data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The graph of the function in the form \\( y=\\frac{b}{|x|-a} \\) (where \\( a, b > 0 \\)) resembles the Chinese character \"ๅ\". It is referred to as the \"ๅ function\", and the point symmetric to its intersection with the y-axis about the origin is called the \"็ฎ point\". A ci... | math | {
"style": "rule",
"ground_truth": "3\\pi"
} | {
"split": "train",
"index": 32849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\(a\\), \\(b\\), \\(c\\), and \\(d\\) are four positive prime numbers such that the product of these four prime numbers is equal to the sum of 55 consecutive positive integers, find the smallest possible value of \\(a + b + c + d\\). Note that the four numbers \\(a\... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 32850,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function\n$$\nf(x)=\\left(1-x^{2}\\right)\\left(x^{2}+b x+c\\right) \\text{ for } x \\in [-1, 1].\n$$\n\nLet $\\mid f(x) \\mid$ have a maximum value of $M(b, c)$. As $b$ and $c$ vary, find the minimum value of $M(b, c)$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "3 - 2\\sqrt{2}"
} | {
"split": "train",
"index": 32851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagonal of square $I$ is $a+b$. The area of square $II$ with twice the area of $I$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(a+b)^2"
} | {
"split": "train",
"index": 32852,
"solution": "1. **Identify the side length of square $I$**: Given that the diagonal of square $I$ is $a+b$, we can use the relationship between the side length $s$ and the diagonal $d$ of a square, which is $d = s\\sqrt{2}$. Therefore, the side length $s$ of square $I$ is:\n ... |
deepscaler | [
{
"role": "user",
"content": "The equation of one of the axes of symmetry for the graph of the function $f(x)=\\sin \\left(x- \\frac {\\pi}{4}\\right)$ $(x\\in\\mathbb{R})$ can be found. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 32853,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an odd function $f(x)$ defined on $R$ satisfying $f(x+1)=f(1-x)$, when $0\\leqslant x\\leqslant 1$, $f(x)=x^{3}$, then $f(2019)=\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 32854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the sequence $00$ , $01$ , $02$ , $03$ , $\\cdots$ , $99$ the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by $1$ (for example, $29$ can be followed by $19$ , $39$ , or $28$ , but not by ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 32855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ be the number of students in Danny's high school. If Maria's high school has $4$ times as many students as Danny's high school, then Maria's high school has $4x$ students. The difference between the number of students in the two high schools is $1800$, so we have the eq... | math | {
"style": "rule",
"ground_truth": "2400"
} | {
"split": "train",
"index": 32856,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three people, A, B, and C, visit three tourist spots, with each person visiting only one spot. Let event $A$ be \"the three people visit different spots,\" and event $B$ be \"person A visits a spot alone.\" Then, the probability $P(A|B)=$ ______. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{2}"
} | {
"split": "train",
"index": 32857,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The natural numbers from 1951 to 1982 are arranged in a certain order one after another. A computer reads two consecutive numbers from left to right (i.e., the 1st and 2nd, the 2nd and 3rd, etc.) until the last two numbers. If the larger number is on the left, the computer swap... | math | {
"style": "rule",
"ground_truth": "1982"
} | {
"split": "train",
"index": 32858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of\n\\[9y + \\frac{1}{y^6}\\] for \\(y > 0\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 32859,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six distinct positive integers are randomly chosen between $1$ and $2006$, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of $5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 32860,
"solution": "1. **Understanding the Problem:**\n We need to find the probability that among six distinct positive integers chosen from the set $\\{1, 2, \\ldots, 2006\\}$, there exists at least one pair whose difference is a multiple of $5$.\n\n2. **Using Modular Arithmetic:*... |
deepscaler | [
{
"role": "user",
"content": "Find $\\tan G$ in the right triangle shown below.\n\n[asy]\n\npair H,F,G;\n\nH = (0,0);\n\nG = (15,0);\n\nF = (0,8);\n\ndraw(F--G--H--F);\n\ndraw(rightanglemark(F,H,G,20));\n\nlabel(\"$H$\",H,SW);\n\nlabel(\"$G$\",G,SE);\n\nlabel(\"$F$\",F,N);\n\nlabel(\"$17$\",(F+G)/2,NE);\n\n... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{15}"
} | {
"split": "train",
"index": 32861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lateral surface of a cylinder unfolds into a square. What is the ratio of its lateral surface area to the base area. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"split": "train",
"index": 32862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of two positive integers $a$ and $b$ is 1001. What is the largest possible value of $\\gcd(a,b)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "143"
} | {
"split": "train",
"index": 32863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find \\[\\dfrac{3}{\\,\\,\\frac{6}{11}\\,\\,}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{2}"
} | {
"split": "train",
"index": 32864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a rectangle and let $\\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\\overline{DM}$ has integer length, and the lengths of $\\overline{MA}, \\overline{MC},$ and $\\overline{MB}$ are consecutive odd positive integers (in this order... | math | {
"style": "rule",
"ground_truth": "24\\sqrt{5}"
} | {
"split": "train",
"index": 32865,
"solution": "1. **Assigning Variables to Lengths**:\n Let the length of $\\overline{DM}$ be $d$. Since $\\overline{MA}, \\overline{MC},$ and $\\overline{MB}$ are consecutive odd positive integers, let $\\overline{MA} = a$, $\\overline{MC} = a+2$, and $\\overline{MB} = a+4$.\n... |
deepscaler | [
{
"role": "user",
"content": "The sum of two numbers is 6. The difference of their squares is 12. What is the positive difference of the two numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 32866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many even integers between 3000 and 6000 have four different digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "784"
} | {
"split": "train",
"index": 32867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The volume of a hemispherical soup bowl is 8 liters. How much soup fills the bowl up to half its height? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"split": "train",
"index": 32868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An assembly line produces an average of $85\\%$ first-grade products. How many products need to be selected so that with a probability of 0.997, the deviation of the frequency of first-grade products from the probability $p=0.85$ does not exceed $0.01$ in absolute value? Let's ... | math | {
"style": "rule",
"ground_truth": "11171"
} | {
"split": "train",
"index": 32869,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number that cannot be written in the form \\(\\frac{2^{a} - 2^{b}}{2^{c} - 2^{d}}\\), where \\(a\\), \\(b\\), \\(c\\), and \\(d\\) are natural numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 32870,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Every positive integer $k$ has a unique factorial base expansion $(f_1,f_2,f_3,\\ldots,f_m)$, meaning that $k=1!\\cdot f_1+2!\\cdot f_2+3!\\cdot f_3+\\cdots+m!\\cdot f_m$, where each $f_i$ is an integer, $0\\le f_i\\le i$, and $0<f_m$. Given that $(f_1,f_2,f_3,\\ldots,f_j)$ is ... | math | {
"style": "rule",
"ground_truth": "495"
} | {
"split": "train",
"index": 32871,
"solution": "Note that $1+\\sum_{k=1}^{n-1} {k\\cdot k!} = 1+\\sum_{k=1}^{n-1} {((k+1)\\cdot k!- k!)} = 1+\\sum_{k=1}^{n-1} {((k+1)!- k!)} = n!$\n Thus for all $m\\in\\mathbb{N}$,\n$(32m+16)!-(32m)! = \\left(1+\\sum_{k=1}^{32m+15} {k\\cdot k!}\\right)-\\left(1+\\sum_{k=1}^{32m-... |
deepscaler | [
{
"role": "user",
"content": "What are the first three digits to the right of the decimal point in the decimal representation of $(10^{2003}+1)^{11/7}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "571"
} | {
"split": "train",
"index": 32872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the distance between the foci of the hyperbola $x^2 - 4x - 9y^2 - 18y = 45.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{40}{3}"
} | {
"split": "train",
"index": 32873,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The slope angle of the line passing through points M(-3, 2) and N(-2, 3) is equal to what angle, measured in radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 32874,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be an equilateral triangle. Extend side $\\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3 \\cdot AB$. Similarly, extend side $\\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3 \\cdot BC$, and extend side $\\overline{CA}$ beyond $A$ to a point $A'$ so ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 32875,
"solution": "1. **Define the side length of $\\triangle ABC$:** Let $AB = BC = CA = x$.\n\n2. **Extend the sides of $\\triangle ABC$:** \n - Extend $AB$ to $B'$ such that $BB' = 3AB$. Thus, $AB' = AB + BB' = x + 3x = 4x$.\n - Extend $BC$ to $C'$ such that $CC' = 3BC$. Thus,... |
deepscaler | [
{
"role": "user",
"content": "One day Max says to Liz, \"Out of the 25 people taking either English or French, you and I are the only two taking both.'' Liz, being mathematically inclined, responds by pointing out that there are exactly twice as many people in the English class as there are in the French cl... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 32876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Marvin had a birthday on Tuesday, May 27 in the leap year $2008$. In what year will his birthday next fall on a Saturday? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"split": "train",
"index": 32877,
"solution": "To determine the next year when Marvin's birthday, May 27, falls on a Saturday after 2008, we need to consider the day of the week progression from 2008 onwards, taking into account whether each year is a leap year or not.\n\n1. **Day Increment Calculation**:\n -... |
deepscaler | [
{
"role": "user",
"content": "Given that \\(7^{-1} \\equiv 55 \\pmod{101}\\), find \\(49^{-1} \\pmod{101}\\), as a residue modulo 101. (Answer should be between 0 and 100, inclusive.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 32878,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the graph of the exponential function $y=f(x)$ passes through the point $(\\frac{1}{2}, \\frac{\\sqrt{2}}{2})$, determine the value of $\\log_{2}f(2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 32879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $a=2b$. Also, $\\sin A$, $\\sin C$, and $\\sin B$ form an arithmetic sequence. \n$(1)$ Find the value of $\\cos A$; \n$(2)$ If the area of $... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"split": "train",
"index": 32880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Out of 500 participants in a remote math olympiad, exactly 30 did not like the problem conditions, exactly 40 did not like the organization of the event, and exactly 50 did not like the method used to determine the winners. A participant is called \"significantly dissatisfied\"... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 32881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the probability mass function of the random variable $X$ is $P(X=k)= \\frac{k}{25}$ for $k=1, 2, 3, 4, 5$, find the value of $P(\\frac{1}{2} < X < \\frac{5}{2})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 32882,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x$, $y$, and $z$ are all non-negative numbers and $x + y + z = 2$, find the minimum value of $\\frac{1}{3}x^{3} + y^{2} + z$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{12}"
} | {
"split": "train",
"index": 32883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be the product of the first $100$ positive odd integers. Find the largest integer $k$ such that $P$ is divisible by $3^k .$\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 32884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( n! \\) is evenly divisible by \\( 1 + 2 + \\cdots + n \\), find the number of positive integers \\( n \\) less than or equal to 50. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 32885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mean of one set of five numbers is 13, and the mean of a separate set of six numbers is 24. What is the mean of the set of all eleven numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 32886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a regular octagon and a square formed by drawing four diagonals of the octagon. The edges of the square have length 1. What is the area of the octagon?\n\nA) \\(\\frac{\\sqrt{6}}{2}\\) \nB) \\(\\frac{4}{3}\\) \nC) \\(\\frac{7}{5}\\) \nD) \\(\\sqrt{2}\\) \n... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 32887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the coefficient of the $x^5$ term in the expansion of $(x+1)(x^2-x-2)^3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 32888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the perimeter of $\\triangle UVZ$ if $UVWX$ is a rectangle that lies flat on a horizontal floor, a vertical semi-circular wall with diameter $XW$ is constructed, point $Z$ is the highest point on this wall, and $UV=20$ and $VW=30$? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "86"
} | {
"split": "train",
"index": 32889,
"solution": "The perimeter of $\\triangle UVZ$ equals $UV+UZ+VZ$. \n We know that $UV=20$. We need to calculate $UZ$ and $VZ$. \n Let $O$ be the point on $XW$ directly underneath $Z$. \n Since $Z$ is the highest point on the semi-circle and $XW$ is the diameter, then $O$ is the... |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then choose a suitable value for $x$ from $2$, $-2$, and $-6$ to substitute and evaluate.<br/>$\\frac{{x}^{3}+2x^{2}}{{x}^{2}-4x+4}รท\\frac{4x+8}{x-2}-\\frac{1}{x-2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 32890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be positive real numbers with $a\\ge b$. Let $\\rho$ be the maximum possible value of $\\frac {a}{b}$ for which the system of equations \\[a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2\\] has a solution in $(x,y)$ satisfying $0\\le x < a$ and $0\\le y < b$. Then... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 32891,
"solution": "Notice that the given equation implies\n$a^2 + y^2 = b^2 + x^2 = 2(ax + by)$\n We have $2by \\ge y^2$, so $2ax \\le a^2 \\implies x \\le \\frac {a}{2}$.\n Then, notice $b^2 + x^2 = a^2 + y^2 \\ge a^2$, so $b^2 \\ge \\frac {3}{4}a^2 \\implies \\rho^2 \\le \\frac {4}... |
deepscaler | [
{
"role": "user",
"content": "A point is selected at random from the portion of the number line shown here. What is the probability that the point is closer to 4 than to 0? Express your answer as a decimal to the nearest tenth.\n\n[asy]unitsize(0.2inch);\ndraw((0,0)--(5,0));\ndraw((0,-0.1)--(0,0.1));\ndraw(... | math | {
"style": "rule",
"ground_truth": ".6"
} | {
"split": "train",
"index": 32892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In parallelogram $ABCD$, if $\\overrightarrow{AE}=2\\overrightarrow{ED}$, $\\overrightarrow{BF}=\\overrightarrow{FC}$, and $\\overrightarrow{AC}=ฮป\\overrightarrow{AE}+\\overrightarrow{AF}$, then $\\lambda =$____. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 32893,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a prime number which is a factor of every sum of three consecutive integers. What is the number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 32894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A child has a set of 96 distinct blocks. Each block is one of 2 materials (plastic, wood), 3 sizes (small, medium, large), 4 colors (blue, green, red, yellow), and 4 shapes (circle, hexagon, square, triangle). How many blocks in the set differ from the 'plastic medium red circl... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 32895,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find minimum of $x+y+z$ where $x$ , $y$ and $z$ are real numbers such that $x \\geq 4$ , $y \\geq 5$ , $z \\geq 6$ and $x^2+y^2+z^2 \\geq 90$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 32896,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the golden ratio $m = \\frac{{\\sqrt{5}-1}}{2}$, calculate the value of $\\frac{{\\sin{42}ยฐ+m}}{{\\cos{42}ยฐ}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 32897,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside a square of side length 1, four quarter-circle arcs are traced with the edges of the square serving as the radii. These arcs intersect pairwise at four distinct points, forming the vertices of a smaller square. This process is repeated for the smaller square, and continu... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{1 - (2 - \\sqrt{3})}"
} | {
"split": "train",
"index": 32898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle in the coordinate plane has vertices at $(0, 0), (1000, 0), (1000, 1000),$ and $(0, 1000)$. Compute the radius $d$ to the nearest tenth such that the probability the point is within $d$ units from any lattice point is $\\tfrac{1}{4}$. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"split": "train",
"index": 32899,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "9. The real quartic $P x^{4}+U x^{3}+M x^{2}+A x+C$ has four different positive real roots. Find the square of the smallest real number $z$ for which the expression $M^{2}-2 U A+z P C$ is always positive, regardless of what the roots of the quartic are. Let's think step b... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 32900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sets \\( M=\\{x, xy, \\lg(xy)\\} \\) and \\( N=\\{0, |x|, y\\} \\), and that \\( M=N \\), find the value of \\( \\left(x+\\frac{1}{y}\\right)+\\left(x^{2}+\\frac{1}{y^{2}}\\right)+\\left(x^{3}+\\frac{1}{y^{3}}\\right)+\\cdots+\\left(x^{2001}+\\frac{1}{y^{2001}}\\right... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 32901,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The shape shown is made up of three similar right-angled triangles. The smallest triangle has two sides of side-length 2, as shown. What is the area of the shape? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 32902,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ where the sum of the first $n$ terms is $S_n = (a+1)n^2 + a$, if the sides of a certain triangle are $a_2$, $a_3$, and $a_4$, then the area of this triangle is ________. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{4} \\sqrt{3}"
} | {
"split": "train",
"index": 32903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of different recommendation plans for the high school given that 3 male and 2 female students are selected as candidates, where both Russian and Japanese exams must include male participants, and 2 spots are available for Russian, 2 for Japanese, and 1 for Spani... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 32904,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Place the sequence $\\{2n+1\\}$ in parentheses sequentially, with the first parenthesis containing one number, the second two numbers, the third three numbers, the fourth four numbers, the fifth one number again, and then continuing in this cycle. Determine the sum of the numbe... | math | {
"style": "rule",
"ground_truth": "2104"
} | {
"split": "train",
"index": 32905,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $16\\left (\\frac{125}{2}+\\frac{25}{4}+\\frac{9}{16}+1\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1125"
} | {
"split": "train",
"index": 32906,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I'm thinking of two whole numbers. Their product is 24 and their sum is 11. What is the larger number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 32907,
"solution": "1. **Define the variables**: Let the two whole numbers be $x$ and $y$.\n\n2. **Set up the equations based on the problem statement**:\n - The product of the numbers is 24: \\( xy = 24 \\)\n - The sum of the numbers is 11: \\( x + y = 11 \\)\n\n3. **Express one ... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=-\\frac{1}{3}x^{3}+bx^{2}+cx+bc$ has an extreme value of $-\\frac{4}{3}$ at $x=1$, find the value of $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 32908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For all real numbers \\( r, s, t \\) satisfying \\( 1 \\leq r \\leq s \\leq t \\leq 4 \\), find the minimum value of \\( (r-1)^{2}+\\left(\\frac{s}{r}-1\\right)^{2} +\\left(\\frac{t}{s}-1\\right)^{2}+\\left(\\frac{4}{t}-1\\right)^{2} \\). Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "4(\\sqrt{2} - 1)^2"
} | {
"split": "train",
"index": 32909,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum number of numbers that can be selected from the set \\( 1, 2, \\ldots, 1963 \\) such that the sum of no two numbers is divisible by their difference? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "655"
} | {
"split": "train",
"index": 32910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a floor is tiled in a similar pattern with a $4 \\times 4$ unit repeated pattern and each of the four corners looks like the scaled down version of the original, determine the fraction of the tiled floor made up of darker tiles, assuming symmetry and pattern are pres... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 32911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In square $ABCD$, points $P$ and $Q$ lie on $\\overline{AD}$ and $\\overline{AB}$ respectively. Segments $\\overline{BP}$ and $\\overline{CQ}$ intersect at point $R$, with $BR = 8$ and $PR = 9$. If $\\triangle BRP$ is a right triangle with $\\angle BRP = 90^\\circ$, what is the... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 32912,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $\\log_{3} \\sqrt{27} + \\lg 25 + \\lg 4 + 7^{\\log_{7} 2} + (-9.8)^{0} = \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{2}"
} | {
"split": "train",
"index": 32913,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For integers $a$ and $b$ consider the complex number \\[\\frac{\\sqrt{ab+2016}}{ab+100}-\\left({\\frac{\\sqrt{|a+b|}}{ab+100}}\\right)i\\]\n Find the number of ordered pairs of integers $(a,b)$ such that this complex number is a real number. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "103"
} | {
"split": "train",
"index": 32914,
"solution": "Similar to Solution 1, but concise:\n First, we set the imaginary expression to $0$, so that $|a+b|=0$ or $a=-b$, of which there are $44\\cdot 2+1=89$ possibilities. But $(a,b)\\ne(\\pm 10,\\mp 10)$ because the denominator would be $0$. So this gives $89-2=87$ solu... |
deepscaler | [
{
"role": "user",
"content": "Given: $\\because 4 \\lt 7 \\lt 9$, $\\therefore 2 \\lt \\sqrt{7} \\lt 3$, $\\therefore$ the integer part of $\\sqrt{7}$ is $2$, and the decimal part is $\\sqrt{7}-2$. The integer part of $\\sqrt{51}$ is ______, and the decimal part of $9-\\sqrt{51}$ is ______. Let's think step... | math | {
"style": "rule",
"ground_truth": "8-\\sqrt{51}"
} | {
"split": "train",
"index": 32915,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first four terms of an arithmetic sequence are $p$, $9$, $3p-q$, and $3p+q$. What is the $2010^{\\text{th}}$ term of this sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8041"
} | {
"split": "train",
"index": 32916,
"solution": "1. **Identify the common difference**: \n Given the terms of the sequence are $p$, $9$, $3p-q$, and $3p+q$, we know that the common difference $d$ between consecutive terms can be calculated as:\n \\[\n d = (3p+q) - (3p-q) = 2q\n \\]\n\n2. **Set up equation... |
deepscaler | [
{
"role": "user",
"content": "A rectangular box has width $12$ inches, length $16$ inches, and height $\\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle wit... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 32917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two distinct numbers are selected from the set $\\{1,2,3,4,\\dots,38\\}$ so that the sum of the remaining $36$ numbers equals the product of these two selected numbers plus one. Find the difference of these two numbers. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 32918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nA right triangle \\(ABC\\) has a perimeter of 54, with the leg \\(AC\\) being greater than 10. A circle with a radius of 6, whose center lies on leg \\(BC\\), is tangent to lines \\(AB\\) and \\(AC\\). Find the area of triangle \\(ABC\\). Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "243/2"
} | {
"split": "train",
"index": 32919,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The foci of the ellipse \\(\\frac{x^2}{25} + \\frac{y^2}{b^2} = 1\\) and the foci of the hyperbola\n\\[\\frac{x^2}{196} - \\frac{y^2}{121} = \\frac{1}{49}\\] coincide. Find \\(b^2\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{908}{49}"
} | {
"split": "train",
"index": 32920,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the expression $\\frac{810 \\times 811 \\times 812 \\times \\cdots \\times 2010}{810^{n}}$ is an integer, find the maximum value of $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "149"
} | {
"split": "train",
"index": 32921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the largest prime number less than 5000 of the form \\( a^n - 1 \\), where \\( a \\) and \\( n \\) are positive integers, and \\( n \\) is greater than 1. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "127"
} | {
"split": "train",
"index": 32922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $(ax+b)(bx+a)=26x^2+\\Box\\cdot x+26$, where $a$, $b$, and $\\Box$ are distinct integers, what is the minimum possible value of $\\Box$, the coefficient of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "173"
} | {
"split": "train",
"index": 32923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability that at most two of the cars have a deceased passenger, given that there are 4 deceased passengers among the travelers on a metro train consisting of 6 cars? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/6"
} | {
"split": "train",
"index": 32924,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If two distinct numbers from the set $\\{ 5, 15, 21, 35, 45, 63, 70, 90 \\}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 105? Express your answer as a common fraction. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 32925,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Chen, Ruan, Lu, Tao, and Yang did push-ups. It is known that Chen, Lu, and Yang together averaged 40 push-ups per person, Ruan, Tao, and Chen together averaged 28 push-ups per person, and Ruan, Lu, Tao, and Yang together averaged 33 push-ups per person. How many push-ups did Ch... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 32926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a positive term geometric sequence ${a_n}$, ${a_5 a_6 =81}$, calculate the value of ${\\log_{3}{a_1} + \\log_{3}{a_5} +...+\\log_{3}{a_{10}}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 32927,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nOn the sides $AB, AC, BC$ of an equilateral triangle $ABC$, with a side length of 2, points $C_{1}, B_{1}, A_{1}$ are chosen respectively.\n\nWhat is the maximum possible value of the sum of the radii of the circles inscribed in the triangles $AB_{1}C_{1}$, $A_{1}BC_{1}$, and... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 32928,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\\left\\{\\begin{array}{l}x=1+\\frac{\\sqrt{2}}{2}t,\\\\ y=\\frac{\\sqrt{2}}{2}t\\end{array}\\right.$ (where $t$ is the parameter). Using $O$ as the pole and the positive half-axis o... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 32929,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct sequences of five letters can be made from the letters in \"PROBLEMS\" if each letter can be used only once and each sequence must begin with \"S\" and not end with \"M\"? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 32930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(z=\\frac{1+i}{\\sqrt{2}}.\\)What is \\(\\left(z^{1^2}+z^{2^2}+z^{3^2}+\\dots+z^{{12}^2}\\right) \\cdot \\left(\\frac{1}{z^{1^2}}+\\frac{1}{z^{2^2}}+\\frac{1}{z^{3^2}}+\\dots+\\frac{1}{z^{{12}^2}}\\right)?\\) Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 32931,
"solution": "1. **Identify the form of \\( z \\) and \\( \\frac{1}{z} \\):**\n Given \\( z = \\frac{1+i}{\\sqrt{2}} \\), we can express \\( z \\) in exponential form using Euler's formula \\( e^{i\\theta} = \\cos(\\theta) + i\\sin(\\theta) \\). Since \\( \\cos(\\frac{\\pi}{4}... |
deepscaler | [
{
"role": "user",
"content": "If $\\left(a+\\frac{1}{a}\\right)^{2}=3$, find $\\left(a+\\frac{1}{a}\\right)^{3}$ in terms of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 32932,
"solution": "0."
} |
deepscaler | [
{
"role": "user",
"content": "Distribute 16 identical books among 4 students so that each student gets at least one book, and each student gets a different number of books. How many distinct ways can this be done? (Answer with a number.) Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 32933,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To make pizza dough, Luca mixes 50 mL of milk for every 250 mL of flour. How many mL of milk does he mix with 750 mL of flour? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 32934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers which divide $5n^{11}-2n^5-3n$ for all positive integers $n$ are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 32935,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute \\(\\frac{x^{10} + 15x^5 + 125}{x^5 + 5}\\) when \\( x=3 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "248.4032"
} | {
"split": "train",
"index": 32936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(a\\), \\(b\\), and \\(c\\) be real numbers such that \\(9a^2 + 4b^2 + 25c^2 = 4\\). Find the maximum value of\n\\[6a + 3b + 10c.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{41}"
} | {
"split": "train",
"index": 32937,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The greatest common divisor of two positive integers is $(x+5)$ and their least common multiple is $x(x+5)$, where $x$ is a positive integer. If one of the integers is 50, what is the smallest possible value of the other one? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 32938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which of the following numbers is equal to 33 million? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33000000"
} | {
"split": "train",
"index": 32939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the location of the military camp is $A(1,1)$, and the general sets off from point $B(4,4)$ at the foot of the mountain, with the equation of the riverbank line $l$ being $x-y+1=0$, find the shortest total distance of the \"General Drinking Horse\" problem. Let's thi... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"split": "train",
"index": 32940,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\overline{AB}$ be a diameter in a circle of radius $10$. Let $\\overline{CD}$ be a chord in the circle that intersects $\\overline{AB}$ at a point $E$ such that $BE=4$ and $\\angle AEC = 30^{\\circ}.$ Compute the sum $CE^2 + DE^2$. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 32941,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are three spheres and a cube. The first sphere is tangent to each face of the cube, the second sphere is tangent to each edge of the cube, and the third sphere passes through each vertex of the cube. What is the ratio of the surface areas of these three spheres? Let's thi... | math | {
"style": "rule",
"ground_truth": "1:2:3"
} | {
"split": "train",
"index": 32942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system $(xOy)$, point $P(1, 2)$ is on a line $l$ with a slant angle of $\\alpha$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis. The equation of curve $C$... | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{7}}{7}"
} | {
"split": "train",
"index": 32943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $m$ is a positive integer, and given that $\\mathop{\\text{lcm}}[40,m]=120$ and $\\mathop{\\text{lcm}}[m,45]=180$, what is $m$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 32944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \\left\\lbrace\\frac{1}{z}|z \\in R\\right\\rbrace$. Then th... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 32945,
"solution": "We can describe the line parallel to the imaginary axis $x=\\frac{1}{2}$ using polar coordinates as $r(\\theta)=\\dfrac{1}{2\\cos{\\theta}},$\n which rearranges to $z=\\left(\\dfrac{1}{2\\cos{\\theta}}\\right)(cis{\\theta})\\implies \\frac{1}{z}=2\\cos{\\theta}cis(... |
deepscaler | [
{
"role": "user",
"content": "Suppose $\\cos S = 0.5$ in the diagram below. What is $ST$?\n\n[asy]\npair P,S,T;\nP = (0,0);\nS = (6,0);\nT = (0,6*tan(acos(0.5)));\ndraw(P--S--T--P);\ndraw(rightanglemark(S,P,T,18));\nlabel(\"$P$\",P,SW);\nlabel(\"$S$\",S,SE);\nlabel(\"$T$\",T,N);\nlabel(\"$10$\",S/2,S);\n[/a... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 32946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl}\n-x - 3 & \\text{if } x \\le 1, \\\\\n\\frac{x}{2} + 1 & \\text{if } x > 1.\n\\end{array}\n\\right.\\]Find the sum of all values of $x$ such that $f(x) = 0$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 32947,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You can arrange 15 balls in the shape of a triangle, but you cannot arrange 96 balls in the shape of a square (missing one ball). Out of how many balls, not exceeding 50, can you arrange them both in the shape of a triangle and a square? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 32948,
"solution": ""
} |
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