data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let $b(x)=x^{2}+x+1$. The polynomial $x^{2015}+x^{2014}+\\cdots+x+1$ has a unique \"base $b(x)$ \" representation $x^{2015}+x^{2014}+\\cdots+x+1=\\sum_{k=0}^{N} a_{k}(x) b(x)^{k}$ where each \"digit\" $a_{k}(x)$ is either the zero polynomial or a nonzero polynomial of degree le... | math | {
"style": "rule",
"ground_truth": "-1006"
} | {
"split": "train",
"index": 6700,
"solution": "Comparing degrees easily gives $N=1007$. By ignoring terms of degree at most 2013, we see $a_{N}(x)\\left(x^{2}+x+1\\right)^{1007} \\in x^{2015}+x^{2014}+O\\left(x^{2013}\\right)$. Write $a_{N}(x)=u x+v$, so $a_{N}(x)\\left(x^{2}+x+1\\right)^{1007} \\in(u x+v)\\left... |
deepscaler | [
{
"role": "user",
"content": "Tanks has a pile of 5 blue cards and 5 red cards. Every morning, he takes a card and throws it down a well. What is the probability that the first card he throws down and the last card he throws down are the same color? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"split": "train",
"index": 6701,
"solution": "Once he has thrown the first card down the well, there are 9 remaining cards, and only 4 have the same color as the card that was thrown down. Therefore, the probability that the last card he throws down has the same color is $\\frac{4}{9}$."
} |
deepscaler | [
{
"role": "user",
"content": "For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $500$ that are neither $3$-nice nor $5$-nice. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "266"
} | {
"split": "train",
"index": 6702,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the hyperbola $C$: $x^{2}- \\frac{y^{2}}{b^{2}}=1$ ($b > 0$) has two asymptotes that intersect with the circle $O$: $x^{2}+y^{2}=2$ at four points sequentially named $A$, $B$, $C$, and $D$. If the area of rectangle $ABCD$ is $b$, then t... | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"split": "train",
"index": 6703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $1,2,\\ldots,9$ are arranged so that the $1$ st term is not $1$ and the $9$ th term is not $9$ . Calculate the probability that the third term is $3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{43}{399}"
} | {
"split": "train",
"index": 6704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given three points in space: A(0,1,5), B(1,5,0), and C(5,0,1), if the vector $\\vec{a}=(x,y,z)$ is perpendicular to both $\\overrightarrow{AB}$ and $\\overrightarrow{AC}$, and the magnitude of vector $\\vec{a}$ is $\\sqrt{15}$, then find the value of $x^2y^2z^2$. Let's think st... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 6705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g_{1}(x) = \\sqrt{2 - x}$, and for integers $n \\geq 2$, define \\[g_{n}(x) = g_{n-1}\\left(\\sqrt{(n+1)^2 - x}\\right).\\] Find the largest value of $n$, denoted as $M$, for which the domain of $g_n$ is nonempty. For this value of $M$, if the domain of $g_M$ consists of a... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 6706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $g(x)$ is a function such that\n\\[ g(xy) + x = xg(y) + g(x) \\] for all real numbers $x$ and $y$. If $g(-2) = 4$, compute $g(-1002)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"split": "train",
"index": 6707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If 7 knicks = 2 knacks and 3 knacks = 4 knocks, how many knicks are equal to 24 knocks? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 6708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $c\\sin\\frac{A+C}{2}=b\\sin C$. \n$(1)$ Find angle $B$; \n$(2)$ Let $BD$ be the altitude from $B$ to side $AC$, and $BD=1$, $b=\\sqrt{3}$. Find the perimeter ... | math | {
"style": "rule",
"ground_truth": "3 + \\sqrt{3}"
} | {
"split": "train",
"index": 6709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $\\frac{1}{2222}$ is expressed as a decimal, what is the sum of the first 60 digits after the decimal point? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "114"
} | {
"split": "train",
"index": 6710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(12 points in total) 4 students are sitting in a row to watch a movie, and there are 6 seats in the row. \n(1) How many seating arrangements are there such that there is exactly one person between students A and B, and there are no empty seats between them? \n(2) How many sea... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 6711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest number of points of intersection that can occur when $2$ different circles and $2$ different straight lines are drawn on the same piece of paper? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 6712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what value of $c$ will the circle with equation $x^2 + 6x + y^2 - 4y + c = 0$ have a radius of length 4? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 6713,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the following expressions:\n\n(1) $\\sin^{4}{\\alpha} + \\tan^{2}{\\alpha} \\cdot \\cos^{4}{\\alpha} + \\cos^{2}{\\alpha}$\n\n(2) $\\frac{\\cos{(180^{\\circ} + \\alpha)} \\cdot \\sin{(\\alpha + 360^{\\circ})}}{\\sin{(-\\alpha - 180^{\\circ})} \\cdot \\cos{(-180^{\\circ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 6714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value $2^8 -1$ is divisible by 3 prime numbers. What is the sum of the three prime numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 6715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Emma has $100$ fair coins. She flips all the coins once. Any coin that lands on tails is tossed again, and the process continues up to four times for coins that sequentially land on tails. Calculate the expected number of coins that finally show heads. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "93.75"
} | {
"split": "train",
"index": 6716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An artist wants to completely cover a rectangle with identically sized squares which do not overlap and do not extend beyond the edges of the rectangle. If the rectangle is \\(60 \\frac{1}{2} \\mathrm{~cm}\\) long and \\(47 \\frac{2}{3} \\mathrm{~cm}\\) wide, what is the minimu... | math | {
"style": "rule",
"ground_truth": "858"
} | {
"split": "train",
"index": 6717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the solution set of the inequality about $x$, $|x-2| < a$ ($a \\in \\mathbb{R}$), be $A$, and $\\frac{3}{2} \\in A$, $-\\frac{1}{2} \\notin A$. \n(1) For any $x \\in \\mathbb{R}$, the inequality $|x-1| + |x-3| \\geq a^2 + a$ always holds true, and $a \\in \\mathbb{N}$. Fin... | math | {
"style": "rule",
"ground_truth": "\\frac{1 + 2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 6718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Natural numbers \\( a, b, c \\) are such that \\( 1 \\leqslant a < b < c \\leqslant 3000 \\). Find the largest possible value of the quantity\n\n$$\n\\gcd(a, b) + \\gcd(b, c) + \\gcd(c, a)\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3000"
} | {
"split": "train",
"index": 6719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the quiz consists of 4 multiple-choice questions, each with 3 choices, calculate the probability that the contestant wins the quiz. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 6720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the side \\( BC \\) of an equilateral triangle \\( ABC \\), points \\( K \\) and \\( L \\) are marked such that \\( BK = KL = LC \\). On the side \\( AC \\), point \\( M \\) is marked such that \\( AM = \\frac{1}{3} AC \\). Find the sum of the angles \\( \\angle AKM \\) and ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 6721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to distribute 7 balls into 4 boxes if the balls are not distinguishable and neither are the boxes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 6722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the beginning of the school year, Jinshi Middle School has 17 classrooms. Teacher Dong Helong takes 17 keys to open the doors, knowing that each key can only open one door, but not knowing which key matches which door. What is the maximum number of attempts he needs to make ... | math | {
"style": "rule",
"ground_truth": "136"
} | {
"split": "train",
"index": 6723,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $1-(1+(1-(1+(1-x))))$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1-x"
} | {
"split": "train",
"index": 6724,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The population of Nosuch Junction at one time was a perfect square. Later, with an increase of $100$, the population was one more than a perfect square. Now, with an additional increase of $100$, the population is again a perfect square. \n\nThe original population is a multipl... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 6725,
"solution": "1. Let $a^2$ be the original population count, $b^2+1$ be the population after an increase of $100$, and $c^2$ be the population after another increase of $100$.\n2. We have the equations:\n - $a^2 + 100 = b^2 + 1$\n - $b^2 + 1 + 100 = c^2$\n\n3. Simplifying the... |
deepscaler | [
{
"role": "user",
"content": "Roberto has four pairs of trousers, seven shirts, and three jackets. How many different outfits can he put together if an outfit consists of a pair of trousers, a shirt, and a jacket? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 6726,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Choose the largest of the following sums, and express it as a fraction in simplest form:\n$$\\frac{1}{4} + \\frac{1}{5}, \\ \\ \\frac{1}{4} + \\frac{1}{6}, \\ \\ \\frac{1}{4} + \\frac{1}{3}, \\ \\ \\frac{1}{4} + \\frac{1}{8}, \\ \\ \\frac{1}{4} + \\frac{1}{7}$$ Let's think step... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{12}"
} | {
"split": "train",
"index": 6727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the functions $f(x)=x^{2}-2x+m\\ln x(m∈R)$ and $g(x)=(x- \\frac {3}{4})e^{x}$.\n(1) If $m=-1$, find the value of the real number $a$ such that the minimum value of the function $φ(x)=f(x)-\\[x^{2}-(2+ \\frac {1}{a})x\\](0 < x\\leqslant e)$ is $2$;\n(2) If $f(x)$ has two e... | math | {
"style": "rule",
"ground_truth": "-e^{- \\frac {1}{4}}"
} | {
"split": "train",
"index": 6728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a\\_n\\}$ with the sum of its first $n$ terms denoted as $S\\_n$. The sequence satisfies the conditions $a\\_1=23$, $a\\_2=-9$, and $a_{n+2}=a\\_n+6\\times(-1)^{n+1}-2$ for all $n \\in \\mathbb{N}^*$.\n(1) Find the general formula for the terms of the seque... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 6729,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose we need to divide 15 dogs into three groups, one with 4 dogs, one with 7 dogs, and one with 4 dogs. We want to form the groups such that Fluffy is in the 4-dog group, Nipper is in the 7-dog group, and Daisy is in the other 4-dog group. How many ways can we arrange the r... | math | {
"style": "rule",
"ground_truth": "18480"
} | {
"split": "train",
"index": 6730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $w,x,y,z$ satisfy \\begin{align*}w+x+y+z&=25,wx+wy+wz+xy+xz+yz&=2y+2z+193\\end{align*} The largest possible value of $w$ can be expressed in lowest terms as $w_1/w_2$ for some integers $w_1,w_2>0$ . Find $w_1+w_2$ . Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 6731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the harmonic mean of the first n terms of the sequence $\\left\\{{a}_{n}\\right\\}$ is $\\dfrac{1}{2n+1}$, and ${b}_{n}= \\dfrac{{a}_{n}+1}{4}$, find the value of $\\dfrac{1}{{b}_{1}{b}_{2}}+ \\dfrac{1}{{b}_{2}{b}_{3}}+\\ldots+ \\dfrac{1}{{b}_{10}{b}_{11}}$. Let's think s... | math | {
"style": "rule",
"ground_truth": "\\dfrac{10}{11}"
} | {
"split": "train",
"index": 6732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the function $f(x) = x^3 + 3\\sqrt{x}$. Evaluate $3f(3) - 2f(9)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1395 + 9\\sqrt{3}"
} | {
"split": "train",
"index": 6733,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As $n$ ranges over the positive integers, what is the maximum possible value for the greatest common divisor of $11n+3$ and $6n+1$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 6734,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain product in a shopping mall sells an average of 30 items per day, with a profit of 50 yuan per item. In order to reduce inventory quickly, the mall decides to take appropriate price reduction measures. After investigation, it was found that for each item, for every 1 y... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 6735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A=\\{0, |x|\\}$ and $B=\\{1, 0, -1\\}$. If $A \\subseteq B$, then $x$ equals \\_\\_\\_\\_\\_\\_; The union of sets $A$ and $B$, denoted $A \\cup B$, equals \\_\\_\\_\\_\\_\\_; The complement of $A$ in $B$, denoted $\\complement_B A$, equals \\_\\_\\_\\_\\_\\_. Let's think ... | math | {
"style": "rule",
"ground_truth": "\\{-1\\}"
} | {
"split": "train",
"index": 6736,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lucy begins a sequence with the first term at 4. Each subsequent term is generated as follows: If a fair coin flip results in heads, she triples the previous term and then adds 3. If it results in tails, she subtracts 3 and divides the result by 3. What is the probability that ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 6737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a configuration of two right triangles, $PQR$ and $PRS$, squares are constructed on three sides of the triangles. The areas of three of the squares are 25, 4, and 49 square units. Determine the area of the fourth square built on side $PS$.\n\n[asy]\ndefaultpen(linewidth(0.7)... | math | {
"style": "rule",
"ground_truth": "53"
} | {
"split": "train",
"index": 6738,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $AB$ be a segment of unit length and let $C, D$ be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set $\\{A,B,C,D\\}.$ Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{125}"
} | {
"split": "train",
"index": 6739,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a particular school with 43 students, each student takes chemistry, biology, or both. The chemistry class is three times as large as the biology class, and 5 students are taking both classes. How many people are in the chemistry class? Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 6740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial $f(x)=x^3-4\\sqrt{3}x^2+13x-2\\sqrt{3}$ has three real roots, $a$ , $b$ , and $c$ . Find $$ \\max\\{a+b-c,a-b+c,-a+b+c\\}. $$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3} + 2\\sqrt{2}"
} | {
"split": "train",
"index": 6741,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three people, A, B, and C, are taking an elevator from the 1st floor to the 3rd to 7th floors of a mall. Each floor can accommodate at most 2 people getting off the elevator. How many ways are there for them to get off the elevator? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 6742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ each be chosen at random from the set $\\{1, 2, 3, \\ldots, 40\\}$. Additionally, let $c$ and $d$ also be chosen at random from the same set. Calculate the probability that the integer $2^c + 5^d + 3^a + 7^b$ has a units digit of $8$. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"split": "train",
"index": 6743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In square ABCD, point E is on AB and point F is on BC such that AE=3EB and BF=FC. Find the ratio of the area of triangle DEF to the area of square ABCD. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{16}"
} | {
"split": "train",
"index": 6744,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the first 1234 terms of the sequence where the number of 2s between consecutive 1s increases by 1 each time? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2419"
} | {
"split": "train",
"index": 6745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a similar tournament setup, the top 6 bowlers have a playoff. First #6 bowls #5, and the loser gets the 6th prize. The winner then bowls #4, and the loser of this match gets the 5th prize. The process continues with the previous winner bowling the next highest ranked bowler ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 6746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the squares of two positive integers is 90. The product of the two integers is 27. What is the sum of the two integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 6747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a natural number $n \\geq 1$, it satisfies: $2002 \\times n$ is a perfect cube, and $n \\div 2002$ is a perfect square. The smallest such $n$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2002^5"
} | {
"split": "train",
"index": 6748,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the expression $4(x^2-2x+2)-7(x^3-3x+1)$ is fully simplified, what is the sum of the squares of the coefficients of the terms? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "235"
} | {
"split": "train",
"index": 6749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the first $2018$ numbers in the sequence $101, 1001, 10001, 100001, \\dots$ are divisible by $101$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "505"
} | {
"split": "train",
"index": 6750,
"solution": "1. **Identify the sequence and its general form**: The sequence given is $101, 1001, 10001, 100001, \\dots$. We can express the $n$-th term of this sequence as $10^n + 1$.\n\n2. **Check divisibility by $101$**: We need to determine how many terms of this sequence up... |
deepscaler | [
{
"role": "user",
"content": "If the area of $\\triangle ABC$ is $64$ square units and the geometric mean (mean proportional) between sides $AB$ and $AC$ is $12$ inches, then $\\sin A$ is equal to Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"split": "train",
"index": 6751,
"solution": "1. **Assign Variables to Sides**: Let $AB = s$ and $AC = r$.\n\n2. **Use the Formula for the Area of a Triangle**: The area of $\\triangle ABC$ can be expressed as:\n \\[\n \\text{Area} = \\frac{1}{2} \\times AB \\times AC \\times \\sin A = \\frac{1}{2} \\times ... |
deepscaler | [
{
"role": "user",
"content": "Set $S = \\{1, 2, 3, ..., 2005\\}$ . If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 6752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of \\(m + n\\) where \\(m\\) and \\(n\\) are integers defined as follows: The positive difference between the roots of the quadratic equation \\(5x^2 - 9x - 14 = 0\\) can be expressed as \\(\\frac{\\sqrt{m}}{n}\\), with \\(m\\) not divisible by the square of any ... | math | {
"style": "rule",
"ground_truth": "366"
} | {
"split": "train",
"index": 6753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four complex numbers lie at the vertices of a square in the complex plane. Three of the numbers are $1+2i, -2+i$, and $-1-2i$. The fourth number is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2-i"
} | {
"split": "train",
"index": 6754,
"solution": "1. **Identify the given points in the complex plane**: The complex numbers given are $1+2i, -2+i$, and $-1-2i$. We can interpret these as points in the Cartesian plane: $A = (1, 2), B = (-2, 1), C = (-1, -2)$.\n\n2. **Determine the slopes of lines AB and BC**:\n -... |
deepscaler | [
{
"role": "user",
"content": "A toy store manager received a large order of Mr. Slinkums just in time for the holidays. The manager places $20\\%$ of them on the shelves, leaving the other 120 Mr. Slinkums in storage. How many Mr. Slinkums were in this order? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 6755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Climbing the first flight of stairs takes Jimmy 20 seconds, and each following flight takes 5 seconds more than the preceding one. How many total seconds does it take to climb the first five flights of stairs? Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 6756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An ethnographer determined that in a primitive tribe he studied, the distribution of lifespan among tribe members can be described as follows: 25% live only up to 40 years, 50% die at 50 years, and 25% live to 60 years. He then randomly selected two individuals to study in more... | math | {
"style": "rule",
"ground_truth": "53.75"
} | {
"split": "train",
"index": 6757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A \\(4 \\times 4\\) Sudoku grid is filled with digits so that each column, each row, and each of the four \\(2 \\times 2\\) sub-grids that compose the grid contains all of the digits from 1 to 4. \n\nFind the total number of possible \\(4 \\times 4\\) Sudoku grids. Let's think ... | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 6758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the quadratic equation in one variable $x$, $x^{2}+mx+n=0$ always has two real roots $x_{1}$ and $x_{2}$. \n$(1)$ When $n=3-m$ and both roots are negative, find the range of real number $m$. \n$(2)$ The inequality $t\\leqslant \\left(m-1\\right)^{2}+\\left(n-1\\right)^{2}... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{8}"
} | {
"split": "train",
"index": 6759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α$ and $β$ are both acute angles, $cosα= \\frac {3}{5}$, and $cos(α+β)=- \\frac {5}{13}$, find the value of $sinβ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {56}{65}"
} | {
"split": "train",
"index": 6760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five friends all brought some cakes with them when they met. Each of them gave a cake to each of the others. They then ate all the cakes they had just been given. As a result, the total number of cakes they had between them decreased by half. How many cakes did the five friends... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 6761,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle touches the sides of an angle at points \\( A \\) and \\( B \\). The distance from a point \\( C \\) lying on the circle to the line \\( AB \\) is 6. Find the sum of the distances from point \\( C \\) to the sides of the angle, given that one of these distances is nine... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 6762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve in prime numbers the equation $x^y - y^x = xy^2 - 19$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2, 3)(2, 7)"
} | {
"split": "train",
"index": 6763,
"solution": "\nTo find the solutions of the equation \\(x^y - y^x = xy^2 - 19\\) in prime numbers, we will begin by analyzing possible small prime candidates, as powers of small primes often have manageable forms that can be verified manually.\n\n**Step 1: Try small primes for \... |
deepscaler | [
{
"role": "user",
"content": "Let triangle $ABC$ be a right triangle with right angle at $C.$ Let $D$ and $E$ be points on $\\overline{AB}$ with $D$ between $A$ and $E$ such that $\\overline{CD}$ and $\\overline{CE}$ trisect $\\angle C.$ If $\\frac{DE}{BE} = \\frac{8}{15},$ then find $\\tan B.$ Let's think ... | math | {
"style": "rule",
"ground_truth": "\\frac{4 \\sqrt{3}}{11}"
} | {
"split": "train",
"index": 6764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A parking lot consists of 2012 parking spots equally spaced in a line, numbered 1 through 2012. One by one, 2012 cars park in these spots under the following procedure: the first car picks from the 2012 spots uniformly randomly, and each following car picks uniformly randomly a... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2062300}"
} | {
"split": "train",
"index": 6765,
"solution": "We see that for 1 to be the last spot, 2 must be picked first (with probability $\\frac{1}{n}$ ), after which spot $n$ is picked. Then, cars from 3 to $n-1$ will be picked until there are only gaps of 1 or 2 remaining. At this point, each of the remaining spots (inc... |
deepscaler | [
{
"role": "user",
"content": "Find $w$, such that $5^65^w=25$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 6766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the complex number \\( z \\) satisfies \\( |z|=1 \\), find the maximum value of \\( \\left| z^3 - 3z - 2 \\right| \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"split": "train",
"index": 6767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a bank has 320 employees with an average annual profit of 200,000 yuan per employee, and each remaining employee's profit increases by 20,000 yuan for every laid-off employee, while the bank must pay 60,000 yuan per year for each laid-off employee, and the number of ... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 6768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\\frac{1}{2}$, a circle with the origin as its center and the short axis of the ellipse as its radius is tangent to the line $\\sqrt{7}x-\\sqrt{5}y+12=0$.\n(1) Find the equa... | math | {
"style": "rule",
"ground_truth": "-\\frac{12}{7}"
} | {
"split": "train",
"index": 6769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the set \\( S = \\{1, 2, \\cdots, 15\\} \\). Define \\( A = \\{a_{1}, a_{2}, a_{3}\\} \\) as a subset of \\( S \\), such that \\( (a_{1}, a_{2}, a_{3}) \\) satisfies \\( 1 \\leq a_{1} < a_{2} < a_{3} \\leq 15 \\) and \\( a_{3} - a_{2} \\leq 6 \\). Find the number of such su... | math | {
"style": "rule",
"ground_truth": "371"
} | {
"split": "train",
"index": 6770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a shaded region inside a regular hexagon. The shaded region is divided into equilateral triangles. What fraction of the area of the hexagon is shaded?\nA) $\\frac{3}{8}$\nB) $\\frac{2}{5}$\nC) $\\frac{3}{7}$\nD) $\\frac{5}{12}$\nE) $\\frac{1}{2}$ Let's think s... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 6771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Penniless Pete's piggy bank has no pennies in it, but it has 100 coins, all nickels, dimes, and quarters, whose total value is $8.35. It does not necessarily contain coins of all three types. What is the difference between the largest and smallest number of dimes that could be ... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 6772,
"solution": "Let $a$, $b$, and $c$ represent the number of nickels, dimes, and quarters, respectively. We are given two conditions:\n1. The total number of coins is 100.\n2. The total value of the coins is $8.35, which is equivalent to 835 cents.\n\nFrom these conditions, we can... |
deepscaler | [
{
"role": "user",
"content": "How can you cut a 5 × 5 square with straight lines so that the resulting pieces can be assembled into 50 equal squares? It is not allowed to leave unused pieces or to overlap them. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 6773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the regular hexagon to the right, how many degrees are in the exterior angle indicated?\n\n[asy]size(101);\ndraw((0,0)--(2,0)--(3,sqrt(3))--(2,2sqrt(3))--(0,2sqrt(3))--(-1,sqrt(3))--cycle);\ndraw((2,0)--(4,0));\npair arrowstart = (2,0) + .75 expi(pi/7);\ndraw(arrowstart--arr... | math | {
"style": "rule",
"ground_truth": "60^\\circ"
} | {
"split": "train",
"index": 6774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $f(x)$ is a function defined for all real $x$, and suppose $f$ is invertible (that is, $f^{-1}(x)$ exists for all $x$ in the range of $f$).\n\nIf the graphs of $y=f(x^2)$ and $y=f(x^4)$ are drawn, at how many points do they intersect? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 6775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an eight-digit number, each digit (except the last one) is greater than the following digit. How many such numbers are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 6776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If there are exactly $3$ integer solutions for the inequality system about $x$: $\\left\\{\\begin{array}{c}6x-5≥m\\\\ \\frac{x}{2}-\\frac{x-1}{3}＜1\\end{array}\\right.$, and the solution to the equation about $y$: $\\frac{y-2}{3}=\\frac{m-2}{3}+1$ is a non-negative number, find... | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 6777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, points $U$, $V$, $W$, $X$, $Y$, and $Z$ lie on a straight line with $UV=VW=WX=XY=YZ=5$. Semicircles with diameters $UZ$, $UV$, $VW$, $WX$, $XY$, and $YZ$ create the shape shown. What is the area of the shaded region?\n[asy]\nsize(5cm); defaultpen(fontsize(9));... | math | {
"style": "rule",
"ground_truth": "\\frac{325}{4}\\pi"
} | {
"split": "train",
"index": 6778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A shooter's probability of hitting the 10, 9, and 8 rings in a single shot are respectively 0.2, 0.3, and 0.1. Express the probability that the shooter scores no more than 8 in a single shot as a decimal. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 6779,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the complex plane, the distance between the points corresponding to the complex numbers $-3+i$ and $1-i$ is $\\boxed{\\text{answer}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{20}"
} | {
"split": "train",
"index": 6780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine how many hours it will take Carl to mow the lawn, given that the lawn measures 120 feet by 100 feet, the mower's swath is 30 inches wide with an overlap of 6 inches, and Carl walks at a rate of 4000 feet per hour. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 6781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Katherine makes Benj play a game called $50$ Cent. Benj starts with $\\$ 0.50 $, and every century thereafter has a $ 50\\% $ chance of doubling his money and a $ 50\\% $ chance of having his money reset to $ \\ $0.50$ . What is the expected value of the amount of money Benj... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 6782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number ending in the digit 2 such that it doubles when this digit is moved to the beginning. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "105263157894736842"
} | {
"split": "train",
"index": 6783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given triangle $\\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, $c\\cos A= \\frac{4}{b}$, and the area of $\\triangle ABC$, $S \\geq 2$. \n(1) Determine the range of possible values for angle $A$. \n(2) Find the maximum value of... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2} + \\frac{\\sqrt{6}}{4}"
} | {
"split": "train",
"index": 6784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 8 young people, among whom 5 are capable of doing English translation work, and 4 are capable of doing computer software design work (including one person who is capable of doing both tasks). Now, 5 young people are to be selected to undertake a task, with 3 people do... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 6785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let an ordered pair of positive integers $(m, n)$ be called *regimented* if for all nonnegative integers $k$ , the numbers $m^k$ and $n^k$ have the same number of positive integer divisors. Let $N$ be the smallest positive integer such that $\\left(2016^{2016}, N\\rig... | math | {
"style": "rule",
"ground_truth": "10086"
} | {
"split": "train",
"index": 6786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "27 points P and a square with side length 1 lie on the same plane. Let the vertices of the square be \\( A, B, C, \\) and \\( D \\) (counterclockwise), and the distances from P to \\( A, B, \\) and \\( C \\) are \\( u, v, \\) and \\( w \\) respectively. If \\( u^2 + v^2 = w^2 \... | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{2}"
} | {
"split": "train",
"index": 6787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two integers are relatively prime if they have no common factors other than 1 or -1. What is the probability that a positive integer less than or equal to 30 is relatively prime to 30? Express your answer as a common fraction. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{15}"
} | {
"split": "train",
"index": 6788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than 10,000 have at most two different digits?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "927"
} | {
"split": "train",
"index": 6789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle $ABC$ with $\\angle B = \\frac{\\pi}{3}$, \n(I) if $AB=8\\sqrt{3}$ and $AC=12$, find the area of $\\triangle ABC$; \n(II) if $AB=4$ and $\\vec{BM} = \\vec{MN} = \\vec{NC}$ with $AN=2\\sqrt{3}BM$, find the length of $AM$. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"split": "train",
"index": 6790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A laptop is originally priced at $800. The store offers a $15\\%$ discount, followed by another $10\\%$ discount on the discounted price. Tom also has a special membership card giving an additional $5\\%$ discount on the second discounted price. What single percent discount wou... | math | {
"style": "rule",
"ground_truth": "27.325\\%"
} | {
"split": "train",
"index": 6791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $F$ is taken on the extension of side $AD$ of parallelogram $ABCD$. $BF$ intersects diagonal $AC$ at $E$ and side $DC$ at $G$. If $EF = 32$ and $GF = 24$, then $BE$ equals:\n[asy] size(7cm); pair A = (0, 0), B = (7, 0), C = (10, 5), D = (3, 5), F = (5.7, 9.5); pair G = in... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 6792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A manufacturer built a machine which will address $500$ envelopes in $8$ minutes. He wishes to build another machine so that when both are operating together they will address $500$ envelopes in $2$ minutes. The equation used to find how many minutes $x$ it would require the se... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}+\\frac{1}{x}=\\frac{1}{2}"
} | {
"split": "train",
"index": 6793,
"solution": "1. **Understanding the problem**: The first machine addresses 500 envelopes in 8 minutes. We need to find the time $x$ it would take for a second machine to address 500 envelopes alone such that both machines working together can address 500 envelopes in 2 minutes.\... |
deepscaler | [
{
"role": "user",
"content": "There are 49 children, each wearing a unique number from 1 to 49 on their chest. Select several children and arrange them in a circle such that the product of the numbers of any two adjacent children is less than 100. What is the maximum number of children you can select? Let's... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 6794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the positive integer that has three digits in both base-10 and base-8, and the sum of its digits in both bases is fourteen. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "455"
} | {
"split": "train",
"index": 6795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In square $EFGH$, $EF$ is 8 centimeters, and $N$ is the midpoint of $\\overline{GH}$. Let $P$ be the intersection of $\\overline{EC}$ and $\\overline{FN}$, where $C$ is a point on segment $GH$ such that $GC = 6$ cm. What is the area ratio of triangle $EFP$ to triangle $EPG$? Le... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 6796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number with 6 as the last digit, such that if the final 6 is moved to the front of the number it is multiplied by 4. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "153846"
} | {
"split": "train",
"index": 6797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers \\(a, b, c, d\\) belong to the interval \\([-12.5, 12.5]\\). 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": "650"
} | {
"split": "train",
"index": 6798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two springs with stiffnesses of $6 \\, \\text{kN} / \\text{m}$ and $12 \\, \\text{kN} / \\text{m}$ are connected in series. How much work is required to stretch this system by 10 cm? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 6799,
"solution": ""
} |
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