data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Find all triples $(x, y, z)$ of nonnegative integers such that\n$$ x^5+x^4+1=3^y7^z $$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0, 0, 0), (1, 1, 0), (2, 0, 2)"
} | {
"split": "train",
"index": 31649,
"solution": "\nGiven the equation:\n\n\\[\nx^5 + x^4 + 1 = 3^y 7^z\n\\]\n\nwe are tasked with finding all triples \\((x, y, z)\\) of nonnegative integers that satisfy this equation.\n\n### Step 1: Analyze Small Values of \\(x\\)\n\nStart by checking small values of \\(x\\) to f... |
deepscaler | [
{
"role": "user",
"content": "Mathematician Wiener, the founder of cybernetics, was asked about his age during his Ph.D. awarding ceremony at Harvard University because he looked very young. Wiener's interesting response was: \"The cube of my age is a four-digit number, and the fourth power of my age is a s... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 31650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the right triangle shown the sum of the distances $BM$ and $MA$ is equal to the sum of the distances $BC$ and $CA$. \nIf $MB = x, CB = h$, and $CA = d$, then $x$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{hd}{2h+d}"
} | {
"split": "train",
"index": 31651,
"solution": "1. **Given Information and Equation Setup:** \n We are given that the sum of the distances $BM$ and $MA$ is equal to the sum of the distances $BC$ and $CA$. This translates to the equation:\n \\[\n BM + MA = BC + CA\n \\]\n Substituting the given values,... |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, point $G$ satisfies $\\overrightarrow{GA}+\\overrightarrow{GB}+\\overrightarrow{GC}=\\overrightarrow{0}$. The line passing through $G$ intersects $AB$ and $AC$ at points $M$ and $N$ respectively. If $\\overrightarrow{AM}=m\\overrightarrow{AB}$ $(m>... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}+\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 31652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many rectangles are there whose four vertices are points on this grid? [asy]\nsize(50);\ndot((0,0));\ndot((5,0));\ndot((10,0));\ndot((0,5));\ndot((0,10));\ndot((5,5));\ndot((5,10));\ndot((10,5));\ndot((10,10));\n[/asy] Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 31653,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( X = \\{1, 2, \\ldots, 2001\\} \\). Find the smallest positive integer \\( m \\) such that in any \\( m \\)-element subset \\( W \\) of \\( X \\), there exist \\( u, v \\in W \\) (where \\( u \\) and \\( v \\) are allowed to be the same) such that \\( u + v \\) is a powe... | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 31654,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose\n$$a(2+i)^4 + b(2+i)^3 + c(2+i)^2 + b(2+i) + a = 0,$$where $a,b,c$ are integers whose greatest common divisor is $1$. Determine $|c|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 31655,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the coefficients in the polynomial $3(3x^{7} + 8x^4 - 7) + 7(x^5 - 7x^2 + 5)$ when it is fully simplified. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 31656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tsrutsuna starts in the bottom left cell of a 7 × 7 square table, while Tsuna is in the upper right cell. The center cell of the table contains cheese. Tsrutsuna wants to reach Tsuna and bring a piece of cheese with him. From a cell Tsrutsuna can only move to the right or the t... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 31657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For all positive integers $n$ greater than 2, the greatest common divisor of $n^5 - 5n^3 + 4n$ is. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 31658,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), \\(BK\\) is the median, \\(BE\\) is the angle bisector, and \\(AD\\) is the altitude. Find the length of side \\(AC\\) if it is known that lines \\(BK\\) and \\(BE\\) divide segment \\(AD\\) into three equal parts and the length of \\(AB\\) is 4. Let's th... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 31659,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A larger grid is considered for the next challenge, where each segment must still only be traversed in a rightward or downward direction. Starting from point $A$, located at the top-left of a 3x3 grid, to point $B$ at the bottom-right. How many different routes can be taken? Le... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 31660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\angle1+\\angle2=180^\\circ$ and $\\angle3=\\angle4,$ find $\\angle4.$ Express your answer in degrees. [asy]\n/* AMC8 1997 #12 Problem */\npair A=(0,0), B=(24,0), C=(48,0), D=(18,24), E=(12,48);\npen p=1mm+black;\ndraw(A--C);\ndraw(A--E);\ndraw(B--E);\ndraw(D--C);\nlabe... | math | {
"style": "rule",
"ground_truth": "35^\\circ"
} | {
"split": "train",
"index": 31661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given six people are arranged in a row from left to right. Only A or B can be placed at the far left, and A cannot be placed at the far right, calculate the total number of different arrangements. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 31662,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the degree measure of an interior angle of a regular pentagon? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108\\text{ degrees}"
} | {
"split": "train",
"index": 31663,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right rectangular prism $Q$ has integral side lengths $a, b, c$ with $a \\le b \\le c$. A plane parallel to one of the faces of $Q$ cuts $Q$ into two prisms, one of which is similar to $Q$, with both having nonzero volumes. The middle side length $b = 3969$. Determine the num... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 31664,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "After the implementation of the \"double reduction\" policy, schools have attached importance to extended services and increased the intensity of sports activities in these services. A sports equipment store seized the opportunity and planned to purchase 300 sets of table tenni... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 31665,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $Q(x)=x^{2}+2x+3$, and suppose that $P(x)$ is a polynomial such that $P(Q(x))=x^{6}+6x^{5}+18x^{4}+32x^{3}+35x^{2}+22x+8$. Compute $P(2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 31666,
"solution": "Note that $Q(-1)=2$. Therefore, $P(2)=P(Q(-1))=1-6+18-32+35-22+8=2$."
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\frac{x^{2}}{25} + \\frac{y^{2}}{9} = 1$, a line $L$ passing through the right focus $F$ of the ellipse intersects the ellipse at points $A$ and $B$, and intersects the $y$-axis at point $P$. Suppose $\\overrightarrow{PA} = λ_{1} \\overrightarrow{AF}$ and $\... | math | {
"style": "rule",
"ground_truth": "-\\frac{50}{9}"
} | {
"split": "train",
"index": 31667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I have chosen five of the numbers $\\{1,2,3,4,5,6,7\\}$. If I told you what their product was, that would not be enough information for you to figure out whether their sum was even or odd. What is their product? Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "420"
} | {
"split": "train",
"index": 31668,
"solution": "Giving you the product of the five numbers is equivalent to telling you the product of the two numbers I didn't choose. The only possible products that are achieved by more than one pair of numbers are $12(\\{3,4\\}$ and $\\{2,6\\})$ and $6(\\{1,6\\}$ and $\\{2,3\\... |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), angle \\(C\\) equals \\(30^\\circ\\), and angle \\(A\\) is acute. A line perpendicular to side \\(BC\\) is drawn, cutting off triangle \\(CNM\\) from triangle \\(ABC\\) (point \\(N\\) lies between vertices \\(B\\) and \\(C\\)). The areas of triangles \\(C... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 31669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the interval [1, 6], three different integers are randomly selected. The probability that these three numbers are the side lengths of an obtuse triangle is ___. 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": 31670,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( p \\) and \\( q \\) are prime numbers, the number of divisors \\( d(a) \\) of a natural number \\( a = p^{\\alpha} q^{\\beta} \\) is given by the formula\n$$\nd(a) = (\\alpha+1)(\\beta+1).\n$$\n\nFor example, \\( 12 = 2^2 \\times 3^1 \\), the number of divisors of 12 is\... | math | {
"style": "rule",
"ground_truth": "450"
} | {
"split": "train",
"index": 31671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When studying the operation of a new type of cyclic thermal engine, it was found that during part of the period it receives heat, and the absolute power of heat supply is expressed by the law: \n\\[ P_{1}(t)=P_{0} \\frac{\\sin (\\omega t)}{100+\\sin (t^{2})}, \\quad 0<t<\\frac{... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 31672,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the average of six data points $a_1, a_2, a_3, a_4, a_5, a_6$ is $\\bar{x}$, and the variance is 0.20, what is the variance of the seven data points $a_1, a_2, a_3, a_4, a_5, a_6, \\bar{x}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{35}"
} | {
"split": "train",
"index": 31673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let triangle $ABC$ with incenter $I$ satisfy $AB = 10$ , $BC = 21$ , and $CA = 17$ . Points $D$ and E lie on side $BC$ such that $BD = 4$ , $DE = 6$ , and $EC = 11$ . The circumcircles of triangles $BIE$ and $CID$ meet again at point $P$ , and line $IP$ m... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 31674,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lateral sides of a right trapezoid are 10 and 8. The diagonal of the trapezoid, drawn from the vertex of the acute angle, bisects this angle. Find the area of the trapezoid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "104"
} | {
"split": "train",
"index": 31675,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A snail crawls from one tree to another. In half a day, it covered \\( l_{1}=5 \\) meters. Then, it got tired of this and turned back, crawling \\( l_{2}=4 \\) meters. It got tired and fell asleep. The next day, the same process repeats. The distance between the trees is \\( s=... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 31676,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular hexagon \\( K L M N O P \\) is inscribed in an equilateral triangle \\( A B C \\) such that the points \\( K, M, O \\) lie at the midpoints of the sides \\( A B, B C, \\) and \\( A C \\), respectively. Calculate the area of the hexagon \\( K L M N O P \\) given that t... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 31677,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If three lines from the family of lines given by \\( C: x \\cos t + (y + 1) \\sin t = 2 \\) enclose an equilateral triangle \\( D \\), what is the area of the region \\( D \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12\\sqrt{3}"
} | {
"split": "train",
"index": 31678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin(\\theta + 3\\pi) = -\\frac{2}{3}$, find the value of $\\frac{\\tan(-5\\pi - \\theta) \\cdot \\cos(\\theta - 2\\pi) \\cdot \\sin(-3\\pi - \\theta)}{\\tan(\\frac{7\\pi}{2} + \\theta) \\cdot \\sin(-4\\pi + \\theta) \\cdot \\cot(-\\theta - \\frac{\\pi}{2})} + 2 \\... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 31679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, lines $CF$ and $AD$ are drawn such that $\\dfrac{CD}{DB}=\\dfrac{2}{3}$ and $\\dfrac{AF}{FB}=\\dfrac{1}{3}$. Let $s = \\dfrac{CQ}{QF}$ where $Q$ is the intersection point of $CF$ and $AD$. Find $s$.\n[asy]\nsize(8cm); \npair A = (0, 0), B = (9, 0), C = (3, ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 31680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two positive integers are written on the blackboard, one being 2002 and the other being a number less than 2002. If the arithmetic mean of the two numbers is an integer $m$, then the following operation can be performed: one of the numbers is erased and replaced by $m$. What is... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 31681,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During the National Day holiday, a fruit company organized 20 cars to transport three types of fruits, $A$, $B$, and $C$, totaling 120 tons for sale in other places. It is required that all 20 cars be fully loaded, each car can only transport the same type of fruit, and each ty... | math | {
"style": "rule",
"ground_truth": "198900"
} | {
"split": "train",
"index": 31682,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\triangle DEF$ is inscribed inside $\\triangle ABC$ such that $D,E,F$ lie on $BC, AC, AB$, respectively. The circumcircles of $\\triangle DEC, \\triangle BFD, \\triangle AFE$ have centers $O_1,O_2,O_3$, respectively. Also, $AB = 23, BC = 25, AC=24$, and $\\stackrel{\\frown}{B... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 31683,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a conference with 35 businessmen, 18 businessmen drank coffee, 15 businessmen drank tea, and 8 businessmen drank juice. Six businessmen drank both coffee and tea, four drank both tea and juice, and three drank both coffee and juice. Two businessmen drank all three beverages.... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 31684,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\\frac{c}{2} = b - a\\cos C$,\n(1) Determine the measure of angle $A$.\n(2) If $a=\\sqrt{15}$ and $b=4$, find the length of side $c$. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "2 - \\sqrt{3}"
} | {
"split": "train",
"index": 31685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought $400$ jelly beans, and when she finished, she had... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 31686,
"solution": "1. **Define Variables:**\n Let $x$ be the number of girls in the class. Then, the number of boys in the class is $x + 2$.\n\n2. **Set Up the Equation:**\n Mrs. Wonderful gave each girl $x$ jelly beans and each boy $x + 2$ jelly beans. The total number of jelly ... |
deepscaler | [
{
"role": "user",
"content": " $ABCD$ is a rectangle with $AB = CD = 2$ . A circle centered at $O$ is tangent to $BC$ , $CD$ , and $AD$ (and hence has radius $1$ ). Another circle, centered at $P$ , is tangent to circle $O$ at point $T$ and is also tangent to $AB$ and $BC$ . If line $AT$ ... | math | {
"style": "rule",
"ground_truth": "3 - 2\\sqrt{2}"
} | {
"split": "train",
"index": 31687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers from 1 to 16500 \n\na) are not divisible by 5;\n\nb) are not divisible by either 5 or 3;\n\nc) are not divisible by either 5, 3, or 11? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8000"
} | {
"split": "train",
"index": 31688,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rodney is now guessing a secret number based on these clues:\n- It is a two-digit integer.\n- The tens digit is even.\n- The units digit is odd.\n- The number is greater than 50. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10}"
} | {
"split": "train",
"index": 31689,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\sqrt{12 +\\!\\sqrt{12 + \\!\\sqrt{12 + \\!\\sqrt{12 + \\cdots}}}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 31690,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, $\\cos C=\\frac{2}{3}$, $AC=4$, $BC=3$, calculate the value of $\\tan B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{5}"
} | {
"split": "train",
"index": 31691,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the function defined piecewise by\n\\[ f(x) = \\left\\{ \\begin{aligned} \n2x + 1 & \\quad \\text{if } x < 1 \\\\\nx^2 & \\quad \\text{if } x \\ge 1 \n\\end{aligned} \\right. \\]\nDetermine the value of \\( f^{-1}(-3) + f^{-1}(-1) + f^{-1}(1) + f^{-1}(3) + f^{-1}(9) \\... | math | {
"style": "rule",
"ground_truth": "1 + \\sqrt{3}"
} | {
"split": "train",
"index": 31692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC,$ $b = 5,$ $c = 4,$ and $\\cos (B - C) = \\frac{31}{32}.$ Find $a.$\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": "6"
} | {
"split": "train",
"index": 31693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $$\\sum_{n=1}^{\\infty} \\frac{2 n+5}{2^{n} \\cdot\\left(n^{3}+7 n^{2}+14 n+8\\right)}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{137}{24}-8 \\ln 2"
} | {
"split": "train",
"index": 31694,
"solution": "First, we manipulate using partial fractions and telescoping: $$\\begin{aligned} \\sum_{n=1}^{\\infty} \\frac{2 n+5}{2^{n} \\cdot\\left(n^{3}+7 n^{2}+14 n+8\\right)} & =\\frac{1}{2} \\cdot \\sum_{n=1}^{\\infty} \\frac{1}{2^{n}}\\left(\\frac{2}{n+1}-\\frac{1}{n+2}-\... |
deepscaler | [
{
"role": "user",
"content": "Two people, A and B, visit the \"2011 Xi'an World Horticultural Expo\" together. They agree to independently choose 4 attractions from numbered attractions 1 to 6 to visit, spending 1 hour at each attraction. Calculate the probability that they will be at the same attraction du... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{6}"
} | {
"split": "train",
"index": 31695,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A dump truck delivered sand to a construction site. The sand formed a conical pile with a diameter of $8$ feet and a height that was $75\\%$ of the diameter. How many cubic feet of sand were in the pile? Express your answer in terms of $\\pi$. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "32 \\pi"
} | {
"split": "train",
"index": 31696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The edge of cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is equal to 1. Construct a cross-section of the cube by a plane that has the maximum perimeter. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"split": "train",
"index": 31697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a scalene triangle, the lengths of the medians $A N$ and $B P$ are 3 and 6, respectively, and the area is $3 \\sqrt{15}$. The length of the third median $C M$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\sqrt{6}"
} | {
"split": "train",
"index": 31698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a pair $ A \\equal{} (x_1, y_1)$ and $ B \\equal{} (x_2, y_2)$ of points on the coordinate plane, let $ d(A,B) \\equal{} |x_1 \\minus{} x_2| \\plus{} |y_1 \\minus{} y_2|$. We call a pair $ (A,B)$ of (unordered) points [i]harmonic[/i] if $ 1 < d(A,B) \\leq 2$. Determine the ... | math | {
"style": "rule",
"ground_truth": "3750"
} | {
"split": "train",
"index": 31699,
"solution": "\nGiven a set of 100 points in the plane, we want to determine the maximum number of harmonic pairs, where a pair \\((A, B)\\) of points is considered harmonic if \\(1 < d(A, B) \\leq 2\\) and \\(d(A, B) = |x_1 - x_2| + |y_1 - y_2|\\).\n\nTo solve this problem, we ... |
deepscaler | [
{
"role": "user",
"content": "A line through the points $(5, -12)$ and $(k, 23)$ is parallel to the line $4x + 6y = 12$. 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": "-47.5"
} | {
"split": "train",
"index": 31700,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The side \\( AB \\) of triangle \\( ABC \\) is longer than side \\( AC \\), and \\(\\angle A = 40^\\circ\\). Point \\( D \\) lies on side \\( AB \\) such that \\( BD = AC \\). Points \\( M \\) and \\( N \\) are the midpoints of segments \\( BC \\) and \\( AD \\) respectively. F... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 31701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD$, $AB = 6$ and $BC = 4$. Points $F$ and $G$ are on $\\overline{CD}$ such that $DF = 2$ and $GC = 1.5$. Lines $AF$ and $BG$ intersect at $E$. Calculate the area of $\\triangle AEB$. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{144}{7}"
} | {
"split": "train",
"index": 31702,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Coordinate System and Parametric Equation\nGiven the ellipse $(C)$: $\\frac{x^{2}}{16} + \\frac{y^{2}}{9} = 1$, which intersects with the positive semi-axis of $x$ and $y$ at points $A$ and $B$ respectively. Point $P$ is any point on the ellipse. Find the maximum area of $\\tri... | math | {
"style": "rule",
"ground_truth": "6(\\sqrt{2} + 1)"
} | {
"split": "train",
"index": 31703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system, an ellipse C passes through points A $(\\sqrt{3}, 0)$ and B $(0, 2)$.\n(I) Find the equation of ellipse C;\n(II) Let P be any point on the ellipse, find the maximum area of triangle ABP, and find the coordinates of point P when the area of ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6} + \\sqrt{3}"
} | {
"split": "train",
"index": 31704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Jeff, Maria, and Lee paid $90, $150, and $210 respectively, find j - m where Jeff gave Lee $j dollars and Maria gave Lee $m dollars to settle the debts such that everyone paid equally. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 31705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Hexagon $ABCDEF$ is divided into five rhombuses, $P, Q, R, S,$ and $T$ , as shown. Rhombuses $P, Q, R,$ and $S$ are congruent, and each has area $\\sqrt{2006}.$ Let $K$ be the area of rhombus $T$ . Given that $K$ is a positive integer, find the number of possible values for $K.... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 31706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cube $A B C D E F G H$ has edge length 100. Point $P$ is on $A B$, point $Q$ is on $A D$, and point $R$ is on $A F$, as shown, so that $A P=x, A Q=x+1$ and $A R=\\frac{x+1}{2 x}$ for some integer $x$. For how many integers $x$ is the volume of triangular-based pyramid $A P Q R$... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 31707,
"solution": "Triangular-based pyramid $A P Q R$ can be thought of as having triangular base $\\triangle A P Q$ and height $A R$. Since this pyramid is built at a vertex of the cube, then $\\triangle A P Q$ is right-angled at $A$ and $A R$ is perpendicular to the base. The area ... |
deepscaler | [
{
"role": "user",
"content": "Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has 27 cards, with every shape-color-sh... | math | {
"style": "rule",
"ground_truth": "117"
} | {
"split": "train",
"index": 31708,
"solution": "Treat the sets as ordered. Then for each of the three criterion, there are $3!=6$ choices if the attribute is different and there are $3$ choices is the attribute is the same. Thus all three attributes combine to a total of $(6+3)^3=729$ possibilities. However if a... |
deepscaler | [
{
"role": "user",
"content": "Given that the positive numbers $x$ and $y$ satisfy the equation $$3x+y+ \\frac {1}{x}+ \\frac {2}{y}= \\frac {13}{2}$$, find the minimum value of $$x- \\frac {1}{y}$$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {1}{2}"
} | {
"split": "train",
"index": 31709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a Cartesian coordinate plane, points \\((4,-1)\\) and \\((-1, 3)\\) are adjacent corners on a square. Calculate the area of this square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 31710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( R \\) be a semicircle with diameter \\( XY \\). A trapezoid \\( ABCD \\) in which \\( AB \\) is parallel to \\( CD \\) is circumscribed about \\( R \\) such that \\( AB \\) contains \\( XY \\). If \\( AD = 4 \\), \\( CD = 5 \\), and \\( BC = 6 \\), determine \\( AB \\).... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 31711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An 8-foot by 10-foot floor is tiled with square tiles of size 1 foot by 1 foot. Each tile has a pattern consisting of four white quarter circles of radius 1/2 foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floo... | math | {
"style": "rule",
"ground_truth": "80 - 20\\pi"
} | {
"split": "train",
"index": 31712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Estimate the time it takes to send $60$ blocks of data over a communications channel if each block consists of $512$ \"chunks\" and the channel can transmit $120$ chunks per second. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 31713,
"solution": "1. **Calculate the total number of chunks**: \n Each block consists of $512$ chunks, and there are $60$ blocks. Therefore, the total number of chunks is:\n \\[\n 60 \\times 512 = 30720\n \\]\n\n2. **Determine the transmission time**:\n The channel can tra... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the parametric equations of curve $C$ are given by $$\\begin{cases} x=2\\cos\\theta \\\\ y=\\sqrt{3}\\sin\\theta \\end{cases}$$ (where $\\theta$ is the parameter). With the origin $O$ as the pole and the positive $x$-axis as the polar a... | math | {
"style": "rule",
"ground_truth": "\\frac{24}{7}"
} | {
"split": "train",
"index": 31714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a right square prism $ABCD-A_{1}B_{1}C_{1}D_{1}$ with a base edge length of $1$, and $AB_{1}$ forms a $60^{\\circ}$ angle with the base $ABCD$, find the distance from $A_{1}C_{1}$ to the base $ABCD$. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 31715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence \\( a_{0}, a_{1}, \\cdots, a_{n} \\) satisfies:\n\\[ \na_{0}=\\sqrt{3}, \\quad a_{n+1}=[a_{n}]+\\frac{1}{\\{a_{n}\\}} \n\\]\nwhere \\( [x] \\) denotes the greatest integer less than or equal to the real number \\( x \\), and \\( \\{x\\}=x-[x] \\). Find the value of... | math | {
"style": "rule",
"ground_truth": "3024 + \\sqrt{3}"
} | {
"split": "train",
"index": 31716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $F_1 = \\left( -3, 1 - \\frac{\\sqrt{5}}{4} \\right)$ and $F_ 2= \\left( -3, 1 + \\frac{\\sqrt{5}}{4} \\right).$ Then the set of points $P$ such that\n\\[|PF_1 - PF_2| = 1\\]form a hyperbola. The equation of this hyperbola can be written as\n\\[\\frac{(y - k)^2}{a^2} - \\... | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{4}"
} | {
"split": "train",
"index": 31717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(12 points) A workshop has a total of 12 workers and needs to equip two types of machines. Each type A machine requires 2 people to operate, consumes 30 kilowatt-hours of electricity per day, and can produce products worth 40,000 yuan; each type B machine requires 3 people to o... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 31718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $\\$0.50$ per mile, and her only expense is gasoline at $\\$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense? Le... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 31719,
"solution": "1. **Calculate the total distance driven**: The driver travels at a speed of $60$ miles per hour for $2$ hours. Therefore, the total distance driven is:\n \\[\n 60 \\text{ miles/hour} \\times 2 \\text{ hours} = 120 \\text{ miles}\n \\]\n\n2. **Calculate the g... |
deepscaler | [
{
"role": "user",
"content": "For a positive integer $n$ , let $v(n)$ denote the largest integer $j$ such that $n$ is divisible by $2^j$ . Let $a$ and $b$ be chosen uniformly and independently at random from among the integers between 1 and 32, inclusive. What is the probability that $v(a) > ... | math | {
"style": "rule",
"ground_truth": "341/1024"
} | {
"split": "train",
"index": 31720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum $2\\frac{1}{7} + 3\\frac{1}{2} + 5\\frac{1}{19}$ is between Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10\\frac{1}{2} \\text{ and } 11"
} | {
"split": "train",
"index": 31721,
"solution": "1. **Convert mixed numbers to improper fractions**:\n - $2\\frac{1}{7} = \\frac{14}{7} + \\frac{1}{7} = \\frac{15}{7}$\n - $3\\frac{1}{2} = \\frac{6}{2} + \\frac{1}{2} = \\frac{7}{2}$\n - $5\\frac{1}{19} = \\frac{95}{19} + \\frac{1}{19} = \\frac{96}{19}$\n\n2... |
deepscaler | [
{
"role": "user",
"content": "Given that the center of the ellipse $E$ is at the origin, the focus is on the $y$-axis with an eccentricity of $e=\\frac{\\sqrt{2}}{2}$, and it passes through the point $P\\left( 1,\\sqrt{2} \\right)$:\n\n(1) Find the equation of the ellipse $E$;\n\n(2) Two pairs of perpendicu... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{3}"
} | {
"split": "train",
"index": 31722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all three-digit numbers that are equal to the sum of all their digits plus twice the square of the sum of their digits. List all possible numbers in ascending order without spaces and enter the resulting concatenated multi-digit number. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "171465666"
} | {
"split": "train",
"index": 31723,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circle inscribed in square $ABCD$ , is tangent to sides $AB$ and $CD$ at points $M$ and $K$ respectively. Line $BK$ intersects this circle at the point $L, X$ is the midpoint of $KL$ . Find the angle $\\angle MXK $ . Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 31724,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $3 \\begin{pmatrix} 2 \\\\ -8 \\end{pmatrix} - 2 \\begin{pmatrix} 1 \\\\ -7 \\end{pmatrix}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 4 \\\\ -10 \\end{pmatrix}"
} | {
"split": "train",
"index": 31725,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An integer $n>1$ is given . Find the smallest positive number $m$ satisfying the following conditions： for any set $\\{a,b\\}$ $\\subset \\{1,2,\\cdots,2n-1\\}$ ,there are non-negative integers $ x, y$ ( not all zero) such that $2n|ax+by$ and $x+y\\leq m.$ Let's think ... | math | {
"style": "rule",
"ground_truth": "n"
} | {
"split": "train",
"index": 31726,
"solution": "\nGiven an integer \\( n > 1 \\), we aim to find the smallest positive number \\( m \\) satisfying the following conditions: for any set \\(\\{a, b\\} \\subset \\{1, 2, \\ldots, 2n-1\\}\\), there exist non-negative integers \\( x \\) and \\( y \\) (not both zero) s... |
deepscaler | [
{
"role": "user",
"content": "If $\\log_6 x=2.5$, the value of $x$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36\\sqrt{6}"
} | {
"split": "train",
"index": 31727,
"solution": "1. We start with the given logarithmic equation:\n \\[\n \\log_6 x = 2.5\n \\]\n\n2. Recognize that $2.5$ can be expressed as a fraction:\n \\[\n \\log_6 x = \\frac{5}{2}\n \\]\n\n3. Using the property of logarithms that allows the exponentiation of bot... |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( ABC \\), the angle bisector \\( BL \\) is drawn. Find the area of the triangle, given that \\( AL=2 \\), \\( BL=3\\sqrt{10} \\), and \\( CL=3 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15\\sqrt{15}}{4}"
} | {
"split": "train",
"index": 31728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the largest integer $k$ such that $2004^k$ divides $2004!$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 31729,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB = BC$, and $\\overline{BD}$ is an altitude. Point $E$ is on the extension of $\\overline{AC}$ such that $BE =\n10$. The values of $\\tan \\angle CBE$, $\\tan \\angle DBE$, and $\\tan \\angle ABE$ form a geometric progression, and the values of $\\cot \\a... | math | {
"style": "rule",
"ground_truth": "\\frac{50}{3}"
} | {
"split": "train",
"index": 31730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 10 cards, labeled from 1 to 10. Three cards denoted by $ a,\\ b,\\ c\\ (a > b > c)$ are drawn from the cards at the same time.\r\nFind the probability such that $ \\int_0^a (x^2 \\minus{} 2bx \\plus{} 3c)\\ dx \\equal{} 0$ . Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "1/30"
} | {
"split": "train",
"index": 31731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ and $Q$ be the midpoints of sides $AB$ and $BC,$ respectively, of $\\triangle ABC.$ Suppose $\\angle A = 30^{\\circ}$ and $\\angle PQC = 110^{\\circ}.$ Find $\\angle B$ in degrees. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 31732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $7^{4x}=343$, what is the value of $7^{4x-3}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 31733,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Joy has $30$ thin rods, one each of every integer length from $1 \\text{ cm}$ through $30 \\text{ cm}$. She places the rods with lengths $3 \\text{ cm}$, $7 \\text{ cm}$, and $15 \\text{cm}$ on a table. She then wants to choose a fourth rod that she can put with these three to ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 31734,
"solution": "1. **Identify the range for the fourth rod**: To form a quadrilateral, the sum of the lengths of any three sides must be greater than the length of the fourth side. This is known as the triangle inequality theorem. We apply this to the three rods of lengths $3 \\te... |
deepscaler | [
{
"role": "user",
"content": "Given that the point $P$ on the ellipse $\\frac{x^{2}}{64} + \\frac{y^{2}}{28} = 1$ is 4 units away from the left focus, find the distance from point $P$ to the right directrix. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 31735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Parallelepiped $ABCDEFGH$ is generated by vectors $\\overrightarrow{AB},$ $\\overrightarrow{AD},$ and $\\overrightarrow{AE},$ as shown below.\n\n[asy]\nimport three;\n\nsize(220);\ncurrentprojection = orthographic(0.5,0.3,0.2);\n\ntriple I = (1,0,0), J = (0,1,0), K = (0,0,1), O... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 31736,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "10 pairs of distinct shoes are mixed in a bag. If 4 shoes are randomly drawn, how many possible outcomes are there for the following results?\n(1) None of the 4 shoes form a pair;\n(2) Among the 4 shoes, there is one pair, and the other two shoes do not form a pair;\n(3) The 4 ... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 31737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the $MNRK$ trapezoid with the lateral side $RK = 3$ if the distances from the vertices $M$ and $N$ to the line $RK$ are $5$ and $7$ , respectively. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 31738,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $x$ if\n\\[3 \\arctan \\frac{1}{4} + \\arctan \\frac{1}{20} + \\arctan \\frac{1}{x} = \\frac{\\pi}{4}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1985"
} | {
"split": "train",
"index": 31739,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\\\(a > b\\\\), the quadratic trinomial \\\\(a{x}^{2}+2x+b \\geqslant 0 \\\\) holds for all real numbers, and there exists \\\\(x_{0} \\in \\mathbb{R}\\\\), such that \\\\(ax_{0}^{2}+2{x_{0}}+b=0\\\\), then the minimum value of \\\\(\\dfrac{a^{2}+b^{2}}{a-b}\\\\) is \\_\... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{2}"
} | {
"split": "train",
"index": 31740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cindy leaves school at the same time every day. If she cycles at \\(20 \\ \\text{km/h}\\), she arrives home at 4:30 in the afternoon. If she cycles at \\(10 \\ \\text{km/h}\\), she arrives home at 5:15 in the afternoon. At what speed, in \\(\\text{km/h}\\), must she cycle to ar... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 31741,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five years ago, there were 25 trailer homes on Maple Street with an average age of 12 years. Since then, a group of brand new trailer homes was added, and 5 old trailer homes were removed. Today, the average age of all the trailer homes on Maple Street is 11 years. How many new... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 31742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which of the following is a possible value of $x$ if given two different numbers on a number line, the number to the right is greater than the number to the left, and the positions of $x, x^{3}$ and $x^{2}$ are marked on a number line? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{5}"
} | {
"split": "train",
"index": 31743,
"solution": "From the number line shown, we see that $x<x^{3}<x^{2}$. \n If $x>1$, then successive powers of $x$ are increasing (that is, $x<x^{2}<x^{3}$ ). \n Since this is not the case, then it is not true that $x>1$. \n If $x=0$ or $x=1$, then successive powers of $x$ are eq... |
deepscaler | [
{
"role": "user",
"content": "The square $BCDE$ is inscribed in circle $\\omega$ with center $O$ . Point $A$ is the reflection of $O$ over $B$ . A \"hook\" is drawn consisting of segment $AB$ and the major arc $\\widehat{BE}$ of $\\omega$ (passing through $C$ and $D$ ). Assume $BCDE$ h... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 31744,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The teacher fills some numbers into the circles in the diagram (each circle can and must contain only one number). The sum of the three numbers in each of the left and right closed loops is 30, and the sum of the four numbers in each of the top and bottom closed loops is 40. If... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 31745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum number of checkers that can be placed on a $6 \\times 6$ board such that no three checkers (specifically, the centers of the cells they occupy) are on the same line (regardless of the angle of inclination)? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 31746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate\n\n$$\n\\int_{L}\\left(z^{2}+2z \\bar{z}\\right)d z\n$$\n\nwhere \\( L \\) is the arc of the circle \\( |z|=1 \\) with \\( \\arg z \\in[0, \\pi] \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{14}{3}"
} | {
"split": "train",
"index": 31747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD$, $AB=2$, $BC=4$, and points $E$, $F$, and $G$ are located as follows: $E$ is the midpoint of $\\overline{BC}$, $F$ is the midpoint of $\\overline{CD}$, and $G$ is one fourth of the way down $\\overline{AD}$ from $A$. If point $H$ is the midpoint of $\\overli... | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{4}"
} | {
"split": "train",
"index": 31748,
"solution": ""
} |
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