data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "If the function \\( f(x) = (x^2 - 1)(x^2 + ax + b) \\) satisfies \\( f(x) = f(4 - x) \\) for any \\( x \\in \\mathbb{R} \\), what is the minimum value of \\( f(x) \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"split": "train",
"index": 30149,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sphere is inscribed in a right cone with base radius $15$ cm and height $30$ cm. The radius of the sphere can be expressed as $b\\sqrt{d} - g$ cm, where $g = b + 6$. What is the value of $b + d$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12.5"
} | {
"split": "train",
"index": 30150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right rectangular prism has integer side lengths $a$ , $b$ , and $c$ . If $\\text{lcm}(a,b)=72$ , $\\text{lcm}(a,c)=24$ , and $\\text{lcm}(b,c)=18$ , what is the sum of the minimum and maximum possible volumes of the prism?\n\n*Proposed by Deyuan Li and Andrew Milas* Le... | math | {
"style": "rule",
"ground_truth": "3024"
} | {
"split": "train",
"index": 30151,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) How many natural numbers are there that are less than 5000 and have no repeated digits?\n(2) From the digits 1 to 9, five digits are selected each time to form a 5-digit number without repeated digits.\n (i) If only odd digits can be placed in odd positions, how many such... | math | {
"style": "rule",
"ground_truth": "1800"
} | {
"split": "train",
"index": 30152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find \\( k \\) such that \\((a+b)(b+c)(c+a) = (a+b+c)(ab+bc+ca) + k \\cdot abc\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 30153,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the focus of the ellipse $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a>b>0)$ be $F_{1}$, $F_{2}$, and $P$ be a point on the ellipse with $\\angle F_1PF_2=\\frac{π}{3}$. If the radii of the circumcircle and incircle of $\\triangle F_{1}PF_{2}$ are $R$ and $r$ respectively, and when... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 30154,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer that leaves a remainder of 4 when divided by 5 and a remainder of 6 when divided by 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 30155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least integer whose square is 12 more than three times 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": 30156,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The bases \\(AB\\) and \\(CD\\) of trapezoid \\(ABCD\\) are equal to 41 and 24 respectively, and its diagonals are perpendicular to each other. Find the dot product of vectors \\(\\overrightarrow{AD}\\) and \\(\\overrightarrow{BC}\\). Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "984"
} | {
"split": "train",
"index": 30157,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that\n$$\n\\sqrt{9-8 \\sin 50^{\\circ}}=a+b \\sin c^{\\circ}\n$$\nfor exactly one set of positive integers \\((a, b, c)\\), where \\(0 < c < 90\\). Find the value of \\(\\frac{b+c}{a}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 30158,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $b$ men take $c$ days to lay $f$ bricks, then the number of days it will take $c$ men working at the same rate to lay $b$ bricks, is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{b^2}{f}"
} | {
"split": "train",
"index": 30159,
"solution": "1. **Understanding the problem**: We are given that $b$ men can lay $f$ bricks in $c$ days. We need to find the number of days ($x$) it will take $c$ men to lay $b$ bricks.\n\n2. **Setting up the rate equation**: The rate at which bricks are laid can be expressed a... |
deepscaler | [
{
"role": "user",
"content": "The water tank in the diagram below is in the shape of an inverted right circular cone. The radius of its base is 16 feet, and its height is 96 feet. The water in the tank is $25\\%$ of the tank's capacity. The height of the water in the tank can be written in the form $a\\sqr... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 30160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $7 \\times 7$ board is either empty or contains an invisible $2 \\times 2$ ship placed \"on the cells.\" Detectors can be placed on certain cells of the board and then activated simultaneously. An activated detector signals if its cell is occupied by the ship. What is the min... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 30161,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For certain pairs $(m,n)$ of positive integers with $m\\geq n$ there are exactly $50$ distinct positive integers $k$ such that $|\\log m - \\log k| < \\log n$. Find the sum of all possible values of the product $mn$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 30162,
"solution": "We have $\\log m - \\log k = \\log \\left( \\frac mk \\right)$, hence we can rewrite the inequality as follows: \\[- \\log n < \\log \\left( \\frac mk \\right) < \\log n\\] We can now get rid of the logarithms, obtaining: \\[\\frac 1n < \\frac mk < n\\] And this ca... |
deepscaler | [
{
"role": "user",
"content": "There is a cube of size \\(10 \\times 10 \\times 10\\) made up of small unit cubes. A grasshopper is sitting at the center \\(O\\) of one of the corner cubes. It can jump to the center of a cube that shares a face with the one in which the grasshopper is currently located, prov... | math | {
"style": "rule",
"ground_truth": "\\frac{27!}{(9!)^3}"
} | {
"split": "train",
"index": 30163,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $m>0$ and $|x+1|+|2x-1|\\geq m$ always holds, $a, b, c \\in \\mathbb{R}$ satisfy $a^2+2b^2+3c^2=m$. Find the minimum value of $a+2b+3c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 30164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\sin x \\cos x - \\sqrt{3} \\cos (x+\\pi) \\cos x, x \\in \\mathbb{R}$.\n(Ⅰ) Find the minimal positive period of $f(x)$;\n(Ⅱ) If the graph of the function $y = f(x)$ is translated by $\\overrightarrow{b}=\\left( \\frac{\\pi}{4}, \\frac{\\sqrt{3}}{2} ... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 30165,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The marble statue of Abraham Lincoln in the Lincoln Memorial has a height of 60 feet. A scale model of the statue has a height of 4 inches. How many feet of the statue does one inch of the model represent? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 30166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $M$ and $N$ are located on side $AC$ of triangle $ABC$, and points $K$ and $L$ are on side $AB$, with $AM : MN : NC = 1 : 3 : 1$ and $AK = KL = LB$. It is known that the area of triangle $ABC$ is 1. Find the area of quadrilateral $KLNM$. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "7/15"
} | {
"split": "train",
"index": 30167,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given several numbers, one of them, $a$ , is chosen and replaced by the three numbers $\\frac{a}{3}, \\frac{a}{3}, \\frac{a}{3}$ . This process is repeated with the new set of numbers, and so on. Originally, there are $1000$ ones, and we apply the process several times. A n... | math | {
"style": "rule",
"ground_truth": "667"
} | {
"split": "train",
"index": 30168,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the quotient when $8x^3+16x^2-7x+4$ is divided by $2x+5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4x^2 -2x + \\frac{3}{2}"
} | {
"split": "train",
"index": 30169,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The measures of the angles of a triangle are in the ratio 5:6:7. What is the number of degrees in the largest of these angles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70^\\circ"
} | {
"split": "train",
"index": 30170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\left\\{a_{n}\\right\\}$ satisfies $a_{1} = 1$, and for each $n \\in \\mathbf{N}^{*}$, $a_{n}$ and $a_{n+1}$ are the roots of the equation $x^{2} + 3n x + b_{n} = 0$. Find $\\sum_{k=1}^{20} b_{k}$. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "6385"
} | {
"split": "train",
"index": 30171,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In what ratio does the angle bisector of the acute angle divide the area of a right trapezoid inscribed in a circle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1:1"
} | {
"split": "train",
"index": 30172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nThere are 11 children sitting in a circle playing a game. They are numbered clockwise from 1 to 11. The game starts with child number 1, and each child has to say a two-digit number. The number they say cannot have a digit sum of 6 or 9, and no child can repeat a number that ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 30173,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ be the set of all numbers which can be represented as the sum of three consecutive positive integers. What is the greatest common divisor of all numbers in $A$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 30174,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If two distinct members of the set $\\{ 2, 4, 10, 12, 15, 20, 50 \\}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 100? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 30175,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Now a ball is launched from a vertex of an equilateral triangle with side length 5. It strikes the opposite side after traveling a distance of $\\sqrt{19}$. How many times does the ball bounce before it returns to a vertex? (The final contact with a vertex does not count as a b... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 30176,
"solution": "The key idea is that, instead of reflecting the line $AY$ off of $BC$, we will reflect $ABC$ about $BC$ and extend $AY$ beyond $\\triangle ABC$. We keep doing this until the extension of $AY$ hits a vertex of one of our reflected triangles. This is illustrated in t... |
deepscaler | [
{
"role": "user",
"content": "Given $A=\\{a, b, c\\}$ and $B=\\{0, 1, 2\\}$, determine the number of mappings $f: A \\to B$ that satisfy the condition $f(a) + f(b) > f(c)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 30177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T$ be a subset of $\\{1,2,3,...,100\\}$ such that no pair of distinct elements in $T$ has a sum divisible by $5$. What is the maximum number of elements in $T$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 30178,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $c=2$, $C=\\dfrac{\\pi }{3}$, and $\\sin B=2\\sin A$, find the area of $\\triangle ABC$. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 30179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive real numbers $a,$ $b,$ and $c,$ compute the maximum value of\n\\[\\frac{abc(a + b + c)}{(a + b)^2 (b + c)^2}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 30180,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any natural number $n$ , expressed in base $10$ , let $S(n)$ denote the sum of all digits of $n$ . Find all positive integers $n$ such that $n^3 = 8S(n)^3+6S(n)n+1$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 30181,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $Q$ is chosen inside $\\triangle DEF$ such that lines drawn through $Q$ parallel to $\\triangle DEF$'s sides decompose it into three smaller triangles $u_1$, $u_2$, and $u_3$, which have areas $3$, $12$, and $15$ respectively. Determine the area of $\\triangle DEF$. Let... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 30182,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Hou Yi shot three arrows at each of three targets. On the first target, he scored 29 points, and on the second target, he scored 43 points. How many points did he score on the third target? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 30183,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The positive divisors of the integer 630 (including 1 and 630) total how many? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 30184,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive odd integers greater than 1 and less than 200 are square-free? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 30185,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( p, q, r, \\) and \\( s \\) be positive real numbers such that\n\\[\n\\begin{array}{c@{\\hspace{3pt}}c@{\\hspace{3pt}}c@{\\hspace{3pt}}c@{\\hspace{3pt}}c}\np^2+q^2&=&r^2+s^2&=&2512, \\\\\npr&=&qs&=&1225.\n\\end{array}\n\\]\nIf \\( T = p+q+r+s \\), compute the value of \\... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 30186,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$ and $b$ are positive integers such that $ab - 7a + 6b = 559$, what is the minimal possible value of $|a - b|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "587"
} | {
"split": "train",
"index": 30187,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ann and Anne are in bumper cars starting 50 meters apart. Each one approaches the other at a constant ground speed of $10 \\mathrm{~km} / \\mathrm{hr}$. A fly starts at Ann, flies to Anne, then back to Ann, and so on, back and forth until it gets crushed when the two bumper car... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 30188,
"solution": "Suppose that at a given instant the fly is at Ann and the two cars are $12 d$ apart. Then, while each of the cars travels $4 d$, the fly travels $8 d$ and meets Anne. Then the fly turns around, and while each of the cars travels $d$, the fly travels $3 d$ and meets... |
deepscaler | [
{
"role": "user",
"content": "If the positive real numbers \\( x \\) and \\( y \\) satisfy \\( x - 2 \\sqrt{y} = \\sqrt{2x - y} \\), then the maximum value of \\( x \\) is ____ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 30189,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the measure, in degrees, of the acute angle formed by the minute hand and the hour hand on a standard clock when it indicates $9$:$40$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 30190,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The six-digit number \\( 2PQRST \\) is multiplied by 3, and the result is the six-digit number \\( PQRST2 \\). What is the sum of the digits of the original number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 30191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Q. A light source at the point $(0, 16)$ in the co-ordinate plane casts light in all directions. A disc(circle along ith it's interior) of radius $2$ with center at $(6, 10)$ casts a shadow on the X-axis. The length of the shadow can be written in the form $m\\sqrt{n}$ ... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 30192,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "BdMO National 2016 Higher Secondary \n\n<u>**Problem 4:**</u>\nConsider the set of integers $ \\left \\{ 1, 2, ......... , 100 \\right \\} $ . Let $ \\left \\{ x_1, x_2, ......... , x_{100} \\right \\}$ be some arbitrary arrangement of the integers $ \\left \\{ 1, 2, ...... | math | {
"style": "rule",
"ground_truth": "198"
} | {
"split": "train",
"index": 30193,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be real numbers greater than 1 such that\n\\[(\\log_2 x)^4 + (\\log_3 y)^4 + 8 = 8 (\\log_2 x)(\\log_3 y).\\]Compute $x^{\\sqrt{2}} + y^{\\sqrt{2}}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 30194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $PQR$ has vertices at $P(7, 5)$, $Q(1, -3)$, and $R(4, 4)$. The point $S$ with coordinates $(x, y)$ is chosen inside the triangle so that the three smaller triangles $PQS$, $PRS$, and $QRS$ all have equal areas. What is the value of $12x + 3y$? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 30195,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum of\n\\[\\sqrt{x + 27} + \\sqrt{13 - x} + \\sqrt{x}\\]for $0 \\le x \\le 13.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 30196,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a road of length $A B = 8 \\text{ km}$, buses travel in both directions at a speed of $12 \\text{ km/h}$. The first bus from each location starts at 6 o'clock, with subsequent buses departing every 10 minutes.\n\nA pedestrian starts walking from $A$ to $B$ at $\\frac{81}{4}$... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 30197,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $x^5$, $x+\\frac{1}{x}$ and $1+\\frac{2}{x} + \\frac{3}{x^2}$ are multiplied, the product is a polynomial of degree. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 30198,
"solution": "1. **Identify the expressions and their multiplication:** We are given three expressions to multiply: $x^5$, $x + \\frac{1}{x}$, and $1 + \\frac{2}{x} + \\frac{3}{x^2}$.\n\n2. **Multiply the first two expressions:** \n \\[\n x^5 \\left(x + \\frac{1}{x}\\right) ... |
deepscaler | [
{
"role": "user",
"content": "How many triangles can be formed using the vertices of a regular dodecagon (a 12-sided polygon)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "220"
} | {
"split": "train",
"index": 30199,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, 100 workers can produce 300 widgets and 200 whoosits. In two hours, 60 workers can pr... | math | {
"style": "rule",
"ground_truth": "450"
} | {
"split": "train",
"index": 30200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The limit of $\\frac {x^2-1}{x-1}$ as $x$ approaches $1$ as a limit is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 30201,
"solution": "To find the limit of the function $\\frac{x^2-1}{x-1}$ as $x$ approaches 1, we start by simplifying the expression.\n\n1. **Factorize the Numerator**: \n The numerator $x^2 - 1$ is a difference of squares, which can be factored as:\n \\[\n x^2 - 1 = (x + 1)(x... |
deepscaler | [
{
"role": "user",
"content": "Let \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) be a differentiable function such that \\( f(0) = 0 \\), \\( f(1) = 1 \\), and \\( |f'(x)| \\leq 2 \\) for all real numbers \\( x \\). If \\( a \\) and \\( b \\) are real numbers such that the set of possible values of \\( \\i... | math | {
"style": "rule",
"ground_truth": "3/4"
} | {
"split": "train",
"index": 30202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A board game spinner is divided into three parts labeled $A$, $B$ and $C$. The probability of the spinner landing on $A$ is $\\frac{1}{3}$ and the probability of the spinner landing on $B$ is $\\frac{5}{12}$. What is the probability of the spinner landing on $C$? Express your... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 30203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three frogs in a swamp jumped one after another. Each one landed exactly in the middle of the segment between the other two. The jump length of the second frog is 60 cm. Find the jump length of the third frog. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 30204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $0 \\le a,$ $b,$ $c,$ $d \\le 1.$ Find the possible values of the expression\n\\[\\sqrt{a^2 + (1 - b)^2} + \\sqrt{b^2 + (1 - c)^2} + \\sqrt{c^2 + (1 - d)^2} + \\sqrt{d^2 + (1 - a)^2}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "[2 \\sqrt{2},4]"
} | {
"split": "train",
"index": 30205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f(x) = mx^2 + (2n + 1)x - m - 2 \\) (where \\( m, n \\in \\mathbb{R} \\) and \\( m \\neq 0 \\)) have at least one root in the interval \\([3, 4]\\). Find the minimum value of \\( m^2 + n^2 \\). Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "0.01"
} | {
"split": "train",
"index": 30206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pentagon ABCDE has a vertical line of symmetry. What is the $y$-coordinate of vertex C so that the area of the pentagon is 40 square units? [asy]\nunitsize(2mm);\ndefaultpen(linewidth(.7pt)+fontsize(8pt));\ndotfactor=4;\n\npair A=(0,0), B=(0,4), C=(2,16), D=(4,4), E=(4,0);\npai... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 30207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $C$: $\\begin{cases}x=2\\cos θ \\\\\\\\ y=\\sqrt{3}\\sin θ\\end{cases}$, find the value of $\\frac{1}{m}+\\frac{1}{n}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 30208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that $\\underbrace{2^{2 \\cdot 2}}_{n}>3^{3^{3^{3}}}$. (The notation $\\underbrace{2^{2^{2}}}_{n}$, is used to denote a power tower with $n 2$ 's. For example, $\\underbrace{2^{22^{2}}}_{n}$ with $n=4$ would equal $2^{2^{2^{2}}}$.) Le... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 30209,
"solution": "Clearly, $n \\geq 5$. When we take $n=5$, we have $$2^{2^{2^{2^{2}}}}=2^{2^{16}}<3^{3^{27}}=3^{3^{3^{3}}}.$$ On the other hand, when $n=6$, we have $$2^{2^{2^{2^{2^{2}}}}}=2^{2^{65536}}=4^{2^{65535}}>4^{4^{27}}>3^{3^{27}}=3^{3^{3^{3}}}.$$ Our answer is thus $n=6$."... |
deepscaler | [
{
"role": "user",
"content": "A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different desser... | math | {
"style": "rule",
"ground_truth": "729"
} | {
"split": "train",
"index": 30210,
"solution": "1. **Identify the constraints and requirements:**\n - The dessert each day is either cake, pie, ice cream, or pudding.\n - The same dessert may not be served two days in a row.\n - There must be cake on Friday.\n\n2. **Calculate the total number of unrestrict... |
deepscaler | [
{
"role": "user",
"content": "What is the ratio of the numerical value of the area, in square units, of an equilateral triangle of side length 8 units to the numerical value of its perimeter, in units? Express your answer as a common fraction in simplest radical form. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 30211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A frustum of a right circular cone is formed by cutting a smaller cone from a larger cone. Suppose the frustum has a lower base radius of 8 inches, an upper base radius of 2 inches, and a height of 5 inches. Calculate the total surface area of the frustum. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "10\\pi \\sqrt{61} + 68\\pi"
} | {
"split": "train",
"index": 30212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of an equilateral triangle ABC is 36. Points P, Q, R are located on BC, AB, and CA respectively, such that BP = 1/3 BC, AQ = QB, and PR is perpendicular to AC. Find the area of triangle PQR. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 30213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a table, there are 210 different cards each with a number from the sequence \\(2, 4, 6, \\ldots, 418, 420\\) (each card has exactly one number, and each number appears exactly once). In how many ways can 2 cards be chosen so that the sum of the numbers on the chosen cards is... | math | {
"style": "rule",
"ground_truth": "3135"
} | {
"split": "train",
"index": 30214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that the number of birch trees in a certain mixed forest plot ranges from $13\\%$ to $14\\%$ of the total number of trees. Find the minimum possible total number of trees in this plot. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 30215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Count the number of sequences $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ of integers such that $a_{i} \\leq 1$ for all $i$ and all partial sums $\\left(a_{1}, a_{1}+a_{2}\\right.$, etc.) are non-negative. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "132"
} | {
"split": "train",
"index": 30216,
"solution": "$C$ (length +1$)=C(6)=132$."
} |
deepscaler | [
{
"role": "user",
"content": "Mike has two containers. One container is a rectangular prism with width 2 cm, length 4 cm, and height 10 cm. The other is a right cylinder with radius 1 cm and height 10 cm. Both containers sit on a flat surface. Water has been poured into the two containers so that the height... | math | {
"style": "rule",
"ground_truth": "7.2"
} | {
"split": "train",
"index": 30217,
"solution": "Suppose that the height of the water in each container is $h \\mathrm{~cm}$. Since the first container is a rectangular prism with a base that is 2 cm by 4 cm, then the volume of the water that it contains, in $\\mathrm{cm}^{3}$, is $2 \\times 4 \\times h=8h$. Sinc... |
deepscaler | [
{
"role": "user",
"content": "The hypotenuse of a right triangle measures $6\\sqrt{2}$ inches and one angle is $45^{\\circ}$. What is the number of square inches in the area of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 30218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( f(x)=\\frac{2x+3}{x-1} \\), the graph of the function \\( y=g(x) \\) is symmetric with the graph of the function \\( y=f^{-1}(x+1) \\) with respect to the line \\( y=x \\). Find \\( g(3) \\). Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"split": "train",
"index": 30219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A grocer stacks oranges in a pyramid-like stack whose rectangular base is $5$ oranges by $8$ oranges. Each orange above the first level rests in a pocket formed by four oranges below. The stack is completed by a single row of oranges. How many oranges are in the stack? Let's... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 30220,
"solution": "1. **Calculate the number of oranges in each layer**:\n - The base layer (1st layer) is a rectangle of $5$ oranges by $8$ oranges. Thus, the number of oranges in the 1st layer is:\n \\[\n 5 \\times 8 = 40\n \\]\n - Each subsequent layer reduces in s... |
deepscaler | [
{
"role": "user",
"content": "Calculate: $99\\times 101=\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9999"
} | {
"split": "train",
"index": 30221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all positive real solutions $x$ to the equation \\[2\\cos2x \\left(\\cos2x - \\cos{\\left( \\frac{2014\\pi^2}{x} \\right) } \\right) = \\cos4x - 1,\\]where $x$ is measured in radians. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "1080 \\pi"
} | {
"split": "train",
"index": 30222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square sheet of paper has an area of $12 \\text{ cm}^2$. The front is white and the back is black. When the paper is folded so that point $A$ rests on the diagonal and the visible black area is equal to the visible white area, how far is point A from its original position? Gi... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{6}"
} | {
"split": "train",
"index": 30223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the infinite sum $\\sum_{n=1}^{\\infty}\\frac{n}{n^4+4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac 3 8"
} | {
"split": "train",
"index": 30224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A jar contains $29\\frac{5}{7}$ tablespoons of peanut butter. If one serving of peanut butter is 2 tablespoons, how many servings of peanut butter does the jar contain? Express your answer as a mixed number. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "14\\frac{6}{7}"
} | {
"split": "train",
"index": 30225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Expand $(2z^2 + 5z - 6)(3z^3 - 2z + 1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6z^5+15z^4-22z^3-8z^2+17z-6"
} | {
"split": "train",
"index": 30226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $3\\sin \\left(-3\\pi +\\theta \\right)+\\cos \\left(\\pi -\\theta \\right)=0$, then the value of $\\frac{sinθcosθ}{cos2θ}$ is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{8}"
} | {
"split": "train",
"index": 30227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $ \\dfrac {3\\pi}{4} < \\alpha < \\pi$, $\\tan \\alpha+ \\dfrac {1}{\\tan \\alpha}=- \\dfrac {10}{3}$. \n$(1)$ Find the value of $\\tan \\alpha$; \n$(2)$ Find the value of $ \\dfrac {5\\sin ^{2} \\dfrac {\\alpha}{2}+8\\sin \\dfrac {\\alpha}{2}\\cos \\dfrac {\\alpha}{2}+... | math | {
"style": "rule",
"ground_truth": "- \\dfrac {5}{4}"
} | {
"split": "train",
"index": 30228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an opaque bag, there are a total of 50 glass balls in red, black, and white colors. Except for the color, everything else is the same. After several trials of drawing balls, Xiaogang found that the probability of drawing a red or black ball is stable at 15% and 45%, respecti... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 30229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a \"sing, read, speak, and spread\" performance activity participated by six units, including units A and B, each unit's program is arranged together. If a lottery method is used to randomly determine the order of performance for each unit (numbered 1, 2, …, 6), calculate: ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 30230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a straight stretch of one-way, two-lane highway, vehicles obey a safety rule: the distance from the back of one vehicle to the front of another is exactly one vehicle length for each 20 kilometers per hour of speed or fraction thereof. Suppose a sensor on the roadside counts... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 30231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the figure. She plants one flower per square foot in each region. Asters cost $\\$1$ each... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 30232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A metal bar at a temperature of $20^{\\circ} \\mathrm{C}$ is placed in water at a temperature of $100^{\\circ} \\mathrm{C}$. After thermal equilibrium is established, the temperature becomes $80^{\\circ} \\mathrm{C}$. Then, without removing the first bar, another identical meta... | math | {
"style": "rule",
"ground_truth": "68"
} | {
"split": "train",
"index": 30233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two concentric circles have radii of 24 and 36 units, respectively. A shaded region is formed between these two circles. A new circle is to be drawn such that its diameter is equal to the area of the shaded region. What must the diameter of this new circle be? Express your answ... | math | {
"style": "rule",
"ground_truth": "720 \\pi"
} | {
"split": "train",
"index": 30234,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $\\left(\\frac{1}{2}\\right)^{8} \\cdot \\left(\\frac{3}{4}\\right)^{-3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{108}"
} | {
"split": "train",
"index": 30235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The function $f(x) = 2\\cos x \\sin (x - A) (x \\in \\mathbb{R})$ reaches its minimum value at $x = \\frac{11\\pi}{12}$.\n1. Find the measure of angle $A$.\n2. If $a = 7$ and... | math | {
"style": "rule",
"ground_truth": "10\\sqrt{3}"
} | {
"split": "train",
"index": 30236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=1-2\\sin ^{2}(x+ \\frac {π}{8})+2\\sin (x+ \\frac {π}{8})\\cos (x+ \\frac {π}{8})$.\n(1) Find the smallest positive period and the monotonically increasing interval of $f(x)$;\n(2) Find the maximum and minimum values of $f(x)$ on the interval $[- \\frac... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 30237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four standard, six-sided dice are to be rolled. If the product of their values turns out to be an even number, what is the probability their sum is odd? 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{8}{15}"
} | {
"split": "train",
"index": 30238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bottled water company has designed a new cup for its dispenser. The cup will be a right circular cone with a three-inch radius. How tall will the cup need to be to hold 93 cubic inches of water? Express your answer to the nearest whole number. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 30239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $ \\sin \\alpha = \\frac{1}{3} $, and $ 0 < \\alpha < \\pi $, then $ \\tan \\alpha = $_____, and $ \\sin \\frac{\\alpha}{2} + \\cos \\frac{\\alpha}{2} = $_____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{3}}{3}"
} | {
"split": "train",
"index": 30240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ioana has three ropes whose lengths are 39 inches, 52 inches and 65 inches. She wants to cut the ropes into equal length pieces for magic tricks. No rope is to be wasted. What is the greatest number of inches possible in the length of each piece? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 30241,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x)$ and $g(x)$ are polynomials such that $f(x) + g(x) = -2 + x,$ then what is $g(x)$ if $f(x) = x^3 - 2x - 2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-x^3 + 3x"
} | {
"split": "train",
"index": 30242,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider all polynomials of the form\n\\[x^7 + b_6 x^6 + b_5 x^5 + \\dots + b_2 x^2 + b_1 x + b_0,\\]\nwhere \\( b_i \\in \\{0,1\\} \\) for all \\( 0 \\le i \\le 6 \\). Find the number of such polynomials that have exactly two different integer roots, -1 and 0. Let's think step... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 30243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A strip of size $1 \\times 10$ is divided into unit squares. The numbers $1, 2, \\ldots, 10$ are written in these squares. First, the number 1 is written in one of the squares, then the number 2 is written in one of the neighboring squares, then the number 3 is written in one o... | math | {
"style": "rule",
"ground_truth": "512"
} | {
"split": "train",
"index": 30244,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), side \\(BC\\) is equal to 5. A circle passes through vertices \\(B\\) and \\(C\\) and intersects side \\(AC\\) at point \\(K\\), where \\(CK = 3\\) and \\(KA = 1\\). It is known that the cosine of angle \\(ACB\\) is \\(\\frac{4}{5}\\). Find the ratio of t... | math | {
"style": "rule",
"ground_truth": "\\frac{10\\sqrt{10} + 25}{9}"
} | {
"split": "train",
"index": 30245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The positive integer divisors of 175, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 7? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 30246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 2012 distinct points in the plane, each of which is to be coloured using one of \\( n \\) colours so that the number of points of each colour are distinct. A set of \\( n \\) points is said to be multi-coloured if their colours are distinct. Determine \\( n \\) that m... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 30247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find a number such that when it is multiplied by its reverse, the product is 78445. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "145"
} | {
"split": "train",
"index": 30248,
"solution": ""
} |
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