data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "At the Hardey Fitness Center, the management did a survey of their membership. The average age of the female members was 40 years old. The average age of the male members was 25 years old. The average age of the entire membership was 30 years old. What is the ratio of the femal... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 27849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The functions $f(x) = x^2-2x + m$ and $g(x) = x^2-2x + 4m$ are evaluated when $x = 4$. What is the value of $m$ if $2f(4) = g(4)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 27850,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Use the Horner's method to calculate the polynomial $f(x) = 3x^6 + 5x^5 + 6x^4 + 79x^3 - 8x^2 + 35x + 12$ when $x = -4$, find the value of $v_3$. \n(2) Convert the hexadecimal number $210_{(6)}$ into a decimal number. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"split": "train",
"index": 27851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The game of rock-scissors is played just like rock-paper-scissors, except that neither player is allowed to play paper. You play against a poorly-designed computer program that plays rock with $50 \\%$ probability and scissors with $50 \\%$ probability. If you play optimally ag... | math | {
"style": "rule",
"ground_truth": "\\frac{163}{256}"
} | {
"split": "train",
"index": 27852,
"solution": "Since rock will always win against scissors, the optimum strategy is for you to always play rock; then, you win a game if and only if the computer plays scissors. Let $p_{n}$ be the probability that the computer plays scissors $n$ times; we want $p_{0}+p_{1}+p_{2}+... |
deepscaler | [
{
"role": "user",
"content": "There are three saline solutions with concentrations of 5%, 8%, and 9%, labeled A, B, and C, weighing 60g, 60g, and 47g respectively. We need to prepare 100g of a saline solution with a concentration of 7%. What is the maximum and minimum amount of solution A (5% concentration)... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 27853,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are nine parts in a bag, including five different genuine parts and four different defective ones. These parts are being drawn and inspected one by one. If the last defective part is found exactly on the fifth draw, calculate the total number of different sequences of dra... | math | {
"style": "rule",
"ground_truth": "480"
} | {
"split": "train",
"index": 27854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Yann writes down the first $n$ consecutive positive integers, $1,2,3,4, \\ldots, n-1, n$. He removes four different integers $p, q, r, s$ from the list. At least three of $p, q, r, s$ are consecutive and $100<p<q<r<s$. The average of the integers remaining in the list is 89.562... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 27855,
"solution": "When Yann removes 4 of the $n$ integers from his list, there are $n-4$ integers left. Suppose that the sum of the $n-4$ integers left is $T$. The average of these $n-4$ integers is $89.5625=89.5+0.0625=89+\\frac{1}{2}+\\frac{1}{16}=89 \\frac{9}{16}=\\frac{1433}{16}... |
deepscaler | [
{
"role": "user",
"content": "How many $7$-digit palindromes (numbers that read the same backward as forward) can be formed using the digits $2$, $2$, $3$, $3$, $5$, $5$, $5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 27856,
"solution": "To solve this problem, we need to determine how many $7$-digit palindromes can be formed using the digits $2, 2, 3, 3, 5, 5, 5$. A $7$-digit palindrome has the form $\\overline{abcdcba}$, where the digits are the same reading from left to right and right to left.\n... |
deepscaler | [
{
"role": "user",
"content": "Points \\(O\\) and \\(I\\) are the centers of the circumscribed and inscribed circles of triangle \\(ABC\\), \\(M\\) is the midpoint of the arc \\(AC\\) of the circumscribed circle (which does not contain \\(B\\)). It is known that \\(AB = 15\\), \\(BC = 7\\), and \\(MI = MO\\)... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 27857,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the circle defined by \\(x^2 + 4x + y^2 + 10y + 13 = 0\\) that lies above the line \\(y = -2\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"split": "train",
"index": 27858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer $x$ that, when multiplied by $900$, produces a product that is a multiple of $1152$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 27859,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $1998$ is written as a product of two positive integers whose difference is as small as possible, then the difference is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 27860,
"solution": "1. **Objective**: Find two positive integers whose product is $1998$ and whose difference is minimized.\n\n2. **Calculate the approximate square root of $1998$**:\n \\[\n \\sqrt{1998} \\approx \\sqrt{2000} = \\sqrt{4 \\times 500} = 2 \\times \\sqrt{500} \\appro... |
deepscaler | [
{
"role": "user",
"content": "A portion of the graph of $f(x)=ax^3+bx^2+cx+d$ is shown below.\n\nWhat is the value of $8a-4b+2c-d$?\n\n[asy]\nimport graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=4.25,ymin=-9.25,ymax=4.25;\n\npen cqcq... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 27861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $3\\sin \\alpha - 2\\cos \\alpha = 0$, find the value of the following expressions:\n$$(1)\\ \\frac{\\cos \\alpha - \\sin \\alpha}{\\cos \\alpha + \\sin \\alpha} + \\frac{\\cos \\alpha + \\sin \\alpha}{\\cos \\alpha - \\sin \\alpha};$$\n$$(2)\\ \\sin^2\\alpha - 2\\si... | math | {
"style": "rule",
"ground_truth": "\\frac{28}{13}"
} | {
"split": "train",
"index": 27862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate:\n$$\n\\left(10^{4}-9^{4}+8^{4}-7^{4}+\\cdots+2^{4}-1^{4}\\right)+\\left(10^{2}+9^{2}+5 \\times 8^{2}+5 \\times 7^{2}+9 \\times 6^{2}+9 \\times 5^{2}+13 \\times 4^{2}+13 \\times 3^{2}\\right) =\n$$ Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "7615"
} | {
"split": "train",
"index": 27863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of two consecutive page numbers is $18{,}360.$ What is the sum of the two page numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "271"
} | {
"split": "train",
"index": 27864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When each edge of a cube is increased by $50\\%$, by what percent is the surface area of the cube increased? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125\\%"
} | {
"split": "train",
"index": 27865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the given functions: $$\\begin{array}{ccc}\nf(x) & = & 5x^2 - \\frac{1}{x}+ 3\\\\\ng(x) & = & x^2-k\n\\end{array}$$If $f(2) - g(2) = 2$, 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": "k = \\frac{-33}{2}"
} | {
"split": "train",
"index": 27866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle has sides of lengths 6 cm and 8 cm that create a 45-degree angle between them. Calculate the length of the third side. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5.67"
} | {
"split": "train",
"index": 27867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the distinct prime factors of $5^5 - 5^3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 27868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c,$ and $d$ be real numbers such that $ab = 2$ and $cd = 18.$ Find the minimum value of\n\\[(ac)^2 + (bd)^2.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 27869,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $4 \\in \\{a^2-3a, a\\}$, then the value of $a$ equals ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 27870,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A$ and $B$ are on a circle of radius $5$ and $AB=6$. Point $C$ is the midpoint of the minor arc $AB$. What is the length of the line segment $AC$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"split": "train",
"index": 27871,
"solution": "1. **Identify the Geometry and Define Points**: \n Let $O$ be the center of the circle with radius $5$. Since $A$ and $B$ are on the circle, $OA = OB = 5$. Let $D$ be the midpoint of $\\overline{AB}$. Since $C$ is the midpoint of the minor arc $AB$, $OC$ is perpe... |
deepscaler | [
{
"role": "user",
"content": "Determine the largest value of $S$ such that any finite collection of small squares with a total area $S$ can always be placed inside a unit square $T$ in such a way that no two of the small squares share an interior point. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 27872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is that largest positive integer $n$ for which $n^3+100$ is divisible by $n+10$?\n ~ pi_is_3.14\n ~ momeme Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "890"
} | {
"split": "train",
"index": 27873,
"solution": "Let $n+10=k$, then $n=k-10$. Then $n^3+100 = k^3-30k^2+300k-900$ Therefore, $900$ must be divisible by $k$, which is largest when $k=900$ and $n=\\boxed{890}$"
} |
deepscaler | [
{
"role": "user",
"content": "Gloria's grandmother cycled a total of $\\frac{3}{6} + \\frac{4}{4} + \\frac{3}{3} + \\frac{2}{8}$ hours at different speeds, and she cycled a total of $\\frac{3}{5} + \\frac{4}{5} + \\frac{3}{5} + \\frac{2}{5}$ hours at a speed of 5 miles per hour. Calculate the difference in ... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 27874,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence {a_n} satisfies a_1 = 3, and a_1 + a_3 + a_5 = 21, find the value of a_3 + a_5 + a_7. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 27875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ that satisfy $|\\overrightarrow{a}| = 1$, $|\\overrightarrow{b}| = 2$, and $\\overrightarrow{a} \\cdot \\overrightarrow{b} = -\\sqrt{3}$, find the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Let's ... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{6}"
} | {
"split": "train",
"index": 27876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=a^{x}-(k+1)a^{-x}$ where $a > 0$ and $a\\neq 1$, which is an odd function defined on $\\mathbb{R}$.\n1. Find the value of $k$.\n2. If $f(1)= \\frac {3}{2}$, and the minimum value of $g(x)=a^{2x}+a^{-2x}-2mf(x)$ on $[0,+\\infty)$ is $-6$, find the value ... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {2}"
} | {
"split": "train",
"index": 27877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)= \\begin{cases} ax^{2}-2x-1, & x\\geqslant 0\\\\ x^{2}+bx+c, & x < 0\\end{cases}$ is an even function, and the line $y=t$ intersects the graph of $y=f(x)$ from left to right at four distinct points $A$, $B$, $C$, $D$. If $AB=BC$, then the value of the r... | math | {
"style": "rule",
"ground_truth": "- \\dfrac {7}{4}"
} | {
"split": "train",
"index": 27878,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\alpha$ and $\\beta$ are acute angles, and $\\sin \\alpha = \\frac{\\sqrt{5}}{5}$, $\\cos \\beta = \\frac{3\\sqrt{10}}{10}$, then $\\sin (\\alpha + \\beta) =$____, $\\alpha + \\beta =$____. 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": 27879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum $E(1)+E(2)+E(3)+\\cdots+E(200)$ where $E(n)$ denotes the sum of the even digits of $n$, and $5$ is added to the sum if $n$ is a multiple of $10$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "902"
} | {
"split": "train",
"index": 27880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given any two positive real numbers $x$ and $y$, then $x \\, \\Diamond \\, y$ is a positive real number defined in terms of $x$ and $y$ by some fixed rule. Suppose the operation $x \\, \\Diamond \\, y$ satisfies the equations $(xy) \\, \\Diamond \\, y=x(y \\, \\Diamond \\, y)$... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 27881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $x$ satisfies $x^{3}+x^{2}+x+1=0$. What are all possible values of $x^{4}+2 x^{3}+2 x^{2}+2 x+1 ?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 27882,
"solution": "$x^{4}+2 x^{3}+2 x^{2}+2 x+1=(x+1)\\left(x^{3}+x^{2}+x+1\\right)=0$ is the only possible solution."
} |
deepscaler | [
{
"role": "user",
"content": "A scalene triangle has side lengths which are prime numbers and the length of its perimeter is also prime. What is its smallest possible perimeter? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 27883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $a$ so that the lines with the given equations are perpendicular. \\begin{align*}\ny &= 2x+13 \\\\\n6y+ax &= 6.\n\\end{align*} Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 27884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set $A=\\{x|ax^{2}-x+1=0,a\\in\\mathbb{R},x\\in\\mathbb{R}\\}$. If the proposition \"Set $A$ contains only one element\" is true, find the value of $a$. 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": 27885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the unique positive integer $n$ such that\n\\[2 \\cdot 2^2 + 3 \\cdot 2^3 + 4 \\cdot 2^4 + \\dots + n \\cdot 2^n = 2^{n + 10}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "513"
} | {
"split": "train",
"index": 27886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Students are in classroom with $n$ rows. In each row there are $m$ tables. It's given that $m,n \\geq 3$ . At each table there is exactly one student. We call neighbours of the student students sitting one place right, left to him, in front of him and behind him. Each stud... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 27887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$, $B$, $C$, and $D$ be points on a circle such that $AB = 11$ and $CD = 19.$ Point $P$ is on segment $AB$ with $AP = 6$, and $Q$ is on segment $CD$ with $CQ = 7$. The line through $P$ and $Q$ intersects the circle at $X$ and $Y$. If $PQ = 27$, find $XY$. Let's think step... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 27888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the domains of functions f(x) and g(x) are both 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, g(2) = 4, find the sum of the values of f(k) for k from 1 to 22. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "-24"
} | {
"split": "train",
"index": 27889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a regular tetrahedron A-BCD with an edge length of 1, and $\\overrightarrow{AE} = 2\\overrightarrow{EB}$, $\\overrightarrow{AF} = 2\\overrightarrow{FD}$, calculate $\\overrightarrow{EF} \\cdot \\overrightarrow{DC}$. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"split": "train",
"index": 27890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean of $x$ and $y$ equal to $12^{10}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "409"
} | {
"split": "train",
"index": 27891,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square floor is tiled with congruent square tiles. The tiles on the two diagonals of the floor are black. The rest of the tiles are white. If there are 101 black tiles, then the total number of tiles is Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "2601"
} | {
"split": "train",
"index": 27892,
"solution": "1. **Identify the pattern of black tiles**: The problem states that the tiles on the two diagonals of the square floor are black. This means that each diagonal is filled with black tiles.\n\n2. **Determine the number of black tiles on each diagonal**: \n - If the... |
deepscaler | [
{
"role": "user",
"content": "Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \\minus{} 1$ digits $ 1$ and one digit $ 7$ is prime. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "n=1,2"
} | {
"split": "train",
"index": 27893,
"solution": "\nWe are tasked with finding all natural numbers \\( n \\) such that every natural number with \\( n - 1 \\) digits being \\( 1 \\) and one digit being \\( 7 \\) is prime. To explore this, consider the structure of such numbers.\n\nFor a given \\( n \\), any number... |
deepscaler | [
{
"role": "user",
"content": "Triangle ABC is an equilateral triangle and O is the center of its inscribed circle. If the area of the circle is $4\\pi$ sq cm, what is the area, in square centimeters, of triangle ABC? Express your answer in simplest radical form.\n\nNote: The area of a circle with radius $r$... | math | {
"style": "rule",
"ground_truth": "12\\sqrt 3"
} | {
"split": "train",
"index": 27894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the redesigned version of Mathland's licensing system, all automobile license plates still contain four symbols. The first is a vowel (A, E, I, O, U), the second a non-vowel (any consonant), and the last two symbols are now digits (0 through 9). What is the probability that ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10,500}"
} | {
"split": "train",
"index": 27895,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nPeter's most listened-to CD contains eleven tracks. His favorite is the eighth track. When he inserts the CD into the player and presses one button, the first track starts, and by pressing the button seven more times, he reaches his favorite song. If the device is in \"random... | math | {
"style": "rule",
"ground_truth": "7/11"
} | {
"split": "train",
"index": 27896,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For integers a, b, c, and d the polynomial $p(x) =$ $ax^3 + bx^2 + cx + d$ satisfies $p(5) + p(25) = 1906$ . Find the minimum possible value for $|p(15)|$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 27897,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line that passes through the two foci of the hyperbola $$\\frac {x^{2}}{a^{2}} - \\frac {y^{2}}{b^{2}} = 1 (a > 0, b > 0)$$ and is perpendicular to the x-axis intersects the hyperbola at four points, forming a square. Find the eccentricity of this hyperbola. Let's think step ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} + 1}{2}"
} | {
"split": "train",
"index": 27898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of \n\\[(13 - x)(11 - x)(13 + x)(11 + x) + 1000.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "424"
} | {
"split": "train",
"index": 27899,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A tetrahedron is formed using the vertices of a cube. How many such distinct tetrahedrons can be formed? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 27900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles, one of radius 5 inches, the other of radius 2 inches, are tangent at point P. Two bugs start crawling at the same time from point P, one crawling along the larger circle at $3\\pi$ inches per minute, the other crawling along the smaller circle at $2.5\\pi$ inches p... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 27901,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The pages of a book are numbered $1_{}^{}$ through $n_{}^{}$. When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of $1986_{}^{}$. What was the number of the page that was added twice? Let's think step ... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 27902,
"solution": "Denote the page number as $x$, with $x < n$. The sum formula shows that $\\frac{n(n + 1)}{2} + x = 1986$. Since $x$ cannot be very large, disregard it for now and solve $\\frac{n(n+1)}{2} = 1986$. The positive root for $n \\approx \\sqrt{3972} \\approx 63$. Quickly... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $b\\sin 2C=c\\sin B$. \n$(1)$ Find the value of angle $C$; \n$(2)$ If $\\sin \\left(B- \\frac {\\pi}{3}\\right)= \\frac {3}{5}$, find the value of $\\s... | math | {
"style": "rule",
"ground_truth": "\\frac {4 \\sqrt {3}-3}{10}"
} | {
"split": "train",
"index": 27903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two arithmetic sequences $\\{a_n\\}$ and $\\{b_n\\}$, the sums of the first $n$ terms are $S_n$ and $T_n$, respectively. For any positive integer $n$, it holds that $$\\frac {S_{n}}{T_{n}} = \\frac {3n+5}{2n+3}$$, then $$\\frac {a_{7}}{b_{7}} = \\_\\_\\_\\_\\_\\_ ．$$ Let'... | math | {
"style": "rule",
"ground_truth": "\\frac {44}{29}"
} | {
"split": "train",
"index": 27904,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin(a + \\frac{\\pi}{4}) = \\sqrt{2}(\\sin \\alpha + 2\\cos \\alpha)$, determine the value of $\\sin 2\\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{5}"
} | {
"split": "train",
"index": 27905,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 27906,
"solution": "1. **Define Variables:**\n Let $p$ be Pete's current age and $c$ be Claire's current age.\n\n2. **Translate the Problem into Equations:**\n - From the statement \"Two years ago Pete was three times as old as his cousin Claire,\" we have:\n \\[\n p - 2 =... |
deepscaler | [
{
"role": "user",
"content": "A function \\( f(n) \\) defined for positive integers satisfies:\n\\[ f(n) = \\begin{cases} \nn - 3 & \\text{if } n \\geq 1000 \\\\ \nf[f(n + 7)] & \\text{if } n < 1000 \n\\end{cases} \\]\nDetermine \\( f(90) \\). Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "999"
} | {
"split": "train",
"index": 27907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{137}{a}=0.1 \\dot{2} 3 \\dot{4}$, find the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1110"
} | {
"split": "train",
"index": 27908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ewan writes out a sequence where he counts by 11s starting at 3. Which number will appear in Ewan's sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "113"
} | {
"split": "train",
"index": 27909,
"solution": "Ewan's sequence starts with 3 and each following number is 11 larger than the previous number. Since every number in the sequence is some number of 11s more than 3, this means that each number in the sequence is 3 more than a multiple of 11. Furthermore, every such... |
deepscaler | [
{
"role": "user",
"content": "The sum of the positive divisors of a positive integer of the form $2^i3^j$ is equal to $600$. What is $i + j$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 27910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In our daily life, for a pair of new bicycle tires, the rear tire wears out faster than the front tire. Through testing, it is found that the front tire of a general bicycle is scrapped after traveling 11,000 kilometers, while the rear tire is scrapped after traveling 9,000 kil... | math | {
"style": "rule",
"ground_truth": "4950"
} | {
"split": "train",
"index": 27911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the set of numbers $\\{1, 10, 10^2, 10^3, \\ldots, 10^{10}\\}$. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 27912,
"solution": "To solve the problem, we need to find the ratio of the largest element in the set $\\{1, 10, 10^2, 10^3, \\ldots, 10^{10}\\}$ to the sum of all other elements in the set. The largest element in this set is $10^{10}$.\n\n1. **Calculate the sum of the other elements:... |
deepscaler | [
{
"role": "user",
"content": "Suppose rectangle $F O L K$ and square $L O R E$ are on the plane such that $R L=12$ and $R K=11$. Compute the product of all possible areas of triangle $R K L$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "414"
} | {
"split": "train",
"index": 27913,
"solution": "There are two possible configurations. If $R L=12$, the side length of the square is $6 \\sqrt{2}$. Now $$121=R K^{2}=R E^{2}+E K^{2}=(6 \\sqrt{2})^{2}+E K^{2}$$ so $E K=7$. Then the possible values of $L K$ are $6 \\sqrt{2} \\pm 7$. Note that the area of $\\triang... |
deepscaler | [
{
"role": "user",
"content": "The area of trapezoid $ABCD$ is $164 \\text{cm}^2$. The altitude is $8 \\text{cm}$, $AB$ is $10 \\text{cm}$, and $CD$ is $17 \\text{cm}$. What is $BC$, in centimeters? [asy]\n/* AMC8 2003 #21 Problem */\nsize(2inch,1inch);\ndraw((0,0)--(31,0)--(16,8)--(6,8)--cycle);\ndraw((11... | math | {
"style": "rule",
"ground_truth": "10\\text{ cm}"
} | {
"split": "train",
"index": 27914,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A person rides a bicycle from place A to place B. If they increase their speed by 6 km/h, they can arrive 5 minutes earlier; if they decrease their speed by 5 km/h, they will be 6 minutes late. What is the distance between place A and place B in kilometers? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 27915,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\\%$. Find the maximum discount rate that can be offered. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "12\\%"
} | {
"split": "train",
"index": 27916,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Given $\\cos (α+ \\frac {π}{6})- \\sin α= \\frac {3 \\sqrt {3}}{5}$, find the value of $\\sin (α+ \\frac {5π}{6})$;\n(2) Given $\\sin α+ \\sin β= \\frac {1}{2}, \\cos α+ \\cos β= \\frac { \\sqrt {2}}{2}$, find the value of $\\cos (α-β)$. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "-\\frac {5}{8}"
} | {
"split": "train",
"index": 27917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $a>0$ , let $f(a)=\\lim_{t\\to\\+0} \\int_{t}^1 |ax+x\\ln x|\\ dx.$ Let $a$ vary in the range $0 <a< +\\infty$ , find the minimum value of $f(a)$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\ln 2}{2}"
} | {
"split": "train",
"index": 27918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\(\\frac{x+ \\sqrt{2}i}{i}=y+i\\), where \\(x\\), \\(y\\in\\mathbb{R}\\), and \\(i\\) is the imaginary unit, find \\(|x-yi|\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 27919,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the domains of functions f(x) and g(x) are both 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, g(2) = 4, determine the value of \\sum _{k=1}^{22}f(k). Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "-24"
} | {
"split": "train",
"index": 27920,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Masha looked at the drawing and said: \"There are seven rectangles here: one big one and six small ones.\" \"There are also various middle-sized rectangles here,\" said her mother. How many rectangles are there in total in this drawing? Explain your answer. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 27921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x, y, z$ be real numbers so that \\begin{align*}\ny+z & = 13, \\\\\nz+x & = 14, \\\\\nx+y & = 15.\n\\end{align*} Find $\\sqrt{xyz(x+y+z)}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 27922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "This year is 2017, and the sum of the digits of the year is 10. Find the sum of all the years in this century whose digits sum to 10. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18396"
} | {
"split": "train",
"index": 27923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jonah’s five cousins are visiting and there are four identical rooms for them to stay in. If any number of cousins can occupy any room, how many different ways can the cousins be arranged among the rooms? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 27924,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many ordered pairs $(b,c)$ of positive integers does neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 27925,
"solution": "To determine the number of ordered pairs $(b,c)$ of positive integers such that neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions, we analyze the discriminants of these quadratic equations.\n\n1. **Analyze the discriminants:**\n - For $x^2 + ... |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C:\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with left focus $F$ and a chord perpendicular to the major axis of length $6\\sqrt{2}$, a line passing through point $P(2,1)$ with slope $-1$ intersects $C$ at points $A$ and $B$, where $P$ is the m... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{2} + 6"
} | {
"split": "train",
"index": 27926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each positive digit $D$ and positive integer $k$, we use the symbol $D_{(k)}$ to represent the positive integer having exactly $k$ digits, each of which is equal to $D$. For example, $2_{(1)}=2$ and $3_{(4)}=3333$. There are $N$ quadruples $(P, Q, R, k)$ with $P, Q$ and $R$... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 27927,
"solution": "Suppose that $D$ is a digit and $k$ is a positive integer. Then $D_{(k)}=\\underbrace{D D \\cdots D D}_{k \\text { times }}=D \\cdot \\underbrace{11 \\cdots 11}_{k \\text { times }}=D \\cdot \\frac{1}{9} \\cdot \\underbrace{99 \\cdots 99}_{k \\text { times }}=D \\c... |
deepscaler | [
{
"role": "user",
"content": "Let set $\\mathcal{A}$ be a 90-element subset of $\\{1,2,3,\\ldots,100\\},$ and let $S$ be the sum of the elements of $\\mathcal{A}.$ Find the number of possible values of $S.$\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "901"
} | {
"split": "train",
"index": 27928,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two positive integers \\( x \\) and \\( y \\) have \\( xy=24 \\) and \\( x-y=5 \\). What is the value of \\( x+y \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 27929,
"solution": "The positive integer divisors of 24 are \\( 1,2,3,4,6,8,12,24 \\). The pairs of divisors that give a product of 24 are \\( 24 \\times 1,12 \\times 2,8 \\times 3 \\), and \\( 6 \\times 4 \\). We want to find two positive integers \\( x \\) and \\( y \\) whose produc... |
deepscaler | [
{
"role": "user",
"content": "Given a parabola \\(C\\) with the center of ellipse \\(E\\) as its focus, the parabola \\(C\\) passes through the two foci of the ellipse \\(E\\), and intersects the ellipse \\(E\\) at exactly three points. Find the eccentricity of the ellipse \\(E\\). Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{5}}{5}"
} | {
"split": "train",
"index": 27930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the $\\emph{positive}$ real number(s) $x$ such that $\\frac{1}{2}\\left( 3x^2-1\\right) = \\left( x^2-50x-10\\right)\\left( x^2+25x+5\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25 + 2\\sqrt{159}"
} | {
"split": "train",
"index": 27931,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( ABC \\), angle \\( B \\) is right. The midpoint \\( M \\) is marked on side \\( BC \\), and there is a point \\( K \\) on the hypotenuse such that \\( AB = AK \\) and \\(\\angle BKM = 45^{\\circ}\\). Additionally, there are points \\( N \\) and \\( L \\) on side... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"split": "train",
"index": 27932,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x \\neq y$ be positive reals satisfying $x^3+2013y=y^3+2013x$ , and let $M = \\left( \\sqrt{3}+1 \\right)x + 2y$ . Determine the maximum possible value of $M^2$ .\n\n*Proposed by Varun Mohan* Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "16104"
} | {
"split": "train",
"index": 27933,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the following equation by completing the square: $$64x^2+96x-81 = 0.$$ Rewrite the equation in the form \\((ax + b)^2 = c\\), where \\(a\\), \\(b\\), and \\(c\\) are integers and \\(a > 0\\). What is the value of \\(a + b + c\\)? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "131"
} | {
"split": "train",
"index": 27934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The distance between two cities on a map is 15 inches. If the scale is 0.25 inches = 3 miles, how many miles apart are the actual cities? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "180\\text{ miles}"
} | {
"split": "train",
"index": 27935,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $1 \\times 3$ rectangle is inscribed in a semicircle with the longer side on the diameter. What is the area of the semicircle?\nA) $\\frac{9\\pi}{8}$\nB) $\\frac{12\\pi}{8}$\nC) $\\frac{13\\pi}{8}$\nD) $\\frac{15\\pi}{8}$\nE) $\\frac{16\\pi}{8}$ Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "\\frac{13\\pi}{8}"
} | {
"split": "train",
"index": 27936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ and $y$ are positive integers with $3^{x} 5^{y} = 225$, what is the value of $x + y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 27937,
"solution": "Since $15^{2}=225$ and $15=3 \\cdot 5$, then $225=15^{2}=(3 \\cdot 5)^{2}=3^{2} \\cdot 5^{2}$. Therefore, $x=2$ and $y=2$, so $x+y=4$."
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the number of ways to write $2010$ in the form $2010 = a_3 \\cdot 10^3 + a_2 \\cdot 10^2 + a_1 \\cdot 10 + a_0$, where the $a_i$'s are integers, and $0 \\le a_i \\le 99$. An example of such a representation is $1\\cdot 10^3 + 3\\cdot 10^2 + 67\\cdot 10^1 + 40\\cdot 1... | math | {
"style": "rule",
"ground_truth": "202"
} | {
"split": "train",
"index": 27938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the modular inverse of \\( 31 \\), modulo \\( 45 \\).\n\nExpress your answer as an integer from \\( 0 \\) to \\( 44 \\), inclusive. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 27939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a, b$ be positive reals with $a>b>\\frac{1}{2} a$. Place two squares of side lengths $a, b$ next to each other, such that the larger square has lower left corner at $(0,0)$ and the smaller square has lower left corner at $(a, 0)$. Draw the line passing through $(0, a)$ and... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 27940,
"solution": "Let $t=\\frac{a}{b} \\in(1,2)$; we will rewrite the sum $a+b$ as a function of $t$. The area condition easily translates to $\\frac{a^{2}-a b+2 b^{2}}{2}=2013$, or $b^{2}\\left(t^{2}-t+2\\right)=4026 \\Longleftrightarrow b=\\sqrt{\\frac{4026}{t^{2}-t+2}}$. Thus $a+... |
deepscaler | [
{
"role": "user",
"content": "What is the largest possible value of \\(| |a_1 - a_2| - a_3| - \\ldots - a_{1990}|\\), where \\(a_1, a_2, \\ldots, a_{1990}\\) is a permutation of \\(1, 2, 3, \\ldots, 1990\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1989"
} | {
"split": "train",
"index": 27941,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "King Middle School has $1200$ students. Each student takes $5$ classes a day. Each teacher teaches $4$ classes. Each class has $30$ students and $1$ teacher. How many teachers are there at King Middle School? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 27942,
"solution": "1. **Calculate the total number of classes taken by all students**: Each student takes 5 classes per day, and there are 1200 students. Therefore, the total number of classes taken by all students is:\n \\[\n 5 \\times 1200 = 6000 \\text{ classes}\n \\]\n\n2. ... |
deepscaler | [
{
"role": "user",
"content": "A certain product in a shopping mall sells an average of 70 items per day, with a profit of $50 per item. In order to reduce inventory quickly, the mall decides to take appropriate price reduction measures. After investigation, it was found that for each item, for every $1 decr... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 27943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function $f(x)=2x-\\cos x$, and $\\{a_n\\}$ be an arithmetic sequence with a common difference of $\\dfrac{\\pi}{8}$. If $f(a_1)+f(a_2)+\\ldots+f(a_5)=5\\pi$, calculate $\\left[f(a_3)\\right]^2-a_2a_3$. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "\\dfrac{13}{16}\\pi^2"
} | {
"split": "train",
"index": 27944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbb{N}$ be the set of positive integers, and let $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ be a function satisfying $f(1)=1$ and for $n \\in \\mathbb{N}, f(2 n)=2 f(n)$ and $f(2 n+1)=2 f(n)-1$. Determine the sum of all positive integer solutions to $f(x)=19$ that do n... | math | {
"style": "rule",
"ground_truth": "1889"
} | {
"split": "train",
"index": 27945,
"solution": "For $n=2^{a_{0}}+2^{a_{1}}+\\cdots+2^{a_{k}}$ where $a_{0}>a_{1}>\\cdots>a_{k}$, we can show that $f(n)=2^{a_{0}}-2^{a_{1}}-\\cdots-2^{a_{k}}=2^{a_{0}+1}-n$ by induction: the base case $f(1)=1$ clearly holds; for the inductive step, when $n$ is even we note that $f... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\ln x+\\frac{1}{2}x^2-ax$, where $a\\in \\mathbb{R}$, it has two extreme points at $x=x_1$ and $x=x_2$, with $x_1 < x_2$.\n\n(I) When $a=3$, find the extreme values of the function $f(x)$.\n\n(II) If $x_2\\geqslant ex_1$ ($e$ is the base of the natural... | math | {
"style": "rule",
"ground_truth": "1-\\frac{e}{2}+\\frac{1}{2e}"
} | {
"split": "train",
"index": 27946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A ladder is leaning against a house with its lower end 15 feet from the house. When the lower end is pulled 9 feet farther from the house, the upper end slides 13 feet down. How long is the ladder (in feet)? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 27947,
"solution": "Of course the house makes a right angle with the ground, so we can use the Pythagorean theorem. Let $x$ be the length of the ladder and $y$ be the original height at which it touched the house. Then we are given $x^{2}=15^{2}+y^{2}=24^{2}+(y-13)^{2}$. Isolating $y$... |
deepscaler | [
{
"role": "user",
"content": "A coordinate paper is folded once such that the point \\((0,2)\\) overlaps with the point \\((4,0)\\). If the point \\((7,3)\\) overlaps with the point \\((m, n)\\), what is the value of \\(m+n\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6.8"
} | {
"split": "train",
"index": 27948,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.