
"split": "train",
"index": 37477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the equation below, $A$ and $B$ are consecutive positive integers, and $A$, $B$, and $A+B$ represent number bases: \\[132_A+43_B=69_{A+B}.\\]What is $A+B$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 37478,
"solution": "1. **Convert the given equation to base 10:** \n The equation given is $132_A + 43_B = 69_{A+B}$. We need to express each number in base 10. \n - For $132_A$, it represents $1 \\cdot A^2 + 3 \\cdot A + 2$.\n - For $43_B$, it represents $4 \\cdot B + 3$.\n ... |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be positive real numbers. Find the minimum value of\n\\[\\left( x + \\frac{1}{y} \\right) \\left( x + \\frac{1}{y} - 1024 \\right) + \\left( y + \\frac{1}{x} \\right) \\left( y + \\frac{1}{x} - 1024 \\right).\\] Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "-524288"
} | {
"split": "train",
"index": 37479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ and $c$ be the roots of\n\\[x^3 - 5x + 7 = 0.\\]Find the monic polynomial, in $x,$ whose roots are $a - 2,$ $b - 2,$ and $c - 2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^3 + 6x^2 + 7x + 5"
} | {
"split": "train",
"index": 37480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When placing each of the digits $2,4,5,6,9$ in exactly one of the boxes of this subtraction problem, what is the smallest difference that is possible?\n\\[\\begin{array}{cccc} & \\boxed{} & \\boxed{} & \\boxed{} \\\\ - & & \\boxed{} & \\boxed{} \\\\ \\hline \\end{array}\\] Let... | math | {
"style": "rule",
"ground_truth": "149"
} | {
"split": "train",
"index": 37481,
"solution": "To find the smallest possible difference between a three-digit number $a$ and a two-digit number $b$ using the digits $2, 4, 5, 6, 9$ exactly once each, we need to minimize $a$ and maximize $b$. \n\n1. **Minimize $a$:** To minimize a three-digit number $a = \\overl... |
deepscaler | [
{
"role": "user",
"content": "An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first mo... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 37482,
"solution": "Note that we don't care which exact vertex the ant is located at, just which level (either top face or bottom face). Consider the ant to be on any of the two levels and having moved at least one move. Define $p_i$ to be the probability that after $i$ moves, the ant... |
deepscaler | [
{
"role": "user",
"content": "There are two values of $a$ for which the equation $4x^2+ax+8x+9=0$ has only one solution for $x$. What is the sum of those values of $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"split": "train",
"index": 37483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\nf(n) =\n\\begin{cases}\nn^2+1 & \\text{if }n\\text{ is odd} \\\\\n\\dfrac{n}{2} & \\text{if }n\\text{ is even}\n\\end{cases}.\n$\n\nFor how many integers $n$ from 1 to 100, inclusive, does $f ( f (\\dotsb f (n) \\dotsb )) = 1$ for some number of applications of $f$? Let'... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 37484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One more than $11$ times a certain prime $p$ is another prime $q$. What is the value of $q$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 37485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square pyramid has a base edge of 32 inches and an altitude of 1 foot. A square pyramid whose altitude is one-fourth of the original altitude is cut away at the apex of the original pyramid. The volume of the remaining frustum is what fractional part of the volume of the orig... | math | {
"style": "rule",
"ground_truth": "\\frac{63}{64}"
} | {
"split": "train",
"index": 37486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a regular hexagon \\( ABCDEF \\), the diagonals \\( AC \\) and \\( CE \\) are divided by interior points \\( M \\) and \\( N \\) in the following ratio: \\( AM : AC = CN : CE = r \\). If the points \\( B, M, N \\) are collinear, find the ratio \\( r \\). Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 37487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\textstyle\\binom{2k}k$ results in a number that ends in two zeros, find the smallest positive integer $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 37488,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let vectors $\\overrightarrow{a}$, $\\overrightarrow{b}$, $\\overrightarrow{c}$ satisfy $|\\overrightarrow{a}|=|\\overrightarrow{b}|=1$, $\\overrightarrow{a}\\cdot \\overrightarrow{b}= \\frac{1}{2}$, and $(\\overrightarrow{a}- \\overrightarrow{c})\\cdot(\\overrightarrow{b}- \\o... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}+1}{2}"
} | {
"split": "train",
"index": 37489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A club consisting of $11$ men and $12$ women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as $1$ member or as many as $23$ members. Let $N$ be... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 37490,
"solution": "Let $k$ be the number of women selected. Then, the number of men not selected is $11-(k-1)=12-k$. Note that the sum of the number of women selected and the number of men not selected is constant at $12$. Each combination of women selected and men not selected corre... |
deepscaler | [
{
"role": "user",
"content": "Given point P(-2,0) and line l: (1+3λ)x + (1+2λ)y = 2+5λ (λ ∈ ℝ), find the maximum value of the distance from point P to line l. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"split": "train",
"index": 37491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many functions $f:\\{1,2, \\ldots, 10\\} \\rightarrow\\{1,2, \\ldots, 10\\}$ satisfy the property that $f(i)+f(j)=11$ for all values of $i$ and $j$ such that $i+j=11$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100000"
} | {
"split": "train",
"index": 37492,
"solution": "To construct such a function $f$, we just need to choose a value for $f(x)$ from $\\{1,2, \\ldots, 10\\}$ for each $x \\in\\{1,2, \\ldots, 10\\}$. But the condition that $f(i)+f(j)=11$ whenever $i+j=11$ means that $$\\begin{aligned} f(10) & =11-f(1) \\\\ f(9) & =11... |
deepscaler | [
{
"role": "user",
"content": "For $1 \\le n \\le 100$, how many integers are there such that $\\frac{n}{n+1}$ is a repeating decimal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "86"
} | {
"split": "train",
"index": 37493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice, Bob, and Conway are playing rock-paper-scissors. Each player plays against each of the other $2$ players and each pair plays until a winner is decided (i.e. in the event of a tie, they play again). What is the probability that each player wins exactly once? Let's think... | math | {
"style": "rule",
"ground_truth": "1/4"
} | {
"split": "train",
"index": 37494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box contains five cards, numbered 1, 2, 3, 4, and 5. Three cards are selected randomly without replacement from the box. What is the probability that 4 is the largest value selected? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"split": "train",
"index": 37495,
"solution": "To find the probability that 4 is the largest value selected, we need to consider two scenarios:\n1. The number 5 is not selected.\n2. The numbers selected are all less than or equal to 4.\n\n#### Step 1: Calculate the probability of not selecting 5.\nThe probabili... |
deepscaler | [
{
"role": "user",
"content": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl}\n-x + 3 & \\text{if } x \\le 0, \\\\\n2x - 5 & \\text{if } x > 0.\n\\end{array}\n\\right.\\]How many solutions does the equation $f(f(x)) = 4$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 37496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $n$ in the equation $n + (n + 1) + (n + 2) = 9$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 37497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Once in a restaurant ***Dr. Strange*** found out that there were 12 types of food items from 1 to 12 on the menu. He decided to visit the restaurant 12 days in a row and try a different food everyday. 1st day, he tries one of the items from the first two. On the 2nd day, he eat... | math | {
"style": "rule",
"ground_truth": "2048"
} | {
"split": "train",
"index": 37498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mr. Fast eats a pizza slice in 5 minutes, Mr. Slow eats a pizza slice in 10 minutes, and Mr. Steady eats a pizza slice in 8 minutes. How long does it take for them to eat 24 pizza slices together? Express your answer in minutes. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 37499,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular box measures $a \\times b \\times c$, where $a$, $b$, and $c$ are integers and $1\\leq a \\leq b \\leq c$. The volume and the surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 37500,
"solution": "1. **Equating Volume and Surface Area**: Given a rectangular box with dimensions $a \\times b \\times c$, the volume is $abc$ and the surface area is $2(ab + bc + ca)$. Setting these equal gives:\n \\[\n 2(ab + bc + ca) = abc.\n \\]\n Dividing both sides by... |
deepscaler | [
{
"role": "user",
"content": "The distances from point \\( P \\), which lies inside an equilateral triangle, to its vertices are 3, 4, and 5. Find the area of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9 + \\frac{25\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 37501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that the numbers \\(x, y, z \\) form an arithmetic progression in that order with a common difference \\(\\alpha = \\arcsin \\frac{\\sqrt{7}}{4}\\), and the numbers \\(\\frac{1}{\\sin x}, \\frac{4}{\\sin y}, \\frac{1}{\\sin z}\\) also form an arithmetic progression ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{13}"
} | {
"split": "train",
"index": 37502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$, $b$, $c$ be positive numbers, and $a+b+9c^2=1$. Find the maximum value of $\\sqrt{a}+ \\sqrt{b}+ \\sqrt{3}c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{21}}{3}"
} | {
"split": "train",
"index": 37503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function $f$ has the property that $f(3x-1)=x^2+x+1$ for all real numbers $x$. What is $f(5)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 37504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $A$ lies on the line $y=\\frac{8}{15} x-6$, and point $B$ 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": "1334/255"
} | {
"split": "train",
"index": 37505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the side lengths 2, 3, 5, 7, and 11, how many different triangles with exactly two equal sides can be formed? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 37506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle can be inscribed in a trapezoid with bases of lengths 3 and 5, and a circle can also be circumscribed about it. Calculate the area of the pentagon formed by the radii of the inscribed circle that are perpendicular to the legs of the trapezoid, its smaller base, and the... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{15}}{2}"
} | {
"split": "train",
"index": 37507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A spinner has eight congruent sections, each labeled with numbers 1 to 8. Jane and her brother each spin this spinner once. Jane wins if the non-negative difference of their numbers is less than three; otherwise, her brother wins. Determine the probability of Jane winning. Expr... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{32}"
} | {
"split": "train",
"index": 37508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the matrix $\\mathbf{M}$ such that\n\\[\\mathbf{M} \\begin{pmatrix} -3 & 4 & 0 \\\\ 5 & -7 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} = \\mathbf{I}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -7 & -4 & 0 \\\\ -5 & -3 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix}"
} | {
"split": "train",
"index": 37509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $| \\overrightarrow{a}|=| \\overrightarrow{b}|=| \\overrightarrow{c}|=1$, and $ \\overrightarrow{a}+ \\overrightarrow{b}+ \\sqrt {3} \\overrightarrow{c}=0$, find the value of $ \\overrightarrow{a} \\overrightarrow{b}+ \\overrightarrow{b} \\overrightarrow{c}+ \\overri... | math | {
"style": "rule",
"ground_truth": "\\dfrac {1}{2}- \\sqrt {3}"
} | {
"split": "train",
"index": 37510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[b^2 f(a) = a^2 f(b)\\]for all real numbers $a$ and $b.$ If $f(2) \\neq 0,$ find\n\\[\\frac{f(5) - f(1)}{f(2)}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 37511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given point A (-2, 1) and circle C: $(x-2)^2+(y-2)^2=1$, a ray of light is emitted from point A to the x-axis and then reflects in the direction of the tangent to the circle. The distance traveled by the ray of light from point A to the tangent point is ______. Let's think step... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{6}"
} | {
"split": "train",
"index": 37512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The length of the segment between the points $(2a, a-4)$ and $(4, -1)$ is $2\\sqrt{10}$ units. What is the product of all possible values for $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 37513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Dorothea has a $3 \\times 4$ grid of dots. She colors each dot red, blue, or dark gray. Compute the number of ways Dorothea can color the grid such that there is no rectangle whose sides are parallel to the grid lines and whose vertices all have the same color. Let's think step... | math | {
"style": "rule",
"ground_truth": "284688"
} | {
"split": "train",
"index": 37514,
"solution": "To find an appropriate estimate, we will lower bound the number of rectangles. Let $P(R)$ be the probability a random 3 by 4 grid will have a rectangle with all the same color in the grid. Let $P(r)$ be the probability that a specific rectangle in the grid will hav... |
deepscaler | [
{
"role": "user",
"content": "There are 500 machines, each using 6 parts of the same type. These parts must be replaced by the end of the week if they fail. Out of all new parts, 10% fail by the end of the first week, 30% fail by the end of the second week, and the remaining 60% fail by the end of the third... | math | {
"style": "rule",
"ground_truth": "3000"
} | {
"split": "train",
"index": 37515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integer values of $n$ satisfy $-50 < n^3 < 50$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 37516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that 9 boys and 15 girls line up in a row, but the arrangement must start with a boy and end with a girl. Let $T$ be the number of places in the row where a boy and a girl are standing next to each other. Calculate the average value of $T$ when all possible orders of th... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 37517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of $8^{\\text{th}}$-graders to $6^{\\text{th}}$-graders is $5:3$, and the the ratio of $8^{\\text{th}}$-graders to $7^{\\text{th}}$-graders is $8:5$. What is the smallest... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 37518,
"solution": "\n#### Step 1: Establish the ratios\nWe are given two ratios:\n- The ratio of $8^\\text{th}$-graders to $6^\\text{th}$-graders is $5:3$.\n- The ratio of $8^\\text{th}$-graders to $7^\\text{th}$-graders is $8:5$.\n\n#### Step 2: Equalize the number of $8^\\text{th}$... |
deepscaler | [
{
"role": "user",
"content": "In a triangle with integer side lengths, one side is twice as long as a second side, and the length of the third side is 17. What is the greatest possible perimeter of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 37519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an increasing sequence $\\{a_n\\}$ that satisfies $a_{n+1}a_{n-1} = a_n^2$ (for $n \\geq 2$, $n \\in \\mathbb{N}$), the sum of the first 10 terms equals 50, and the sum of the first 15 terms is 210, calculate the sum of the first 5 terms ($S_5$). Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 37520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer \\overline{A B C}, where $A, B, C$ are digits, satisfies $\\overline{A B C}=B^{C}-A$. Find $\\overline{A B C}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "127"
} | {
"split": "train",
"index": 37521,
"solution": "The equation is equivalent to $100 A+10 B+C=B^{C}-A$. Suppose $A=0$, so that we get $10 B+C=B^{C}$. Reducing $\\bmod B$, we find that $C$ must be divisible by $B$. $C \\neq 0$, since otherwise $10 B=1$, contradiction, so $C \\geq B$. Thus $10 B+C \\geq B^{B}$ for d... |
deepscaler | [
{
"role": "user",
"content": "In the arithmetic sequence $\\{a_n\\}$, $S_{10} = 4$, $S_{20} = 20$. What is $S_{30}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 37522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Queen High School has $1500$ students, and each student takes $6$ classes per day. Each teacher teaches $5$ classes, with each class having $25$ students and $1$ teacher. How many teachers are there at Queen High School? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 37523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $(x^2 - k)(x + k) = x^3 + k(x^2 - x - 5)$ and $k\\neq 0$, what is the value of $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 37524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular prism has dimensions of 1 by 1 by 2. Calculate the sum of the areas of all triangles whose vertices are also vertices of this rectangular prism, and express the sum in the form $m + \\sqrt{n} + \\sqrt{p}$, where $m, n,$ and $p$ are integers. Find $m + n + p$. Let'... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 37525,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Anton ran down a moving escalator and counted 30 steps. Then he decided to run up the same escalator at the same speed relative to the escalator and counted 150 steps. How many steps did he count while descending with a policeman on a stationary escalator? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 37526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The side of rhombus \\(ABCD\\) is equal to 5. A circle with a radius of 2.4 is inscribed in this rhombus.\n\nFind the distance between the points where this circle touches the sides \\(AB\\) and \\(BC\\), if the diagonal \\(AC\\) is less than the diagonal \\(BD\\). Let's think ... | math | {
"style": "rule",
"ground_truth": "3.84"
} | {
"split": "train",
"index": 37527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(P(x)\\) be a quadratic polynomial with real coefficients. Suppose that \\(P(1)=20, P(-1)=22\\), and \\(P(P(0))=400\\). Compute the largest possible value of \\(P(10)\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2486"
} | {
"split": "train",
"index": 37528,
"solution": "Let \\(P(x)=ax^{2}+bx+c\\). The given equations give us: \\(a+b+c=20\\) and \\(a-b+c=22\\). Hence \\(b=-1, a+c=21\\), and so the final equation gives us \\(ac^{2}=400\\). Substituting \\(a=21-c\\) and solving the cubic in \\(c\\), we get \\(c=-4,5,20\\). Of these, ... |
deepscaler | [
{
"role": "user",
"content": "A segment with endpoints at $A(2, -2)$ and $B(14, 4)$ is extended through $B$ to point $C$. If $BC = \\frac{1}{3} \\cdot AB$, what are the coordinates for point $C$? Express your answer as an ordered pair. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "(18,6)"
} | {
"split": "train",
"index": 37529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a convex polygon with 1992 sides, the minimum number of interior angles that are not acute is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1989"
} | {
"split": "train",
"index": 37530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the minimum number of equilateral triangles, of side length 1 unit, needed to cover an equilateral triangle of side length 10 units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 37531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Medians $\\overline{DP}$ and $\\overline{EQ}$ of $\\triangle DEF$ are perpendicular. If $DP= 15$ and $EQ = 20$, then what is ${DF}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{20\\sqrt{13}}{3}"
} | {
"split": "train",
"index": 37532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A whole block of modeling clay is a right rectangular prism six inches by two inches by one inch. How many whole blocks need to be unwrapped to mold a cylindrical sculpture seven inches high and four inches in diameter? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 37533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $12345 + 23451 + 34512 + 45123 + 51234$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "166665"
} | {
"split": "train",
"index": 37534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the signal of the trainer, two ponies simultaneously started running uniformly along the outer circumference of the circus arena in opposite directions. The first pony ran slightly faster than the second and, by the time they met, had run 5 meters more than the second pony. ... | math | {
"style": "rule",
"ground_truth": "6.25"
} | {
"split": "train",
"index": 37535,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Chandra now has five bowls and five glasses, and each expands to a new set of colors: red, blue, yellow, green, and purple. However, she dislikes pairing the same colors; thus, a bowl and glass of the same color cannot be paired together like a red bowl with a red glass. How ma... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 37536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many three-digit perfect cubes are divisible by $9?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 37537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two values of $k$ for which the equation $x^{2}+2kx+7k-10=0$ has two equal real roots (that is, has exactly one solution for $x$). What is the sum of these values of $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 37538,
"solution": "The equation $x^{2}+2kx+7k-10=0$ has two equal real roots precisely when the discriminant of this quadratic equation equals 0. The discriminant, $\\Delta$, equals $\\Delta=(2k)^{2}-4(1)(7k-10)=4k^{2}-28k+40$. For the discriminant to equal 0, we have $4k^{2}-28k+40=... |
deepscaler | [
{
"role": "user",
"content": "How many of the 20 perfect squares $1^{2}, 2^{2}, 3^{2}, \\ldots, 19^{2}, 20^{2}$ are divisible by 9? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 37539,
"solution": "A perfect square is divisible by 9 exactly when its square root is divisible by 3. In other words, $n^{2}$ is divisible by 9 exactly when $n$ is divisible by 3. In the list $1,2,3, \\ldots, 19,20$, there are 6 multiples of 3. Therefore, in the list $1^{2}, 2^{2}, 3... |
deepscaler | [
{
"role": "user",
"content": "Ramon sells two enchiladas and three tacos for $\\$$2.50 and he sells three enchiladas and two tacos for $\\$$2.70. Assuming a fixed price per item, what is the cost, in dollars, of three enchiladas and four tacos? Express your answer as a decimal to the nearest hundredth. Let'... | math | {
"style": "rule",
"ground_truth": "\\$3.54"
} | {
"split": "train",
"index": 37540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Melinda has three empty boxes and $12$ textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in ran... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 37541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, a moving point $M(x,y)$ always satisfies the relation $2 \\sqrt {(x-1)^{2}+y^{2}}=|x-4|$.\n$(1)$ What is the trajectory of point $M$? Write its standard equation.\n$(2)$ The distance from the origin $O$ to the line $l$: $y=kx+m$ is $1$.... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{7}}{5}"
} | {
"split": "train",
"index": 37542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive value of $x$ such that $x + 8765$ results in a palindrome? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 37543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$F, G, H, I,$ and $J$ are collinear in that order such that $FG = 2, GH = 1, HI = 3,$ and $IJ = 7$. If $P$ can be any point in space, what is the smallest possible value of $FP^2 + GP^2 + HP^2 + IP^2 + JP^2$? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "102.8"
} | {
"split": "train",
"index": 37544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many rectangles can be formed by the vertices of a cube? (Note: square is also a special rectangle). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 37545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Molly and Jake play a game where Molly picks an integer between 1 and 720 inclusive, and Jake divides 720 by that integer and states whether the quotient is an integer. How many integers could Molly pick such that Jake’s quotient is an integer? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 37546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four concentric circles are drawn with radii of 1, 3, 5 and 7. The inner circle is painted black, the ring around it is white, the next ring is black and the outer ring is white. What is the ratio of the black area to the white area? Express your answer as a common fraction. Le... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{32}"
} | {
"split": "train",
"index": 37547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A student, Leo, needs to earn 30 study points for a special credit. For the first 6 points, he needs to complete 1 project each. For the next 6 points, he needs 2 projects each; for the next 6 points, 3 projects each, and so on. Determine the minimum number of projects Leo need... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 37548,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.