data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Determine all pairs $(h, s)$ of positive integers with the following property: If one draws $h$ horizontal lines and another $s$ lines which satisfy (i) they are not horizontal, (ii) no two of them are parallel, (iii) no three of the $h+s$ lines are concurrent, then the number ... | math | {
"style": "rule",
"ground_truth": "(995,1),(176,10),(80,21)"
} | {
"split": "train",
"index": 26649,
"solution": "Let $a_{h, s}$ the number of regions formed by $h$ horizontal lines and $s$ another lines as described in the problem. Let $\\mathcal{F}_{h, s}$ be the union of the $h+s$ lines and pick any line $\\ell$. If it intersects the other lines in $n$ (distinct!) points th... |
deepscaler | [
{
"role": "user",
"content": "The arithmetic mean, geometric mean, and harmonic mean of $a$, $b$, $c$ are $8$, $5$, $3$ respectively. What is the value of $a^2+b^2+c^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "326"
} | {
"split": "train",
"index": 26650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many real numbers $x$ satisfy the equation $3^{2x+2}-3^{x+3}-3^x+3=0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 26651,
"solution": "1. **Substitute and Simplify**: Let $a = 3^x$. Then, we can rewrite the given equation $3^{2x+2} - 3^{x+3} - 3^x + 3 = 0$ using the substitution:\n \\[\n 3^{2x+2} = (3^x)^2 \\cdot 3^2 = 9a^2, \\quad 3^{x+3} = 3^x \\cdot 3^3 = 27a, \\quad 3^x = a\n \\]\n Sub... |
deepscaler | [
{
"role": "user",
"content": "Given the set $\\{-25, -4, -1, 3, 5, 9\\}$, find the largest quotient and the smallest product that can be formed using two numbers chosen from the set. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-225"
} | {
"split": "train",
"index": 26652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular hexagon `LMNOPQ` has sides of length 4. Find the area of triangle `LNP`. Express your answer in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{3}"
} | {
"split": "train",
"index": 26653,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\tan x+\\tan y=25$ and $\\cot x + \\cot y=30$, what is $\\tan(x+y)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 26654,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express $0.000 000 04$ in scientific notation. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 \\times 10^{-8}"
} | {
"split": "train",
"index": 26655,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One US dollar is the equivalent of 7 Namibian dollars. One US dollar is also the equivalent of 6 Chinese yuan. If a painting costs 105 Namibian dollars, how much would it cost in Chinese yuan? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 26656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 150 different cards on the table with the numbers $2, 4, 6, \\ldots, 298, 300$ (each card has exactly one number, and each number appears exactly once). In how many ways can you choose 2 cards so that the sum of the numbers on the chosen cards is divisible by $5$? Let... | math | {
"style": "rule",
"ground_truth": "2235"
} | {
"split": "train",
"index": 26657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Choose positive integers $b_1, b_2, \\dotsc$ satisfying\n\\[1=\\frac{b_1}{1^2} > \\frac{b_2}{2^2} > \\frac{b_3}{3^2} > \\frac{b_4}{4^2} > \\dotsb\\]\nand let $r$ denote the largest real number satisfying $\\tfrac{b_n}{n^2} \\geq r$ for all positive integers $n$. What are the po... | math | {
"style": "rule",
"ground_truth": "0 \\leq r \\leq \\frac{1}{2}"
} | {
"split": "train",
"index": 26658,
"solution": "\nLet \\( r \\) denote the largest real number satisfying \\(\\frac{b_n}{n^2} \\geq r\\) for all positive integers \\( n \\), where \\( b_1, b_2, \\dotsc \\) are positive integers satisfying\n\\[\n1 = \\frac{b_1}{1^2} > \\frac{b_2}{2^2} > \\frac{b_3}{3^2} > \\frac{... |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), \\(\\angle A\\) is the smallest angle, \\(\\angle B\\) is the largest angle, and \\(2 \\angle B = 5 \\angle A\\). If the maximum value of \\(\\angle B\\) is \\(m^{\\circ}\\) and the minimum value of \\(\\angle B\\) is \\(n^{\\circ}\\), then f... | math | {
"style": "rule",
"ground_truth": "175"
} | {
"split": "train",
"index": 26659,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that $abc = 216$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{216}"
} | {
"split": "train",
"index": 26660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Zhang has three watches. The first watch runs 2 minutes fast every hour, the second watch runs 6 minutes fast, and the third watch runs 16 minutes fast. If the minute hands of the three watches are currently all pointing in the same direction, after how many hours will the... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 26661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $c=2$ and $A \\neq B$.\n\n1. Find the value of $\\frac{a \\sin A - b \\sin B}{\\sin (A-B)}$.\n2. If the area of $\\triangle ABC$ is $1$ and $\\tan C = 2$, find the value... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5} + 1"
} | {
"split": "train",
"index": 26662,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cone is inscribed in a right rectangular prism as shown. The base of the prism has dimensions, where one side is exactly twice the length of the other ($a$ and $2a$). The cone's base fits perfectly into the base of the prism making one side of the rectangle the... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{24}"
} | {
"split": "train",
"index": 26663,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cylindrical glass is half full of lemonade. The ratio of lemon juice to water in the lemonade is 1:11. If the glass is 6 inches tall and has a diameter of 2 inches, what is the volume of lemon juice in the glass? Express your answer as a decimal to the nearest hundredth. Let'... | math | {
"style": "rule",
"ground_truth": ".79"
} | {
"split": "train",
"index": 26664,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers from 1 to 200, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability it is neither a perfect square, a perfect cube, nor a sixth power? Express your answer as a common fraction. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "\\frac{183}{200}"
} | {
"split": "train",
"index": 26665,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the values of $a$, $b$, and $r$ of the circle given by the equation $x^2 + 14y + 65 = -y^2 - 8x$. Let $a$ and $b$ be the coordinates of the center of the circle, and $r$ be its radius. What is the sum $a + b + r$? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "-11"
} | {
"split": "train",
"index": 26666,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nIn a shooting contest, 8 targets are arranged in two columns with 3 targets and one column with 2 targets. The rules are:\n\n- The shooter can freely choose which column to shoot at.\n- He must attempt the lowest target not yet hit.\n\na) If the shooter ignores the second rul... | math | {
"style": "rule",
"ground_truth": "560"
} | {
"split": "train",
"index": 26667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\cos 72^\\circ.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-1 + \\sqrt{5}}{4}"
} | {
"split": "train",
"index": 26668,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A school organizes a social practice activity during the summer vacation and needs to allocate 8 grade 10 students to company A and company B evenly. Among these students, two students with excellent English scores cannot be allocated to the same company, and neither can the th... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 26669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One hundred concentric circles with radii $1, 2, 3, \\dots, 100$ are drawn in a plane. The interior of the circle of radius $1$ is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio ... | math | {
"style": "rule",
"ground_truth": "301"
} | {
"split": "train",
"index": 26670,
"solution": "We want to find $\\frac{\\sum\\limits_{n=1}^{50} (4n-1)\\pi}{10000\\pi}=\\frac{\\sum\\limits_{n=1}^{50} (4n-1)}{10000}=\\frac{(\\sum\\limits_{n=1}^{50} (4n) )-50}{10000}=\\frac{101}{200} \\rightarrow 101+200=\\boxed{301}$"
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos \\alpha =-\\dfrac{3}{4}, \\sin \\beta =\\dfrac{2}{3}$, with $\\alpha$ in the third quadrant and $\\beta \\in (\\dfrac{\\pi }{2}, \\pi )$.\n\n(I) Find the value of $\\sin 2\\alpha$;\n\n(II) Find the value of $\\cos (2\\alpha + \\beta )$. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "-\\dfrac{\\sqrt{5} + 6\\sqrt{7}}{24}"
} | {
"split": "train",
"index": 26671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During a fireworks display, a body is launched upwards with an initial velocity of $c=90 \\mathrm{m/s}$. We hear its explosion $t=5$ seconds later. At what height did it explode if the speed of sound is $a=340 \\mathrm{m/s}$? (Air resistance is neglected.) Let's think step by s... | math | {
"style": "rule",
"ground_truth": "289"
} | {
"split": "train",
"index": 26672,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(ABCD\\) be a trapezium with \\(AD\\) parallel to \\(BC\\) and \\(\\angle ADC = 90^\\circ\\). Given that \\(M\\) is the midpoint of \\(AB\\) with \\(CM = \\frac{13}{2} \\text{ cm}\\) and \\(BC + CD + DA = 17 \\text{ cm}\\), find the area of the trapezium \\(ABCD\\) in \\(... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 26673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$A B C$ is a triangle with points $E, F$ on sides $A C, A B$, respectively. Suppose that $B E, C F$ intersect at $X$. It is given that $A F / F B=(A E / E C)^{2}$ and that $X$ is the midpoint of $B E$. Find the ratio $C X / X F$. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 26674,
"solution": "Let $x=A E / E C$. By Menelaus's theorem applied to triangle $A B E$ and line $C X F$, $$1=\\frac{A F}{F B} \\cdot \\frac{B X}{X E} \\cdot \\frac{E C}{C A}=\\frac{x^{2}}{x+1}$$ Thus, $x^{2}=x+1$, and $x$ must be positive, so $x=(1+\\sqrt{5}) / 2$. Now apply Menelau... |
deepscaler | [
{
"role": "user",
"content": "There are three candidates standing for one position as student president and 130 students are voting. Sally has 24 votes so far, while Katie has 29 and Alan has 37. How many more votes does Alan need to be certain he will finish with the most votes? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 26675,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\\mathcal{T}.$ Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "2520"
} | {
"split": "train",
"index": 26676,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP=\\frac mn$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.\n Let's think step... | math | {
"style": "rule",
"ground_truth": "164"
} | {
"split": "train",
"index": 26677,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sphere intersects the $xy$-plane in a circle centered at $(2,4,0)$ with radius 1. The sphere also intersects the $yz$-plane in a circle centered at $(0,4,-7),$ with radius $r.$ Find $r.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{46}"
} | {
"split": "train",
"index": 26678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find\n\\[\\sum_{n = 1}^\\infty \\frac{2^n}{1 + 2^n + 2^{n + 1} + 2^{2n + 1}}.\\] 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": 26679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a magic square, the sum of the three entries in any row, column, or diagonal is the same value. The figure shows four of the entries of a magic square. Find $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 26680,
"solution": "Use the table from above. Obviously $c = 114$. Hence $a+e = 115$. Similarly, $1+a = 96 + e \\Rightarrow a = 95+e$.\n Substitute that into the first to get $2e = 20 \\Rightarrow e=10$, so $a=105$, and so the value of $x$ is just $115+x = 210 + 105 \\Rightarrow x = \... |
deepscaler | [
{
"role": "user",
"content": "Suppose $x$ and $y$ are positive real numbers such that $x^2 - 3xy + 4y^2 = 12$. Find the maximum possible value of $x^2 + 3xy + 4y^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 26681,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There exist unique nonnegative integers $A, B$ between 0 and 9, inclusive, such that $(1001 \\cdot A+110 \\cdot B)^{2}=57,108,249$. Find $10 \\cdot A+B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 26682,
"solution": "We only need to bound for $A B 00$; in other words, $A B^{2} \\leq 5710$ but $(A B+1)^{2} \\geq 5710$. A quick check gives $A B=75$. (Lots of ways to get this...)"
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with its upper vertex at $(0,2)$ and an eccentricity of $\\frac{\\sqrt{5}}{3}$.\n(1) Find the equation of ellipse $C$;\n(2) From a point $P$ on the ellipse $C$, draw two tangent lines to the circle... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 26683,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first and thirteenth terms of an arithmetic sequence are $\\frac79$ and $\\frac45$, respectively. What is the seventh term? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{71}{90}"
} | {
"split": "train",
"index": 26684,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Complex numbers $p$, $q$, and $r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48$, find $|pq + pr + qr|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "768"
} | {
"split": "train",
"index": 26685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Given $\\tan(\\alpha+\\beta)= \\frac{2}{5}$ and $\\tan\\left(\\beta- \\frac{\\pi}{4}\\right)= \\frac{1}{4}$, find the value of $\\frac{\\cos\\alpha+\\sin\\alpha}{\\cos\\alpha-\\sin\\alpha}$; \n(2) Given $\\alpha$ and $\\beta$ are acute angles, and $\\cos(\\alpha+\\beta)= \... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 26686,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular tile measures 3 inches by 4 inches. What is the fewest number of these tiles that are needed to completely cover a rectangular region that is 2 feet by 5 feet? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 26687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two years ago there were 20 trailer homes on Elm Street with an average age of 18 years. At that time, a group of brand new trailer homes was then added to Elm Street. Today, the average age of all the trailer homes on Elm Street is 14 years. How many new trailer homes were add... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 26688,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the $xy$-plane, the segment with endpoints $(-5,0)$ and $(25,0)$ is the diameter of a circle. If the point $(x,15)$ is on the circle, then $x=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 26689,
"solution": "1. **Identify the center and radius of the circle**: \n The endpoints of the diameter of the circle are given as $(-5,0)$ and $(25,0)$. The center of the circle, $C$, is the midpoint of the diameter. Using the midpoint formula:\n \\[\n C = \\left(\\frac{-5 + ... |
deepscaler | [
{
"role": "user",
"content": "In a circle of radius $42$, two chords of length $78$ intersect at a point whose distance from the center is $18$. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lengths, and the area of either of the... | math | {
"style": "rule",
"ground_truth": "378"
} | {
"split": "train",
"index": 26690,
"solution": "Let the center of the circle be $O$, and the two chords be $\\overline{AB}, \\overline{CD}$ and intersecting at $E$, such that $AE = CE < BE = DE$. Let $F$ be the midpoint of $\\overline{AB}$. Then $\\overline{OF} \\perp \\overline{AB}$.\n[asy] size(200); pathpen =... |
deepscaler | [
{
"role": "user",
"content": "In $\\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $b=2$, $c=2\\sqrt{2}$, and $C=\\frac{\\pi}{4}$, find the area of $\\Delta ABC$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} +1"
} | {
"split": "train",
"index": 26691,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that 0.5 liters is approximately 1.05 pints, how many pints are in one liter? Express your answer as a decimal to the nearest tenth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2.1"
} | {
"split": "train",
"index": 26692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$ is the smallest positive two-digit multiple of $3$, and $b$ is the smallest positive three-digit multiple of $4$, then find $a+b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 26693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A math field day competition is held in a room with many tables, and there are 6 stools at each table. Each stool has 3 legs, and each table has 4 legs. If there is a total of 484 legs on all the tables and stools in the room, how many tables are in the room? Let's think step b... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 26694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 3 boys and 4 girls, all lined up in a row. How many ways are there for the following situations?\n- $(1)$ Person A is neither at the middle nor at the ends;\n- $(2)$ Persons A and B must be at the two ends;\n- $(3)$ Boys and girls alternate. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 26695,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the year 2009, there is a property that rearranging the digits of the number 2009 cannot yield a smaller four-digit number (numbers do not start with zero). In what subsequent year does this property first repeat again? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "2022"
} | {
"split": "train",
"index": 26696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute \\[\\lfloor \\sqrt{1} \\rfloor + \\lfloor \\sqrt{2} \\rfloor + \\lfloor \\sqrt{3} \\rfloor + \\cdots + \\lfloor \\sqrt{25} \\rfloor.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 26697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\(\\triangle ABC\\) has sides \\(a\\), \\(b\\), and \\(c\\) corresponding to angles \\(A\\), \\(B\\), and \\(C\\) respectively, and knowing that \\(a + b + c = 16\\), find the value of \\(b^2 \\cos^2 \\frac{C}{2} + c^2 \\cos^2 \\frac{B}{2} + 2bc \\cos \\frac{B}{2} \... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 26698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \\) be nonnegative real numbers whose sum is 300. Let \\( M \\) be the maximum of the four numbers \\( x_{1} + x_{2}, x_{2} + x_{3}, x_{3} + x_{4}, \\) and \\( x_{4} + x_{5} \\). Find the least possible value of \\( M \\). Let's think s... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 26699,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=4^{x}-6\\times2^{x}+8$, find the minimum value of the function and the value of $x$ when the minimum value is obtained. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 26700,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α∈( \\dfrac {π}{2},π)$, and $\\sin \\dfrac {α}{2}+\\cos \\dfrac {α}{2}= \\dfrac {2 \\sqrt {3}}{3}$.\n(1) Find the values of $\\sin α$ and $\\cos α$;\n(2) If $\\sin (α+β)=- \\dfrac {3}{5},β∈(0, \\dfrac {π}{2})$, find the value of $\\sin β$. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "\\dfrac {6 \\sqrt {2}+4}{15}"
} | {
"split": "train",
"index": 26701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function \\( y = y_1 + y_2 \\), where \\( y_1 \\) is directly proportional to \\( x^2 \\) and \\( y_2 \\) is inversely proportional to \\( x^2 \\). When \\( x = 1 \\), \\( y = 5 \\); when \\( x = \\sqrt{3} \\), \\( y = 7 \\). Find the value of \\( x \\) when \\( y \\)... | math | {
"style": "rule",
"ground_truth": "\\sqrt[4]{\\frac{3}{2}}"
} | {
"split": "train",
"index": 26702,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, where $a_n \\in \\mathbb{N}^*$, and $S_n = \\frac{1}{8}(a_n+2)^2$. If $b_n = \\frac{1}{2}a_n - 30$, find the minimum value of the sum of the first \\_\\_\\_\\_\\_\\_ terms of the sequence $\\{b_n\\}$. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 26703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "152"
} | {
"split": "train",
"index": 26704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$, $b$, and $c$ be positive integers with $a \\ge b \\ge c$ such that\n$a^2-b^2-c^2+ab=2011$ and\n$a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$.\nWhat is $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "253"
} | {
"split": "train",
"index": 26705,
"solution": "1. **Combine the given equations:**\n \\[\n \\begin{align*}\n a^2 - b^2 - c^2 + ab &= 2011, \\\\\n a^2 + 3b^2 + 3c^2 - 3ab - 2ac - 2bc &= -1997.\n \\end{align*}\n \\]\n Adding these equations, we get:\n \\[\n 2a^2 + 2b^2 + 2c^2 - 4ab - 2ac - 2bc =... |
deepscaler | [
{
"role": "user",
"content": "If point \\( P \\) is the circumcenter of \\(\\triangle ABC\\) and \\(\\overrightarrow{PA} + \\overrightarrow{PB} + \\lambda \\overrightarrow{PC} = \\mathbf{0}\\), where \\(\\angle C = 120^\\circ\\), then find the value of the real number \\(\\lambda\\). Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 26706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\frac{6! + 7!}{5!}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 26707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute without using a calculator: $42!/40!$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1,\\!722"
} | {
"split": "train",
"index": 26708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(This question has a total of 10 points)\nFrom a group consisting of 5 male doctors and 4 female doctors, select 3 doctors to form a medical team. The requirement is that the team must include both male and female doctors. How many different team formation plans are there? Let'... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 26709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the function $f_1$ on the positive integers by setting $f_1(1)=1$ and if $n=p_1^{e_1}p_2^{e_2}\\cdots p_k^{e_k}$ is the prime factorization of $n>1$, then\n\\[f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\\cdots (p_k+1)^{e_k-1}.\\]\nFor every $m\\ge 2$, let $f_m(n)=f_1(f_{m-1}(n... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 26710,
"solution": "1. **Define Interesting and Boring Numbers**: A number $n$ is termed as *interesting* if the sequence $\\{f_k(n)\\}$ is unbounded. Conversely, it is *boring* if the sequence is bounded.\n\n2. **Reduction to Prime Factorization**: For a prime $p$, $f_1(p) = 1$. If $... |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse C₁: $$\\frac {x^{2}}{a^{2}}$$+ $$\\frac {y^{2}}{b^{2}}$$\\=1 (a＞b＞0) with one focus coinciding with the focus of the parabola C₂: y<sup>2</sup>\\=4 $$\\sqrt {2}$$x, and the eccentricity of the ellipse is e= $$\\frac { \\sqrt {6}}{3}$$.\n(I) Find the equation o... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 26711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $ABCD$ has area $200$. Point $E$ lies on side $\\overline{BC}$. Points $F$ and $G$ are the midpoints of $\\overline{AE}$ and $\\overline{DE}$, respectively. Given that quadrilateral $BEGF$ has area $34$, what is the area of triangle $GCD$? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 26712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine how many integers are in the list when Eleanor writes down the smallest positive multiple of 15 that is a perfect square, the smallest positive multiple of 15 that is a perfect cube, and all the multiples of 15 between them. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "211"
} | {
"split": "train",
"index": 26713,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the local frisbee league, teams have 7 members and each of the 4 teams takes turns hosting tournaments. At each tournament, each team selects two members of that team to be on the tournament committee, except the host team, which selects three members. How many possible 9 me... | math | {
"style": "rule",
"ground_truth": "1,\\!296,\\!540"
} | {
"split": "train",
"index": 26714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In square \\( A B C D \\), \\( P \\) and \\( Q \\) are points on sides \\( C D \\) and \\( B C \\), respectively, such that \\( \\angle A P Q = 90^\\circ \\). If \\( A P = 4 \\) and \\( P Q = 3 \\), find the area of \\( A B C D \\). Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "\\frac{256}{17}"
} | {
"split": "train",
"index": 26715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Megan and Hana raced their remote control cars for 100 m. The two cars started at the same time. The average speed of Megan's car was $\\frac{5}{4} \\mathrm{~m} / \\mathrm{s}$. Hana's car finished 5 seconds before Megan's car. What was the average speed of Hana's car? Let's thi... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3} \\mathrm{~m} / \\mathrm{s}"
} | {
"split": "train",
"index": 26716,
"solution": "Megan's car travels 100 m at $\\frac{5}{4} \\mathrm{~m} / \\mathrm{s}$, and so takes $\\frac{100 \\mathrm{~m}}{5 / 4 \\mathrm{~m} / \\mathrm{s}}=\\frac{400}{5} \\mathrm{~s}=80 \\mathrm{~s}$. Hana's car completes the 100 m in 5 s fewer, and so takes 75 s. Thus, the ... |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC,$ we have $\\angle C = 3\\angle A,$ $a = 27,$ and $c = 48.$ What is $b$?\n\nNote: $a$ is the side length opposite $\\angle A,$ etc. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 26717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five different awards are to be given to three students. Each student will receive at least one award. In how many different ways can the awards be distributed? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 26718,
"solution": "1. **Identify the possible distributions of awards:** Since each student must receive at least one award and there are five awards in total, the possible distributions of awards among three students are:\n - One student receives 3 awards, and the other two studen... |
deepscaler | [
{
"role": "user",
"content": "Consider a polynomial $P(x) \\in \\mathbb{R}[x]$ , with degree $2023$ , such that $P(\\sin^2(x))+P(\\cos^2(x)) =1$ for all $x \\in \\mathbb{R}$ . If the sum of all roots of $P$ is equal to $\\dfrac{p}{q}$ with $p, q$ coprime, then what is the product $pq$ ? Let's th... | math | {
"style": "rule",
"ground_truth": "4046"
} | {
"split": "train",
"index": 26719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the right circular cone $P-ABC$, $PA \\perp$ plane $ABC$, $AC \\perp AB$, $PA=AB=2$, $AC=1$. Find the volume of the circumscribed sphere of the cone $P-ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}\\pi"
} | {
"split": "train",
"index": 26720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\\overline{AB}.$ What is the area inside circle $C$ but outside circle $A$ and circle $B?$ Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 26721,
"solution": "1. **Identify the Configuration**: Circles $A$, $B$, and $C$ each have a radius of 1. Circles $A$ and $B$ are tangent to each other, and circle $C$ is tangent to the midpoint $M$ of line segment $\\overline{AB}$. \n\n2. **Positioning of Circle $C$**: Since circle $... |
deepscaler | [
{
"role": "user",
"content": "Find $ 8^8 \\cdot 4^4 \\div 2^{28}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 26722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum area of the part bounded by the parabola $ y\\equal{}a^3x^2\\minus{}a^4x\\ (a>0)$ and the line $ y\\equal{}x$ . 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": 26723,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a monic irreducible polynomial with integral coefficients to be a polynomial with leading coefficient 1 that cannot be factored, and the prime factorization of a polynomial with leading coefficient 1 as the factorization into monic irreducible polynomials. How many not n... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 26724,
"solution": "$x^{8}+x^{4}+1=\\left(x^{8}+2 x^{4}+1\\right)-x^{4}=\\left(x^{4}+1\\right)^{2}-\\left(x^{2}\\right)^{2}=\\left(x^{4}-x^{2}+1\\right)\\left(x^{4}+x^{2}+1\\right)=$ $\\left(x^{4}-x^{2}+1\\right)\\left(x^{2}+x+1\\right)\\left(x^{2}-x+1\\right)$, and $x^{8}+x+1=\\left(... |
deepscaler | [
{
"role": "user",
"content": "Determine the value of $x$ that satisfies $\\sqrt[4]{x\\sqrt{x^4}}=4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2^{8/3}"
} | {
"split": "train",
"index": 26725,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of ways to arrange the numbers 1 through 7 in a circle such that the numbers are increasing along each arc from 1. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 26726,
"solution": "First, we can fix the position of the 1. Then, by the condition that the numbers are increasing along each arc from 1, we know that the 2 must be adjacent to the 1; so we have two options for its placement. Similarly, we have two options for placing each of $3,4,5,... |
deepscaler | [
{
"role": "user",
"content": "Multiply $(2x^3-5y^2)(4x^6+10x^3y^2+25y^4)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8x^9-125y^6"
} | {
"split": "train",
"index": 26727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a, b \\in \\mathbb{R}$. If the line $l: ax+y-7=0$ is transformed by the matrix $A= \\begin{bmatrix} 3 & 0 \\\\ -1 & b\\end{bmatrix}$, and the resulting line is $l′: 9x+y-91=0$. Find the values of the real numbers $a$ and $b$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 26728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\[\n\\frac{\\left(\\left(4.625 - \\frac{13}{18} \\cdot \\frac{9}{26}\\right) : \\frac{9}{4} + 2.5 : 1.25 : 6.75\\right) : 1 \\frac{53}{68}}{\\left(\\frac{1}{2} - 0.375\\right) : 0.125 + \\left(\\frac{5}{6} - \\frac{7}{12}\\right) : (0.358 - 1.4796 : 13.7)}\n\\]\n Let's think s... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{27}"
} | {
"split": "train",
"index": 26729,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$.\n($d(n)$ is the number of divisors of the number n including $1$ and $n$ ). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 26730,
"solution": "\nWe are tasked with finding all odd natural numbers \\( n \\) such that \\( d(n) \\), the number of divisors of \\( n \\), is the largest divisor of \\( n \\) different from \\( n \\) itself. Recall that \\( d(n) \\) counts all divisors of \\( n \\), including \\(... |
deepscaler | [
{
"role": "user",
"content": "A trapezoid $ABCD$ has bases $AD$ and $BC$. If $BC = 60$ units, and altitudes from $B$ and $C$ to line $AD$ divide it into segments of lengths $AP = 20$ units and $DQ = 19$ units, with the length of the altitude itself being $30$ units, what is the perimeter of trapezoid $ABCD$... | math | {
"style": "rule",
"ground_truth": "\\sqrt{1300} + \\sqrt{1261} + 159"
} | {
"split": "train",
"index": 26731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let (a,b,c,d) be an ordered quadruple of not necessarily distinct integers, each one of them in the set {0,1,2,3,4}. Determine the number of such quadruples that make the expression $a \\cdot d - b \\cdot c + 1$ even. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "136"
} | {
"split": "train",
"index": 26732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $a$ and $b \\gt 0$, if $a+2b=1$, then the minimum value of $\\frac{3}{b}+\\frac{1}{a}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7 + 2\\sqrt{6}"
} | {
"split": "train",
"index": 26733,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $\\cos A= \\frac{\\sqrt{3}}{3}$, $c=\\sqrt{3}$, and $a=3\\sqrt{2}$. Find the value of $\\sin C$ and the area of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 26734,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence starts at 2,187,000 and each subsequent number is created by dividing the previous number by 3. What is the last integer in this sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 26735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $l$ is the incenter of $\\triangle ABC$, with $AC=2$, $BC=3$, and $AB=4$. If $\\overrightarrow{AI}=x \\overrightarrow{AB}+y \\overrightarrow{AC}$, then $x+y=$ ______. 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": 26736,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For real numbers \\( x \\) and \\( y \\), define the operation \\( \\star \\) as follows: \\( x \\star y = xy + 4y - 3x \\).\n\nCompute the value of the expression\n\n$$\n((\\ldots)(((2022 \\star 2021) \\star 2020) \\star 2019) \\star \\ldots) \\star 2) \\star 1\n$$ Let's think... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 26737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two arithmetic sequences $\\{a\\_n\\}$ and $\\{b\\_n\\}$ with respective sums of the first $n$ terms $S_n$ and $T_n$, if $\\frac{S_n}{T_n} = \\frac{2n-3}{4n-3}$ holds for any natural number $n$, find the value of $\\frac{a_9}{b_5+b_7} + \\frac{a_3}{b_8+b_4}$. Let's think ... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{41}"
} | {
"split": "train",
"index": 26738,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A hockey team has 6 more red helmets than blue helmets. The ratio of red helmets to blue helmets is $5:3$. What is the total number of red helmets and blue helmets? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 26739,
"solution": "Suppose that the team has $r$ red helmets. Since the team has 6 more red helmets than blue helmets, then the team has $r-6$ blue helmets. Since the ratio of the number of red helmets to the number of blue helmets is $5:3$, then $\\frac{r}{r-6}=\\frac{5}{3}$ and so ... |
deepscaler | [
{
"role": "user",
"content": "Given two curves $y=x^{2}-1$ and $y=1-x^{3}$ have parallel tangents at point $x_{0}$, find the value of $x_{0}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{2}{3}"
} | {
"split": "train",
"index": 26740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a finite sequence \\(P = \\left(p_1, p_2, \\cdots, p_n\\right)\\), the Cesaro sum (named after the mathematician Cesaro) is defined as \\(\\frac{1}{n}(S_1 + S_2 + \\cdots + S_n)\\), where \\(S_k = p_1 + p_2 + \\cdots + p_k\\) for \\(1 \\leq k \\leq n\\). If a sequence \\(\\... | math | {
"style": "rule",
"ground_truth": "991"
} | {
"split": "train",
"index": 26741,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ be a polynomial such that\n\\[f(x^2 + 1) = x^4 + 4x^2.\\]Find $f(x^2 - 1).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^4 - 4"
} | {
"split": "train",
"index": 26742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer with exactly 12 positive integer divisors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 26743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two numbers are independently selected from the set of positive integers less than or equal to 7. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{36}{49}"
} | {
"split": "train",
"index": 26744,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let's consider the sum of five consecutive odd numbers starting from 997 up to 1005. If the sum $997 + 999 + 1001 + 1003 + 1005 = 5100 - M$, find $M$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "95"
} | {
"split": "train",
"index": 26745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that for any positive integer \\( n \\), \\( 9^{2n} - 8^{2n} - 17 \\) is always divisible by \\( m \\), find the largest positive integer \\( m \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2448"
} | {
"split": "train",
"index": 26746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If 20$\\%$ of 10$\\%$ of a number is 12, what is 10$\\%$ of 20$\\%$ of the same number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 26747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On one side of the acute angle \\(A\\), points \\(P\\) and \\(Q\\) are marked such that \\(AP = 4\\), \\(AQ = 12\\). On the other side, points \\(M\\) and \\(N\\) are marked at distances of 6 and 10 from the vertex. Find the ratio of the areas of triangles \\(MNO\\) and \\(PQO\... | math | {
"style": "rule",
"ground_truth": "1:5"
} | {
"split": "train",
"index": 26748,
"solution": ""
} |
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