data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Compute the sum of all positive integers $a \\leq 26$ for which there exist integers $b$ and $c$ such that $a+23 b+15 c-2$ and $2 a+5 b+14 c-8$ are both multiples of 26. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 30249,
"solution": "Assume $b$ and $c$ exist. Considering the two values modulo 13, we find $$\\begin{cases}a+10 b+2 c \\equiv 2 & (\\bmod 13) \\\\ 2 a+5 b+c \\equiv 8 & (\\bmod 13)\\end{cases}$$ Subtracting twice the second equation from the first, we get $-3 a \\equiv-14(\\bmod 13)$... |
deepscaler | [
{
"role": "user",
"content": "In writing the integers from 20 through 199 inclusive, how many times is the digit 7 written? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 30250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many perfect squares are two-digit and divisible by $3?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 30251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of \\(\\frac{1^{2}-3^{2}+5^{2}-7^{2}+\\cdots+97^{2}-99^{2}}{1-3+5-7+\\cdots+97-99}\\) is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 30252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17,$ and $x$ is equal to the mean of those five numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 30253,
"solution": "1. **Calculate the mean of the numbers:** \n The mean of the numbers $4, 6, 8, 17,$ and $x$ is given by:\n \\[\n \\text{Mean} = \\frac{4 + 6 + 8 + 17 + x}{5} = \\frac{35 + x}{5}\n \\]\n\n2. **Identify possible values for the median:** \n The median of t... |
deepscaler | [
{
"role": "user",
"content": "Let $f : \\mathbb{R} \\to \\mathbb{R}$ be a function such that\n\\[f(f(x) + y) = f(x^2 - y) + 4f(x) y\\]for all real numbers $x$ and $y.$\n\nLet $n$ be the number of possible values of $f(3),$ and let $s$ be the sum of all possible values of $f(3).$ Find $n \\times s.$ Let's t... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 30254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the quadratic function $f(x)=ax^{2}+bx+c$, where $a$, $b$, and $c$ are constants, if the solution set of the inequality $f(x) \\geqslant 2ax+b$ is $\\mathbb{R}$, find the maximum value of $\\frac{b^{2}}{a^{2}+c^{2}}$. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}-2"
} | {
"split": "train",
"index": 30255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle \\(ABC\\) has \\(AB = 10\\) and \\(BC:AC = 35:36\\). What is the largest area that this triangle can have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1260"
} | {
"split": "train",
"index": 30256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\tan \\alpha = -\\frac{1}{2}$, find the value of $\\frac{1+2\\sin \\alpha \\cos \\alpha}{\\sin^2 \\alpha - \\cos^2 \\alpha}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"split": "train",
"index": 30257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $\\angle A = 60^\\circ$ and $\\angle B = 45^\\circ$. A line $DE$, with $D$ on $AB$ and $\\angle ADE = 45^\\circ$, divides $\\triangle ABC$ into two pieces of equal area. Determine the ratio $\\frac{AD}{AB}$.\nA) $\\frac{\\sqrt{6}}{4}$ \nB) $\\frac{\\sqrt{7}}... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6} + \\sqrt{2}}{4\\sqrt{2}}"
} | {
"split": "train",
"index": 30258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The students in Mrs. Reed's English class are reading the same $760$-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in $20$ seconds, Bob reads a page in $45$ seconds and Chandra reads a page in $30$ seconds. Chandra and Bob, who each h... | math | {
"style": "rule",
"ground_truth": "456"
} | {
"split": "train",
"index": 30259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Luca mixes 50 mL of milk for every 250 mL of flour to make pizza dough. How much 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 \\text{ mL}"
} | {
"split": "train",
"index": 30260,
"solution": "We divide the 750 mL of flour into portions of 250 mL. We do this by calculating $750 \\div 250 = 3$. Therefore, 750 mL is three portions of 250 mL. Since 50 mL of milk is required for each 250 mL of flour, then $3 \\times 50 = 150 \\text{ mL}$ of milk is required ... |
deepscaler | [
{
"role": "user",
"content": "ABCDEF is a six-digit number. All of its digits are different and arranged in ascending order from left to right. This number is a perfect square.\n\nDetermine what this number is. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "134689"
} | {
"split": "train",
"index": 30261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $y$: $3y+7y = 282-8(y-3)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 30262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ellipse $x^2+4y^2=4$ and the hyperbola $x^2-m(y+2)^2 = 1$ are tangent. Compute $m.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{12}{13}"
} | {
"split": "train",
"index": 30263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During a space experiment conducted by astronauts, they must implement a sequence of 6 procedures. Among them, Procedure A can only occur as the first or the last step, and Procedures B and C must be adjacent when conducted. How many different sequences are there to arrange the... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 30264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three balls are drawn simultaneously from the urn (as described in Problem 4). Find the probability that all the drawn balls are blue (event $B$). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/12"
} | {
"split": "train",
"index": 30265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the smallest base-10 positive integer greater than 5 that is a palindrome when written in both base 2 and 4. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 30266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tom's algebra notebook consists of 50 pages, 25 sheets of paper. Specifically, page 1 and page 2 are the front and back of the first sheet of paper, page 3 and page 4 are the front and back of the second sheet of paper, and so on. One day, Tom left the notebook on the table whi... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 30267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A small town has fewer than 6000 inhabitants. We know that there are $10\\%$ more girls than boys among the children, and $15\\%$ more men than women among the adults. There are $20\\%$ more children than adults in the town.\n\nHow many people live in the town? Let's think step... | math | {
"style": "rule",
"ground_truth": "3311"
} | {
"split": "train",
"index": 30268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $z$ be a non-real complex number with $z^{23}=1$. Compute $$ \\sum_{k=0}^{22} \\frac{1}{1+z^{k}+z^{2 k}} $$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "46 / 3"
} | {
"split": "train",
"index": 30269,
"solution": "First solution: Note that $$ \\sum_{k=0}^{22} \\frac{1}{1+z^{k}+z^{2 k}}=\\frac{1}{3}+\\sum_{k=1}^{22} \\frac{1-z^{k}}{1-z^{3 k}}=\\frac{1}{3}+\\sum_{k=1}^{22} \\frac{1-\\left(z^{24}\\right)^{k}}{1-z^{3 k}}=\\frac{1}{3}+\\sum_{k=1}^{22} \\sum_{\\ell=0}^{7} z^{3 k \... |
deepscaler | [
{
"role": "user",
"content": "Set $B = \\{55, 57, 58, 59, 61\\}.$ Which number in set B has the smallest prime factor? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 30270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an isosceles triangle \\(ABC\\) with base \\(AC\\), point \\(D\\) divides side \\(BC\\) in the ratio \\(3:1\\) starting from vertex \\(B\\), and point \\(E\\) is the midpoint of segment \\(AD\\). It is known that \\(BE = \\sqrt{7}\\) and \\(CE = 3\\). Find the radius of the ... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"split": "train",
"index": 30271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": " $ABC$ is an equilateral triangle and $l$ is a line such that the distances from $A, B,$ and $C$ to $l$ are $39, 35,$ and $13$ , respectively. Find the largest possible value of $AB$ . \n\n*Team #6* Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "58\\sqrt{3}"
} | {
"split": "train",
"index": 30272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $1010101_2 + 111000_2$? Write your answer in base $10$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "141"
} | {
"split": "train",
"index": 30273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ and $A$ denote the perimeter and area respectively of a right triangle with relatively prime integer side-lengths. Find the largest possible integral value of $\\frac{P^{2}}{A}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 30274,
"solution": "Assume WLOG that the side lengths of the triangle are pairwise coprime. Then they can be written as $m^{2}-n^{2}, 2mn, m^{2}+n^{2}$ for some coprime integers $m$ and $n$ where $m>n$ and $mn$ is even. Then we obtain $\\frac{P^{2}}{A}=\\frac{4m(m+n)}{n(m-n)}$. But $n... |
deepscaler | [
{
"role": "user",
"content": "Triangles $\\triangle ABC$ and $\\triangle DEC$ share side $BC$. Given that $AB = 7\\ \\text{cm}$, $AC = 15\\ \\text{cm}$, $EC = 9\\ \\text{cm}$, and $BD = 26\\ \\text{cm}$, what is the least possible integral number of centimeters in $BC$? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 30275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\frac{x^8 + 16x^4 + 64 + 4x^2}{x^4 + 8}$ when $x = 3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "89 + \\frac{36}{89}"
} | {
"split": "train",
"index": 30276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f(x) = |\\log_3 x|$ has a range of $[0,1]$ on the interval $[a, b]$. Find the minimum value of $b - a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 30277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What are the mode and median of the set of ages $29$, $27$, $31$, $31$, $31$, $29$, $29$, and $31$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 30278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$ the median from $A$ is given perpendicular to the median from $B$. If $BC=7$ and $AC=6$, find the length of $AB$.\n$\\textbf{(A)}\\ 4\\qquad \\textbf{(B)}\\ \\sqrt{17} \\qquad \\textbf{(C)}\\ 4.25\\qquad \\textbf{(D)}\\ 2\\sqrt{5} \\qquad \\textbf{(E)}\\ 4.5... | math | {
"style": "rule",
"ground_truth": "\\sqrt{17}"
} | {
"split": "train",
"index": 30279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the regression equation $\\hat{y}$=4.4x+838.19, estimate the ratio of the growth rate between x and y, denoted as \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{22}"
} | {
"split": "train",
"index": 30280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "ABCD is a trapezium inscribed in a circle centered at O. It is given that AB is parallel to CD, angle COD is three times angle AOB, and the ratio of AB to CD is 2:5. Calculate the ratio of the area of triangle BOC to the area of triangle AOB. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 30281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the systematic sampling method to select 32 people for a questionnaire survey from 960 people, determine the number of people among the 32 whose numbers fall within the interval [200, 480]. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 30282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(A B C\\), side \\(B C\\) equals 4, and the median drawn to this side equals 3. Find the length of the common chord of two circles, each of which passes through point \\(A\\) and is tangent to \\(B C\\), with one tangent at point \\(B\\) and the other at point \\(... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 30283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alina and Masha wanted to create an interesting version of the school tour of the Olympiad. Masha proposed several problems and rejected every second problem of Alina (exactly half). Alina also proposed several problems and only accepted every third problem of Masha (exactly on... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 30284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the polar coordinate system, the polar equation of curve $C$ is $\\rho =6\\sin \\theta$, and the polar coordinates of point $P$ are $(\\sqrt{2},\\frac{\\pi }{4})$. Taking the pole as the origin of coordinates and the positive half-axis of the $x$-axis as the polar axis, a pl... | math | {
"style": "rule",
"ground_truth": "3 \\sqrt{2}"
} | {
"split": "train",
"index": 30285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the base four equivalent of $123_{10}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1323_{4}"
} | {
"split": "train",
"index": 30286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When Tanya excluded all numbers from 1 to 333 that are divisible by 3 but not by 7, and all numbers that are divisible by 7 but not by 3, she ended up with 215 numbers. Did she solve the problem correctly? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "205"
} | {
"split": "train",
"index": 30287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Does there exist a fraction equivalent to $\\frac{7}{13}$ such that the difference between the denominator and the numerator is 24? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{28}{52}"
} | {
"split": "train",
"index": 30288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each positive integer \\( n \\), define \\( A_{n} = \\frac{20^{n} + 11^{n}}{n!} \\), where \\( n! = 1 \\times 2 \\times \\cdots \\times n \\). Find the value of \\( n \\) that maximizes \\( A_{n} \\). Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 30289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let X be a set containing 10 elements, and A, B be two disjoint subsets of X, containing 3 and 4 elements respectively. Calculate the number of subsets of X that contain neither A nor B. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "840"
} | {
"split": "train",
"index": 30290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y = x - 5$, let $x = 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5$, we can obtain 10 points on the graph of the function. Randomly select two points $P(a, b)$ and $Q(m, n)$ from these 10 points. What is the probability that $P$ and $Q$ lie on the same inverse prop... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{45}"
} | {
"split": "train",
"index": 30291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest common divisor of $8!$ and $(6!)^2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1440"
} | {
"split": "train",
"index": 30292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given non-zero vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfying $|\\overrightarrow{a}| = |\\overrightarrow{b}| = |\\overrightarrow{a} + \\overrightarrow{b}|$, the cosine of the angle between $\\overrightarrow{a}$ and $2\\overrightarrow{a} - \\overrightarrow{b}... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{7}}{14}"
} | {
"split": "train",
"index": 30293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the complex number $a^2 - 1 + (a - 1)i$ (where $i$ is the imaginary unit) is purely imaginary, calculate the real number $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 30294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a $\\clubsuit$, the second card is a $\\heartsuit$, and the third card is a king? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{2550}"
} | {
"split": "train",
"index": 30295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify completely: $$\\sqrt[3]{80^3 + 100^3 + 120^3}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20\\sqrt[3]{405}"
} | {
"split": "train",
"index": 30296,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that there are three mathematics teachers: Mrs. Germain with 13 students, Mr. Newton with 10 students, and Mrs. Young with 12 students, and 2 students are taking classes from both Mrs. Germain and Mr. Newton and 1 additional student is taking classes from both Mrs. Germai... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 30297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Observe the following equations:<br/>$\\frac{1}{1×2}=1-\\frac{1}{2}=\\frac{1}{2}$;<br/>$\\frac{1}{1×2}+\\frac{1}{2×3}=1-\\frac{1}{2}+\\frac{1}{2}-\\frac{1}{3}=\\frac{2}{3}$;<br/>$\\frac{1}{1×2}+\\frac{1}{2×3}+\\frac{1}{3×4}=1-\\frac{1}{2}+\\frac{1}{2}-\\frac{1}{3}+\\frac{1}{3}-... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{100}"
} | {
"split": "train",
"index": 30298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nThe Gregorian calendar defines a common year as having 365 days and a leap year as having 366 days. The $n$-th year is a leap year if and only if:\n\n1. $n$ is not divisible by 100 and $n$ is divisible by 4, or\n2. $n$ is divisible by 100 and $n$ is divisible by 400.\n\nFor e... | math | {
"style": "rule",
"ground_truth": "365.2425"
} | {
"split": "train",
"index": 30299,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ be a triangle in the plane, and let $D$ be a point outside the plane of $\\triangle ABC$, so that $DABC$ is a pyramid whose faces are all triangles.\n\nSuppose that every edge of $DABC$ has length $20$ or $45$, but no face of $DABC$ is equilateral. Then wha... | math | {
"style": "rule",
"ground_truth": "40 \\sqrt{1925}"
} | {
"split": "train",
"index": 30300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five friends were comparing how much scrap iron they brought to the collection. On average, it was $55 \\mathrm{~kg}$, but Ivan brought only $43 \\mathrm{~kg}$.\n\nWhat is the average amount of iron brought without Ivan?\n\n(Note: By how many kilograms does Ivan's contribution ... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 30301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the continuation of side \\( BC \\) of parallelogram \\( ABCD \\), a point \\( F \\) is taken beyond point \\( C \\). Segment \\( AF \\) intersects diagonal \\( BD \\) at point \\( E \\) and side \\( CD \\) at point \\( G \\), where \\( GF=3 \\) and \\( AE \\) is 1 more than... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 30302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A six digit number (base 10) is squarish if it satisfies the following conditions:\n(i) none of its digits are zero;\n(ii) it is a perfect square; and\n(iii) the first of two digits, the middle two digits and the last two digits of the number are all perfect squares when consid... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 30303,
"solution": "To find the number of squarish numbers, we need to satisfy the following conditions:\n1. The number is a six-digit number.\n2. None of its digits are zero.\n3. It is a perfect square.\n4. The first two digits, the middle two digits, and the last two digits of the n... |
deepscaler | [
{
"role": "user",
"content": "A circle is inscribed in a square, then a square is inscribed in this circle. Following this, a regular hexagon is inscribed in the smaller circle and finally, a circle is inscribed in this hexagon. What is the ratio of the area of the smallest circle to the area of the origina... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{32}"
} | {
"split": "train",
"index": 30304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $6 \\times 9$ rectangle can be rolled to form two different cylinders. Calculate the ratio of the larger volume to the smaller volume. 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{3}{2}"
} | {
"split": "train",
"index": 30305,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression \\[ \\frac{a^2 + 2a}{a^2 + a} \\cdot \\frac{b^2 - 4}{b^2 - 6b + 8} \\cdot \\frac{c^2 + 16c + 64}{c^2 + 12c + 36} \\]\ngiven that \\(c = b - 20\\), \\(b = a + 4\\), \\(a = 2\\), and ensuring none of the denominators are zero. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 30306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the square of four times a positive integer is decreased by twice the integer, the result is $8066$. What is the integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "182"
} | {
"split": "train",
"index": 30307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different counting numbers will each leave a remainder of 5 when divided into 47? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 30308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle with perimeter $7$ has integer sidelengths. What is the maximum possible area of such a triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{7}}{4}"
} | {
"split": "train",
"index": 30309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the margin made on an article costing $C$ dollars and selling for $S$ dollars is $M=\\frac{1}{n}C$, then the margin is given by: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{n+1}S"
} | {
"split": "train",
"index": 30310,
"solution": "Given:\n- Cost of the article, $C$ dollars.\n- Selling price of the article, $S$ dollars.\n- Margin made on the article, $M = \\frac{1}{n}C$.\n\nWe need to express the margin $M$ in terms of the selling price $S$. \n\n1. **Express Cost in terms of Selling Price and... |
deepscaler | [
{
"role": "user",
"content": "Determine the greatest integer less than or equal to \\[\\frac{5^{50} + 3^{50}}{5^{45} + 3^{45}}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3124"
} | {
"split": "train",
"index": 30311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 30312,
"solution": "Line Segment $DE = \\frac{1}{n}$, so $EC = 1 - \\frac{1}{n} = \\frac{n-1}{n}$. Draw line segment $HE$ parallel to the corresponding sides of the small square, $HE$ has length $\\frac{1}{\\sqrt{1985}}$, as it is the same length as the sides of the square. Notice tha... |
deepscaler | [
{
"role": "user",
"content": "The numbers \\(a, b, c, d\\) belong to the interval \\([-7.5, 7.5]\\). Find the maximum value of the expression \\(a + 2b + c + 2d - ab - bc - cd - da\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 30313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $-25 \\le x \\le 25,$ find the maximum value of $\\sqrt{25 + x} + \\sqrt{25 - x}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 30314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $r(x)$ be a monic quartic polynomial such that $r(1) = 5,$ $r(2) = 8,$ $r(3) = 13,$ and $r(4) = 20.$ Find $r(5).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"split": "train",
"index": 30315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vasya is inventing a 4-digit password for a combination lock. He does not like the digit 2, so he does not use it. Moreover, he doesn't like when two identical digits stand next to each other. Additionally, he wants the first digit to match the last one. How many possible combi... | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 30316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A portion of the graph of $y = G(x)$ is shown in red below. The distance between grid lines is $1$ unit.\n\nCompute $G(G(G(G(G(1)))))$.\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nrea... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 30317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expression $\\log_{y^6}{x}\\cdot\\log_{x^5}{y^2}\\cdot\\log_{y^4}{x^3}\\cdot\\log_{x^3}{y^4}\\cdot\\log_{y^2}{x^5}$ can be written as $a\\log_y{x}$ for what constant $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac16"
} | {
"split": "train",
"index": 30318,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive real number $x$ such that\n\\[\\lfloor x^2 \\rfloor - x \\lfloor x \\rfloor = 8.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{89}{9}"
} | {
"split": "train",
"index": 30319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), side \\(BC = 28\\). The angle bisector \\(BL\\) is divided by the intersection point of the angle bisectors of the triangle in the ratio \\(4:3\\) from the vertex. Find the radius of the circumscribed circle around triangle \\(ABC\\) if the radius of the ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 30320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given vectors $\\overrightarrow{m} = (\\cos A, \\sin A)$ and $\\overrightarrow{n} = (\\cos B, -\\sin B)$, and $|\\overrightarrow{m} - \\overrightarrow{n}| = 1$.\n\n(1) Find the... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 30321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The fifth, sixth and seventh terms of an arithmetic sequence are 5, 9 and 13, respectively. What is the sum of the first four terms of the sequence? [asy]\nunitsize(8mm);\ndefaultpen(linewidth(.7pt)+fontsize(12pt));\n\nlabel(\"\\underline{\\quad}, \\underline{\\quad}, \\underli... | math | {
"style": "rule",
"ground_truth": "-20"
} | {
"split": "train",
"index": 30322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f$ from the set $\\mathbb{R}$ of real numbers into $\\mathbb{R}$ which satisfy for all $x, y, z \\in \\mathbb{R}$ the identity \\[f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz).\\] Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "f(x) = 0 \\text{ and } f(x) = x^2"
} | {
"split": "train",
"index": 30323,
"solution": "\nWe need to find all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) satisfying the given functional equation for all \\( x, y, z \\in \\mathbb{R} \\):\n\n\\[\nf(f(x) + f(y) + f(z)) = f(f(x) - f(y)) + f(2xy + f(z)) + 2f(xz - yz).\n\\]\n\nTo solve this, w... |
deepscaler | [
{
"role": "user",
"content": "Let \\( p, q, r, s \\) be distinct real numbers such that the roots of \\( x^2 - 12px - 13q = 0 \\) are \\( r \\) and \\( s \\), and the roots of \\( x^2 - 12rx - 13s = 0 \\) are \\( p \\) and \\( q \\). Find the value of \\( p + q + r + s \\). Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "2028"
} | {
"split": "train",
"index": 30324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The base of a pyramid is a square with each side of length one unit. One of its lateral edges is also one unit long and coincides with the height of the pyramid. What is the largest face angle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 30325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is now 12:00:00 midnight, as read on a 12-hour digital clock. In 122 hours, 39 minutes and 44 seconds the time will be $A:B:C$. What is the value of $A + B + C$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 30326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $n$, define $S(n)$ to be the smallest positive integer divisible by each of the positive integers $1, 2, 3, \\ldots, n$. How many positive integers $n$ with $1 \\leq n \\leq 100$ have $S(n) = S(n+4)$? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 30327,
"solution": "For each positive integer $n, S(n)$ is defined to be the smallest positive integer divisible by each of $1, 2, 3, \\ldots, n$. In other words, $S(n)$ is the least common multiple (lcm) of $1, 2, 3, \\ldots, n$. To calculate the lcm of a set of numbers, we determine... |
deepscaler | [
{
"role": "user",
"content": "Victoria wants to order at least 550 donuts from Dunkin' Donuts for the HMMT 2014 November contest. However, donuts only come in multiples of twelve. Assuming every twelve donuts cost \\$7.49, what is the minimum amount Victoria needs to pay, in dollars? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "344.54"
} | {
"split": "train",
"index": 30328,
"solution": "The smallest multiple of 12 larger than 550 is $552=12 \\cdot 46$. So the answer is $46 \\cdot \\$7.49$. To make the multiplication easier, we can write this as $46 \\cdot(\\$7.5-\\$0.01)=\\$345-\\$0.46=\\$344.54$."
} |
deepscaler | [
{
"role": "user",
"content": "What is the base-10 integer 515 when expressed in base 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2215_6"
} | {
"split": "train",
"index": 30329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph shows the number of minutes studied by both Asha (black bar) and Sasha (grey bar) in one week. On the average, how many more minutes per day did Sasha study than Asha? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 30330,
"solution": "To find how many more minutes per day on average Sasha studied than Asha, we need to calculate the daily differences in their study times and then find the average of these differences.\n\n1. **Calculate the daily differences**: From the problem, the differences be... |
deepscaler | [
{
"role": "user",
"content": "The repeating decimal for $\\frac{3}{11}$ is $0.ababab\\dots$ What is the value of the sum $a+b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 30331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many different ways can four students stand in a straight line if two of the students refuse to stand next to each other? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 30332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In cube \\(ABCDA_1B_1C_1D_1\\) with side length 1, a sphere is inscribed. Point \\(E\\) is located on edge \\(CC_1\\) such that \\(C_1E = \\frac{1}{8}\\). From point \\(E\\), a tangent to the sphere intersects the face \\(AA_1D_1D\\) at point \\(K\\), with \\(\\angle KEC = \\ar... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{8}"
} | {
"split": "train",
"index": 30333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the sum of the squares of nonnegative real numbers $a,b,$ and $c$ is $39$, and $ab + bc + ca = 21$, then what is the sum of $a,b,$ and $c$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 30334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the first term of a geometric sequence be $\\frac{3}{4}$, and let the second term be $15$. What is the smallest $n$ for which the $n$th term of the sequence is divisible by one million? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 30335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the sequence $5, 8, 15, 18, 25, 28, \\cdots, 2008, 2015$, how many numbers have a digit sum that is an even number? (For example, the digit sum of 138 is $1+3+8=12$) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "202"
} | {
"split": "train",
"index": 30336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $(a_1,a_2,a_3,\\ldots,a_{12})$ be a permutation of $(1,2,3,\\ldots,12)$ for which\n$a_1>a_2>a_3>a_4>a_5>a_6 \\mathrm{\\ and \\ } a_6<a_7<a_8<a_9<a_{10}<a_{11}<a_{12}.$\nAn example of such a permutation is $(6,5,4,3,2,1,7,8,9,10,11,12).$ Find the number of such permutations... | math | {
"style": "rule",
"ground_truth": "462"
} | {
"split": "train",
"index": 30337,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a $3 \\times 3$ block of squares as the center area in an array of unit squares. The first ring around this center block contains unit squares that directly touch the block. If the pattern continues as before, how many unit squares are in the $10^{th}$ ring? Let's thin... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 30338,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of 10-tuples $(x_1, x_2, \\dots, x_{10})$ of real numbers such that\n\\[(1 - x_1)^2 + (x_1 - x_2)^2 + (x_2 - x_3)^2 + \\dots + (x_9 - x_{10})^2 + x_{10}^2 = \\frac{1}{11}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 30339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation\n\\[(x - \\sqrt[3]{7})(x - \\sqrt[3]{29})(x - \\sqrt[3]{61}) = \\frac{1}{5}\\]\nhas three distinct solutions $u,$ $v,$ and $w.$ Calculate the value of $u^3 + v^3 + w^3.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "97.6"
} | {
"split": "train",
"index": 30340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Betty goes to the store to get flour and sugar. The amount of flour she buys, in pounds, is at least 6 pounds more than half the amount of sugar, and is no more than twice the amount of sugar. Find the least number of pounds of sugar that Betty could buy. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 30341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alex and Bob have 30 matches. Alex picks up somewhere between one and six matches (inclusive), then Bob picks up somewhere between one and six matches, and so on. The player who picks up the last match wins. How many matches should Alex pick up at the beginning to guarantee tha... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 30342,
"solution": "2."
} |
deepscaler | [
{
"role": "user",
"content": "The length of the chord cut by one of the asymptotes of the hyperbola $\\dfrac{x^2}{a^2} - \\dfrac{y^2}{b^2} = 1 \\ (a > 0, b > 0)$ on the circle $x^2 + y^2 - 6x + 5 = 0$ is $2$. Find the eccentricity of the hyperbola. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 30343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is inscribed in circle $\\omega$. Points $P$ and $Q$ are on side $\\overline{AB}$ with $AP<AQ$. Rays $CP$ and $CQ$ meet $\\omega$ again at $S$ and $T$ (other than $C$), respectively. If $AP=4,PQ=3,QB=6,BT=5,$ and $AS=7$, then $ST=\\frac{m}{n}$, where $m$ and $n$ ... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 30344,
"solution": "By Ptolemy's Theorem applied to quadrilateral $ASTB$, we find \\[AS\\cdot BT+AB\\cdot ST=AT\\cdot BS.\\] Projecting through $C$ we have \\[\\frac{AQ \\cdot PB}{PQ \\cdot AB} = (A,Q; P,B)\\stackrel{C}{=}(A,T; S,B)=\\frac{AT \\cdot BS}{ST \\cdot AB}.\\] Therefore \\[... |
deepscaler | [
{
"role": "user",
"content": "A train travelling at constant speed takes five seconds to pass completely through a tunnel which is $85 \\mathrm{~m}$ long, and eight seconds to pass completely through a second tunnel which is $160 \\mathrm{~m}$ long. What is the speed of the train? Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 30345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\sqrt{\\frac{1}{{49}}}=$____; $|{2-\\sqrt{5}}|=$____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}-2"
} | {
"split": "train",
"index": 30346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $n$, there is a circle around the origin with radius $n$. Rainbow Dash starts off somewhere on the plane, but not on a circle. She takes off in some direction in a straight path. She moves \\frac{\\sqrt{5}}{5}$ units before crossing a circle, then \\sq... | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{170}-9 \\sqrt{5}}{5}"
} | {
"split": "train",
"index": 30347,
"solution": "Note that the distance from Rainbow Dash's starting point to the first place in which she hits a circle is irrelevant, except in checking that this distance is small enough that she does not hit another circle beforehand. It will be clear at the end that our config... |
deepscaler | [
{
"role": "user",
"content": "There are 101 natural numbers written in a circle. It is known that among any three consecutive numbers, there is at least one even number. What is the minimum number of even numbers that can be among the written numbers? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 30348,
"solution": ""
} |
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