data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given that $\\left\\{a_{n}\\right\\}$ is an arithmetic sequence with a nonzero common difference and $\\left\\{b_{n}\\right\\}$ is a geometric sequence where $a_{1}=3, b_{1}=1, a_{2}=b_{2}, 3a_{5}=b_{3}$, and there exist constants $\\alpha$ and $\\beta$ such that for every posi... | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{3} + 3"
} | {
"split": "train",
"index": 26449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that acute angles $\\alpha$ and $\\beta$ satisfy $\\sin\\alpha=\\frac{4}{5}$ and $\\cos(\\alpha+\\beta)=-\\frac{12}{13}$, determine the value of $\\cos \\beta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{16}{65}"
} | {
"split": "train",
"index": 26450,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "David, Delong, and Justin each showed up to a problem writing session at a random time during the session. If David arrived before Delong, what is the probability that he also arrived before Justin? 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": 26451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the surface area of the part of the paraboloid of revolution \\( 3y = x^2 + z^2 \\) that is located in the first octant and bounded by the plane \\( y = 6 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{39 \\pi}{4}"
} | {
"split": "train",
"index": 26452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the base 10 arithmetic problem $H M M T+G U T S=R O U N D$, each distinct letter represents a different digit, and leading zeroes are not allowed. What is the maximum possible value of $R O U N D$? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "16352"
} | {
"split": "train",
"index": 26453,
"solution": "Clearly $R=1$, and from the hundreds column, $M=0$ or 9. Since $H+G=9+O$ or $10+O$, it is easy to see that $O$ can be at most 7, in which case $H$ and $G$ must be 8 and 9, so $M=0$. But because of the tens column, we must have $S+T \\geq 10$, and in fact since $D$ ... |
deepscaler | [
{
"role": "user",
"content": "The vertices and midpoints of the sides of a regular decagon (thus a total of 20 points marked) are noted.\n\nHow many triangles can be formed with vertices at the marked points? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1130"
} | {
"split": "train",
"index": 26454,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two passenger trains, A and B, are 150 meters and 200 meters long, respectively. They are traveling in opposite directions on parallel tracks. A passenger on train A measures that train B passes by his window in 10 seconds. How long does a passenger on train B see train A pass ... | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"split": "train",
"index": 26455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The least positive integer with exactly \\(2023\\) distinct positive divisors can be written in the form \\(m \\cdot 10^k\\), where \\(m\\) and \\(k\\) are integers and \\(10\\) is not a divisor of \\(m\\). What is \\(m+k?\\)\nA) 999846\nB) 999847\nC) 999848\nD) 999849\nE) 9998... | math | {
"style": "rule",
"ground_truth": "999846"
} | {
"split": "train",
"index": 26456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the numbers 1, 2, 3, 4, find the probability that $\\frac{a}{b}$ is not an integer, where $a$ and $b$ are randomly selected numbers from the set $\\{1, 2, 3, 4\\}$. 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": 26457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Menkara has a $4 \\times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch? Let's thin... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 26458,
"solution": "1. **Identify the dimensions of the index card**: Menkara has a $4 \\times 6$ index card. The two possible dimensions are 4 inches and 6 inches.\n\n2. **Calculate the initial area of the card**: \n \\[\n \\text{Area} = \\text{Length} \\times \\text{Width} = 4 \... |
deepscaler | [
{
"role": "user",
"content": "Find the number of addition problems in which a two digit number is added to a second two digit number to give a two digit answer, such as in the three examples: \\[\\begin{tabular}{@{\\hspace{3pt}}c@{\\hspace{3pt}}}2342\\hline 65\\end{tabular}\\,,\\qquad\\begin{tabular}{@{\\hs... | math | {
"style": "rule",
"ground_truth": "3240"
} | {
"split": "train",
"index": 26459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{a} = \\begin{pmatrix} -3 \\\\ 10 \\\\ 1 \\end{pmatrix},$ $\\mathbf{b} = \\begin{pmatrix} 5 \\\\ \\pi \\\\ 0 \\end{pmatrix},$ and $\\mathbf{c} = \\begin{pmatrix} -2 \\\\ -2 \\\\ 7 \\end{pmatrix}.$ Compute\n\\[(\\mathbf{a} - \\mathbf{b}) \\cdot [(\\mathbf{b} - \\ma... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 26460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c$ be real numbers such that $9a^2 + 4b^2 + 25c^2 = 1.$ Find the maximum value of\n\\[8a + 3b + 5c.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{373}}{6}"
} | {
"split": "train",
"index": 26461,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The integer 48178 includes the block of digits 178. How many integers between 10000 and 100000 include the block of digits 178? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "280"
} | {
"split": "train",
"index": 26462,
"solution": "Since 100000 does not include the block of digits 178, each integer between 10000 and 100000 that includes the block of digits 178 has five digits. Such an integer can be of the form $178 x y$ or of the form $x 178 y$ or of the form $x y 178$ for some digits $x$ an... |
deepscaler | [
{
"role": "user",
"content": "Given a sequence \\( a_{1}, a_{2}, \\cdots, a_{n}, \\cdots \\) such that \\( a_{1}=a_{2}=1 \\), \\( a_{3}=2 \\), and for any natural number \\( n \\), \\( a_{n} a_{n+1} a_{n+2} \\neq 1 \\). Additionally, it holds that \\( a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{1} + a_{n+1} + a_{n+2... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 26463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sphere, a cylinder with a square axial section, and a cone. The cylinder and cone have identical bases, and their heights equal the diameter of the sphere. How do the volumes of the cylinder, sphere, and cone compare to each other? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "3 : 2 : 1"
} | {
"split": "train",
"index": 26464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\\times 5$ square array of dots? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 26465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $$1+22+333+4444+55555+666666+7777777+88888888$$is congruent to $n$ modulo 9, where $0\\le n<9$, then what is the value of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 26466,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest number, \\( n \\), which is the product of 3 distinct primes where the mean of all its factors is not an integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "130"
} | {
"split": "train",
"index": 26467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate:<br/>$(1)-7+13-6+20$;<br/>$(2)-2^{3}+\\left(2-3\\right)-2\\times \\left(-1\\right)^{2023}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 26468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point \\( D \\) lies on side \\( BC \\) of triangle \\( ABC \\), and point \\( O \\) is located on segment \\( AD \\) with \\( AO : OD = 9 : 4 \\). A line passing through vertex \\( B \\) and point \\( O \\) intersects side \\( AC \\) at point \\( E \\) with \\( BO : OE = 5 : 6... | math | {
"style": "rule",
"ground_truth": "21 : 44"
} | {
"split": "train",
"index": 26469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a product originally priced at an unknown value is raised by 20% twice consecutively, and another product originally priced at an unknown value is reduced by 20% twice consecutively, ultimately selling both at 23.04 yuan each, determine the profit or loss situation w... | math | {
"style": "rule",
"ground_truth": "5.92"
} | {
"split": "train",
"index": 26470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function \\( f(x) = |x-1| + |x-3| + \\mathrm{e}^x \\) (where \\( x \\in \\mathbf{R} \\)), find the minimum value of the function. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6-2\\ln 2"
} | {
"split": "train",
"index": 26471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What number corresponds to the point \\( P \\) indicated on the given scale? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12.50"
} | {
"split": "train",
"index": 26472,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Margaret started a stamp collection. She collected 8 stamps the first day. Each subsequent day she collected 8 more stamps than she had collected the previous day. If she collected stamps for 5 consecutive days, what was the average number of stamps collected per day? Let's thi... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 26473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An odd function $f(x)$ defined on $R$ satisfies $f(x) = f(2-x)$. When $x \\in [0,1]$, $f(x) = ax^{3} + 2x + a + 1$. Find $f(2023)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 26474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of the solutions of: $|x| = 3(|x| - 2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"split": "train",
"index": 26475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram depicts a bike route through a park, along with the lengths of some of its segments in kilometers. What is the total length of the bike route in kilometers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 26476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest integer $k \\geq 2$ such that for every partition of the set $\\{2, 3,\\hdots, k\\}$ into two parts, at least one of these parts contains (not necessarily distinct) numbers $a$, $b$ and $c$ with $ab = c$. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 26477,
"solution": "\nWe are tasked to find the smallest integer \\( k \\geq 2 \\) such that every partition of the set \\( \\{2, 3, \\ldots, k\\} \\) into two parts results in at least one part containing numbers \\( a \\), \\( b \\), and \\( c \\) such that \\( ab = c \\).\n\nTo sol... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = 2\\sin(\\frac{1}{3}x - \\frac{π}{6})$, where $x \\in \\mathbb{R}$.\n\n(1) Find the value of $f(\\frac{5π}{4})$;\n\n(2) Let $\\alpha, \\beta \\in [0, \\frac{π}{2}], f(3\\alpha + \\frac{π}{2}) = \\frac{10}{13}, f(3\\beta + 2π) = \\frac{6}{5}$, find the ... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{65}"
} | {
"split": "train",
"index": 26478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the coefficient of $x^3$ in the product of the polynomials $$x^4 - 2x^3 + 3x^2 - 4x + 5$$ and $$3x^3 - 4x^2 + x + 6$$ after combining like terms? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 26479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "David has a very unique calculator that performs only two operations, the usual addition $(+)$ and another operation denoted by $*$, which satisfies:\n\n(i) $a * a = a$,\n\n(ii) $a * 0 = 2a$ and \n\n(iii) $(a * b) + (c * d) = (a + c) * (b + d)$,\n\nfor any integers $a$ and $b$.... | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 26480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the units digit of the product of all the odd positive integers between 10 and 110? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 26481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given six balls numbered 1, 2, 3, 4, 5, 6 and boxes A, B, C, D, each to be filled with one ball, with the conditions that ball 2 cannot be placed in box B and ball 4 cannot be placed in box D, determine the number of different ways to place the balls into the boxes. Let's think... | math | {
"style": "rule",
"ground_truth": "252"
} | {
"split": "train",
"index": 26482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the 2011 Zhejiang Province Pilot Module Examination, there were a total of 18 questions. Each examinee was required to choose 6 questions to answer. Examinee A would definitely not choose questions 1, 2, 9, 15, 16, 17, and 18, while Examinee B would definitely not choose que... | math | {
"style": "rule",
"ground_truth": "462"
} | {
"split": "train",
"index": 26483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four red candies and three green candies can be combined to make many different flavors. Flavors are different if the percent red is different, so 3 red / 0 green is the same flavor as 2 red / 0 green; and likewise 4 red / 2 green is the same flavor as 2 red / 1 green. If a fla... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 26484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest natural number in which all the digits are different and each pair of adjacent digits differs by 6 or 7. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60718293"
} | {
"split": "train",
"index": 26485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system on a plane, the parametric equations of curve $C$ are given by $\\begin{cases} x = 2\\cos θ \\\\ y = \\sqrt{3}\\sin θ \\end{cases}$ ($θ$ is the parameter). A polar coordinate system is established with the coordinate origin as the pole and t... | math | {
"style": "rule",
"ground_truth": "\\frac{120}{19}"
} | {
"split": "train",
"index": 26486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence with common difference 3. If $\\frac{S_{3n}}{S_n}$ is a constant that does not depend on $n,$ for all positive integers $n,$ then find the first term. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 26487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If A and B can only undertake the first three tasks, while the other three can undertake all four tasks, calculate the total number of different selection schemes for the team leader group to select four people from five volunteers to undertake four different tasks. Let's think... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 26488,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating 10 pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 26489,
"solution": "1. **Define Variables:**\n Let $b$ represent the number of blueberry jelly beans Camilla originally had, and $c$ represent the number of cherry jelly beans she originally had.\n\n2. **Set Up Initial Equations:**\n According to the problem, Camilla had twice as ... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = 2 \\ln(3x) + 8x$, calculate the value of $$\\lim_{\\Delta x \\to 0} \\frac {f(1-2\\Delta x)-f(1)}{\\Delta x}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-20"
} | {
"split": "train",
"index": 26490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square is divided into nine smaller squares of equal area. The center square is then divided into nine smaller squares of equal area and the pattern continues indefinitely. What fractional part of the figure is shaded? [asy]\nimport olympiad; size(150); defaultpen(linewidth(0... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 26491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given rectangle ABCD, AB=4, BC=8, points K and L are midpoints of BC and AD, respectively, and point M is the midpoint of KL. What is the area of the quadrilateral formed by the rectangle diagonals and segments KM and LM. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 26492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $\\cos(2A-B)+\\sin(A+B)=2$ and $AB=4$. What is $BC$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 26493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As shown in the picture, the knight can move to any of the indicated squares of the $8 \\times 8$ chessboard in 1 move. If the knight starts from the position shown, find the number of possible landing positions after 20 consecutive moves. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 26494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the set $M=\\{1,2,3,...,2020\\}.$ Find the smallest positive integer $k$ such that for any subset $A$ of $M$ with $k$ elements, there exist $3$ distinct numbers $a,b,c$ from $M$ such that $a+b, b+c$ and $c+a$ are all in $A.$ Let's think step by st... | math | {
"style": "rule",
"ground_truth": "1011"
} | {
"split": "train",
"index": 26495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all solutions $x$ of the inequality $$\\frac{5}{24} + \\left|x-\\frac{11}{48}\\right| < \\frac{5}{16}.$$Express your answer in interval notation, simplifying all fractions in your answer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(\\frac{1}{8},\\frac{1}{3}\\right)"
} | {
"split": "train",
"index": 26496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Perfectville, the streets are all $20$ feet wide and the blocks they enclose are all squares of side length $400$ feet, as shown. Sarah runs around the block on the $400$-foot side of the street, while Sam runs on the opposite side of the street. How many more feet than Sa... | math | {
"style": "rule",
"ground_truth": "160"
} | {
"split": "train",
"index": 26497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of white balls and red balls in a jar is in the ratio of $3:2$. If there are 9 white balls, how many red balls are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 26498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression $\\log_{y^8}{x^2}\\cdot\\log_{x^7}{y^3}\\cdot\\log_{y^5}{x^4}\\cdot\\log_{x^4}{y^5}\\cdot\\log_{y^3}{x^7}$ and express it as $b\\log_y{x}$ for some constant $b$. 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": 26499,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cirlce $\\Omega$ is inscribed in triangle $ABC$ with $\\angle BAC=40$ . Point $D$ is inside the angle $BAC$ and is the intersection of exterior bisectors of angles $B$ and $C$ with the common side $BC$ . Tangent form $D$ touches $\\Omega$ in $E$ . FInd $\\an... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 26500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $f(z)$ and $g(z)$ are polynomials in $z$, and the degree of $g(z)$ is less than the degree of $f(z)$. If the degree of $f(z)$ is two, what is the degree of $f(z)+g(z)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 26501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many points of intersection are there between the graphs of the following equations: \\begin{align*}\ny &=|2x + 5|, \\\\\ny &= -|3x - 2|\n\\end{align*} Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 26502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Meghana writes two (not necessarily distinct) primes $q$ and $r$ in base 10 next to each other on a blackboard, resulting in the concatenation of $q$ and $r$ (for example, if $q=13$ and $r=5$, the number on the blackboard is now 135). She notices that three more than the result... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 26503,
"solution": "Trying $p=2$, we see that $p^{2}-3=1$ is not the concatenation of two primes, so $p$ must be odd. Then $p^{2}-3$ is even. Since $r$ is prime and determines the units digit of the concatenation of $q$ and $r, r$ must be 2. Then $p^{2}$ will have units digit 5, which... |
deepscaler | [
{
"role": "user",
"content": "What integer $n$ satisfies $0\\le n<19$ and $$-200\\equiv n\\pmod{19}~?$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 26504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that each of the smaller rectangles has a shorter side of 7 feet and a longer side with a length three times that of the shorter side, and that two smaller rectangles are placed adjacent to each other by their longer sides and the third rectangle is placed adjacent by its... | math | {
"style": "rule",
"ground_truth": "588"
} | {
"split": "train",
"index": 26505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin 2x+2\\cos ^{2}x-1$. \n$(1)$ Find the smallest positive period of $f(x)$; \n$(2)$ When $x∈[0,\\frac{π}{2}]$, find the minimum value of $f(x)$ and the corresponding value of the independent variable $x$. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 26506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Monica tosses a fair 6-sided die. If the roll is a prime number, then she wins that amount of dollars (so that, for example, if she rolls 3, then she wins 3 dollars). If the roll is composite, she wins nothing. Otherwise, she loses 3 dollars. What is the expected value of her... | math | {
"style": "rule",
"ground_truth": "\\$1.17"
} | {
"split": "train",
"index": 26507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many positive integers $x$ is $(x-2)(x-4)(x-6) \\cdots(x-2016)(x-2018) \\leq 0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1514"
} | {
"split": "train",
"index": 26508,
"solution": "We count the positive integers $x$ for which the product $(x-2)(x-4)(x-6) \\cdots(x-2016)(x-2018)$ equals 0 and is less than 0 separately. The product equals 0 exactly when one of the factors equals 0. This occurs exactly when $x$ equals one of $2,4,6, \\ldots, 201... |
deepscaler | [
{
"role": "user",
"content": "Rationalize the denominator of $\\displaystyle \\frac{1}{\\sqrt[3]{3} - \\sqrt[3]{2}}$. With your answer in the form $\\displaystyle \\frac{\\sqrt[3]{A} + \\sqrt[3]{B} + \\sqrt[3]{C}}{D}$, and the fraction in lowest terms, what is $A + B + C + D$? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 26509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers from 1 to 9 are placed in the cells of a \\(3 \\times 3\\) table such that the sum of the numbers on one diagonal equals 7, and the sum on the other diagonal equals 21. What is the sum of the numbers in the five shaded cells? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 26510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $M(m, n)$ is any point on the circle $C: x^2+y^2-4x-14y+45=0$, find the maximum and minimum values of $\\frac{n-3}{m+2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2- \\sqrt{3}"
} | {
"split": "train",
"index": 26511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four students take an exam. Three of their scores are $70, 80,$ and $90$. If the average of their four scores is $70$, then what is the remaining score? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 26512,
"solution": "\n#### Detailed Steps:\n\n1. **Understanding the Problem:**\n We are given the scores of three students as $70, 80,$ and $90$. We also know that the average score of all four students is $70$. We need to find the score of the fourth student, denoted as $r$.\n\n2.... |
deepscaler | [
{
"role": "user",
"content": "The sum of the squares of the first ten binomial coefficients ${C}_{2}^{2}+{C}_{3}^{2}+{C}_{4}^{2}+\\cdots +{C}_{10}^{2}$ can be found. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 26513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Our water polo team has 15 members. I want to choose a starting team consisting of 7 players, one of whom will be the goalie (the other six positions are interchangeable, so the order in which they are chosen doesn't matter). In how many ways can I choose my starting team? Le... | math | {
"style": "rule",
"ground_truth": "45,\\!045"
} | {
"split": "train",
"index": 26514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at ran... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 26515,
"solution": "There are $12 \\cdot 11 = 132$ possible situations ($12$ choices for the initially assigned gate, and $11$ choices for which gate Dave's flight was changed to). We are to count the situations in which the two gates are at most $400$ feet apart.\n If we number the g... |
deepscaler | [
{
"role": "user",
"content": "For the equation $6 x^{2}=(2 m-1) x+m+1$ with respect to $x$, there is a root $\\alpha$ satisfying the inequality $-1988 \\leqslant \\alpha \\leqslant 1988$, and making $\\frac{3}{5} \\alpha$ an integer. How many possible values are there for $m$? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "2385"
} | {
"split": "train",
"index": 26516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of eight-digit integers comprising the eight digits from 1 to 8 such that \\( (i+1) \\) does not immediately follow \\( i \\) for all \\( i \\) that runs from 1 to 7. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16687"
} | {
"split": "train",
"index": 26517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle of radius $r$ has chords $\\overline{AB}$ of length $12$ and $\\overline{CD}$ of length $9$. When $\\overline{AB}$ and $\\overline{CD}$ are extended through $B$ and $C$, respectively, they intersect at point $P$, which is outside of the circle. If $\\angle{APD}=90^\\ci... | math | {
"style": "rule",
"ground_truth": "221"
} | {
"split": "train",
"index": 26518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Completely factor the following expression: $9x^2+3x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3x(3x+1)"
} | {
"split": "train",
"index": 26519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Leah and Jackson run for 45 minutes on a circular track. Leah runs clockwise at 200 m/min in a lane with a radius of 40 meters, while Jackson runs counterclockwise at 280 m/min in a lane with a radius of 55 meters, starting on the same radial line as Leah. Calculate how many ti... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 26520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a convex pentagon $FGHIJ$ where $\\angle F = \\angle G = 100^\\circ$. Let $FI = IJ = JG = 3$ and $GH = HF = 5$. Calculate the area of pentagon $FGHIJ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{4} + 24.62"
} | {
"split": "train",
"index": 26521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. For example, $d(1)=1,d(2)=2,$ and $d(12)=6$. (This function is known as the divisor function.) Let\\[f(n)=\\frac{d(n)}{\\sqrt [3]n}.\\]There is a unique positive integer $N$ such that $f(N... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 26522,
"solution": "1. **Understanding the function $f(n)$**: We define $f(n) = \\frac{d(n)}{\\sqrt[3]{n}}$, where $d(n)$ is the number of divisors of $n$. We need to find the integer $N$ such that $f(N) > f(n)$ for all $n \\neq N$.\n\n2. **Prime factorization and divisor function**: ... |
deepscaler | [
{
"role": "user",
"content": "Triangle $A B C$ has $A B=4, B C=3$, and a right angle at $B$. Circles $\\omega_{1}$ and $\\omega_{2}$ of equal radii are drawn such that $\\omega_{1}$ is tangent to $A B$ and $A C, \\omega_{2}$ is tangent to $B C$ and $A C$, and $\\omega_{1}$ is tangent to $\\omega_{2}$. Find ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{7}"
} | {
"split": "train",
"index": 26523,
"solution": "Denote by $r$ the common radius of $\\omega_{1}, \\omega_{2}$, and let $O_{1}, O_{2}$ be the centers of $\\omega_{1}$ and $\\omega_{2}$ respectively. Suppose $\\omega_{i}$ hits $A C$ at $B_{i}$ for $i=1,2$, so that $O_{1} O_{2}=B_{1} B_{2}=2 r$. Extend angle bisect... |
deepscaler | [
{
"role": "user",
"content": "Equilateral triangle $ABC$ has a side length of $12$. There are three distinct triangles $AD_1E_1$, $AD_2E_2$, and $AD_3E_3$, each congruent to triangle $ABC$, with $BD_1 = BD_2 = BD_3 = 6$. Find $\\sum_{k=1}^3(CE_k)^2$. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 26524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than 1000 are congruent to 6 (mod 11)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "91"
} | {
"split": "train",
"index": 26525,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Li was doing a subtraction problem and mistook the tens digit 7 for a 9 and the ones digit 3 for an 8, resulting in a difference of 76. The correct difference is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 26526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $[x]$ represent the greatest integer less than or equal to $x$. Given that the natural number $n$ satisfies $\\left[\\frac{1}{15}\\right] + \\left[\\frac{2}{15}\\right] + \\left[\\frac{3}{15}\\right] + \\cdots + \\left[\\frac{n-1}{15}\\right] + \\left[\\frac{n}{15}\\right] ... | math | {
"style": "rule",
"ground_truth": "253"
} | {
"split": "train",
"index": 26527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle has three different integer side lengths and a perimeter of 30 units. What is the maximum length of any one side? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 26528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quadratic $8x^2+12x-14$ has two real roots. What is the sum of the squares of these roots? Express your answer as a common fraction in lowest terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{23}{4}"
} | {
"split": "train",
"index": 26529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles are inscribed in an angle of 60 degrees and they touch each other. The radius of the smaller circle is 24. What is the radius of the larger circle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 26530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify and write the result as a common fraction: $$\\sqrt{\\sqrt[3]{\\sqrt{\\frac{1}{4096}}}}$$ 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": 26531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the arithmetic sequence {a<sub>n</sub>} satisfies a<sub>3</sub> − a<sub>2</sub> = 3, a<sub>2</sub> + a<sub>4</sub> = 14.\n(I) Find the general term formula for {a<sub>n</sub>};\n(II) Let S<sub>n</sub> be the sum of the first n terms of the geometric sequence {b<sub>n</sub... | math | {
"style": "rule",
"ground_truth": "-86"
} | {
"split": "train",
"index": 26532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Ebbinghaus forgetting curve describes the law of human brain forgetting new things. Based on this, a research team found that after learning course $A$, 20% of the memorized content is forgotten every week. In order to ensure that the memorized content does not fall below $... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 26533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle with side length 10 cm is inscribed in a circle. The triangle's side is the diameter of the circle. In two opposite sectors of the circle, there are small shaded regions. If the sum of the areas of the two small shaded regions is in the form \\(a\\pi - b... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 26534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rental company owns 100 cars. When the monthly rent for each car is set at 3000 yuan, all cars can be rented out. For every 50 yuan increase in the monthly rent per car, there will be one more car that is not rented out. The maintenance cost for each rented car is 150 yuan pe... | math | {
"style": "rule",
"ground_truth": "4050"
} | {
"split": "train",
"index": 26535,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the convex quadrilateral \\(ABCD\\): \\(AB = AC = AD = BD\\) and \\(\\angle BAC = \\angle CBD\\). Find \\(\\angle ACD\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 26536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A tangent line is drawn to the moving circle $C: x^2 + y^2 - 2ay + a^2 - 2 = 0$ passing through the fixed point $P(2, -1)$. If the point of tangency is $T$, then the minimum length of the line segment $PT$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "\\sqrt {2}"
} | {
"split": "train",
"index": 26537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $r = \\sin \\theta$ is also seen in a circular form but pertains to a vertical alignment. Find the smallest value of $t$ so that when $r = \\sin \\theta$ is plotted for $0 \\le \\theta \\le t,$ the resulting graph forms a complete circle. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"split": "train",
"index": 26538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression: $(a-\\frac{3a}{a+1}$) $÷\\frac{{a}^{2}-4a+4}{a+1}$, then choose a number you like from $-2$, $-1$, and $2$ to substitute for $a$ and calculate the value. 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": 26539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function \\( f:\\{1,2, \\cdots, 10\\} \\rightarrow\\{1,2,3,4,5\\} \\), and for each \\( k=1,2, \\cdots, 9 \\), it is true that \\( |f(k+1)-f(k)| \\geq 3 \\). Find the number of functions \\( f \\) that satisfy these conditions. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 26540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a, b, c, d, e, f, p, q$ are Arabic numerals and $b > c > d > a$, the difference between the four-digit numbers $\\overline{c d a b}$ and $\\overline{a b c d}$ is a four-digit number of the form $\\overline{p q e f}$. If $\\overline{e f}$ is a perfect square and $\\o... | math | {
"style": "rule",
"ground_truth": "1983"
} | {
"split": "train",
"index": 26541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the representation of three two-digit numbers, there are no zeros, and in each of them, both digits are different. Their sum is 41. What could their sum be if the digits in them are swapped? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "113"
} | {
"split": "train",
"index": 26542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of \\((4 + 44 + 444) \\div 4\\) is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "123"
} | {
"split": "train",
"index": 26543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m=n^{4}+x$, where $n \\in \\mathbf{N}$ and $x$ is a two-digit positive integer. Which value of $x$ ensures that $m$ is always a composite number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 26544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The radius of the circumcircle of an acute triangle \\( ABC \\) is 1. It is known that the center of the circle passing through the vertices \\( A \\), \\( C \\), and the orthocenter of triangle \\( ABC \\) lies on this circumcircle. Find the length of side \\( AC \\). Let's th... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 26545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given in the tetrahedron P-ABC, PA is perpendicular to the plane ABC, AB=AC=PA=2, and in triangle ABC, ∠BAC=120°, then the volume of the circumscribed sphere of the tetrahedron P-ABC is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "\\frac{20\\sqrt{5}\\pi}{3}"
} | {
"split": "train",
"index": 26546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petrov booked an apartment in a newly built house, which has five identical entrances. Initially, the entrances were numbered from left to right, and Petrov's apartment number was 636. Later, the developer changed the numbering to the opposite direction (right to left, as shown... | math | {
"style": "rule",
"ground_truth": "985"
} | {
"split": "train",
"index": 26547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be integers such that $ab = 144.$ Find the minimum value of $a + b.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-145"
} | {
"split": "train",
"index": 26548,
"solution": ""
} |
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