data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "There are five unmarked envelopes on a table, each containing a letter for a different person. If the mail is randomly distributed to these five people, with each person getting one letter, what is the probability that exactly three people receive the correct letter? Let's thin... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 17749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\left(\\sqrt{625681 + 1000} - \\sqrt{1000}\\right)^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "626681 - 2 \\cdot \\sqrt{626681} \\cdot 31.622776601683793 + 1000"
} | {
"split": "train",
"index": 17750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{T}$ be the set $\\lbrace1,2,3,\\ldots,12\\rbrace$. Let $m$ be the number of sets of two non-empty disjoint subsets of $\\mathcal{T}$. Calculate the remainder when $m$ is divided by $1000$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "625"
} | {
"split": "train",
"index": 17751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many five-digit natural numbers are divisible by 9, where the last digit is greater than the second last digit by 2? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "800"
} | {
"split": "train",
"index": 17752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the extended Number Wall, numbers are added from adjacent blocks directly below to form the sum in the block above. What number will be in the block labeled '$n$' in this configuration? \n```plaintext\n __n__\n __ __ \n __ __ \n 8 5 3 2\n``` Let's think step... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 17753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit numbers have the property that the second digit is the average of the first and third digits, and the digits are all even? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 17754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many odd divisors does $10!$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 17755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If you write down twice in a row the grade I received in school for Latin, you will get my grandmother's age. What will you get if you divide this age by the number of my kittens? Imagine - you will get my morning grade, increased by fourteen-thirds.\n\nHow old is my grandmothe... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"split": "train",
"index": 17756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a=\\log_8 225$ and $b=\\log_2 15$, then Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "a=2b/3"
} | {
"split": "train",
"index": 17757,
"solution": "1. **Express $a$ and $b$ in terms of logarithms with a common base:**\n Given $a = \\log_8 225$ and $b = \\log_2 15$, we can use the change of base formula to express both logarithms in terms of base $2$:\n \\[\n a = \\log_8 225 = \\frac{\\log_2 225}{\\log_2 ... |
deepscaler | [
{
"role": "user",
"content": "Among all the five-digit numbers formed without repeating digits using 0, 1, 2, 3, and 4, if arranged in ascending order, what position would the number 12340 occupy? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 17758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In isosceles right triangle $ABC$, hypotenuse $AC = 2\\sqrt{2}$. Find the dot product of $\\overrightarrow{AB}$ and $\\overrightarrow{CA}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 17759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Teams A, B, and C need to complete two projects, $A$ and $B$. The workload of project $B$ is $\\frac{1}{4}$ more than the workload of project $A$. If teams A, B, and C work alone, they can finish project $A$ in 20 days, 24 days, and 30 days respectively. To complete these two p... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 17760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular triangular prism \\( A B C A_{1} B_{1} C_{1} \\) is inscribed in a sphere, where \\( A B C \\) is the base and \\( A A_{1}, B B_{1}, C C_{1} \\) are the lateral edges. The segment \\( C D \\) is the diameter of this sphere, and point \\( K \\) is the midpoint of the e... | math | {
"style": "rule",
"ground_truth": "9\\sqrt{2}"
} | {
"split": "train",
"index": 17761,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The tail of a 1-mile long train exits a tunnel exactly 3 minutes after the front of the train entered the tunnel. If the train is moving 60 miles per hour, how many miles long is the tunnel? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\text{ miles}"
} | {
"split": "train",
"index": 17762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the difference between each number in a row and the number immediately to its left in the given diagram is the same, and the quotient of each number in a column divided by the number immediately above it is the same, then $a + b \\times c =\\quad$ Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "540"
} | {
"split": "train",
"index": 17763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In acute triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $b=2$, $B= \\frac{π}{3}$, and $c\\sin A= \\sqrt{3}a\\cos C$, find the area of $\\triangle ABC$. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 17764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points \\( A, B, C, \\) and \\( D \\) lie on a straight line in that order. For a point \\( E \\) outside the line,\n\n\\[ \\angle AEB = \\angle BEC = \\angle CED = 45^\\circ. \\]\n\nLet \\( F \\) be the midpoint of segment \\( AC \\), and \\( G \\) be the midpoint of segment \... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 17765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Container I holds 8 red balls and 4 green balls; containers II and III each hold 2 red balls and 4 green balls. A container is selected at random and then a ball is randomly selected from that container. What is the probability that the ball selected is green? Express your answ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"split": "train",
"index": 17766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a regular tetrahedron has all edge lengths of $\\sqrt {2}$, and all four vertices are on the same spherical surface, find the surface area of this sphere. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\pi"
} | {
"split": "train",
"index": 17767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graphs of $y = -|x-a| + b$ and $y = |x-c| + d$ intersect at points $(2,5)$ and $(8,3)$. Find $a+c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 17768,
"solution": "1. **Identify the vertex of each graph:**\n - The graph of $y = -|x-a| + b$ has a vertex at $(a, b)$, where the graph reaches its maximum value $b$.\n - The graph of $y = |x-c| + d$ has a vertex at $(c, d)$, where the graph reaches its minimum value $d$.\n\n2. ... |
deepscaler | [
{
"role": "user",
"content": "Suppose a regular tetrahedron \\( P-ABCD \\) has all edges equal in length. Using \\(ABCD\\) as one face, construct a cube \\(ABCD-EFGH\\) on the other side of the regular tetrahedron. Determine the cosine of the angle between the skew lines \\( PA \\) and \\( CF \\). Let's thi... | math | {
"style": "rule",
"ground_truth": "\\frac{2 + \\sqrt{2}}{4}"
} | {
"split": "train",
"index": 17769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An investor put $\\$12,\\!000$ into a three-month savings certificate that paid a simple annual interest rate of $8\\%$. After three months, she reinvested the resulting amount in another three-month certificate. The investment value after another three months was $\\$12,\\!435... | math | {
"style": "rule",
"ground_truth": "6.38\\%"
} | {
"split": "train",
"index": 17770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $A B C$ has perimeter 1. Its three altitudes form the side lengths of a triangle. Find the set of all possible values of $\\min (A B, B C, C A)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(\\frac{3-\\sqrt{5}}{4}, \\frac{1}{3}\\right]"
} | {
"split": "train",
"index": 17771,
"solution": "Let $a, b, c$ denote the side lengths $B C, C A$, and $A B$, respectively. Without loss of generality, assume $a \\leq b \\leq c$; we are looking for the possible range of $a$. First, note that the maximum possible value of $a$ is $\\frac{1}{3}$, which occurs when ... |
deepscaler | [
{
"role": "user",
"content": "If several students participate in three competitions where the champion earns 5 points, the runner-up earns 3 points, and the third-place finisher earns 1 point, and there are no ties, what is the minimum score a student must achieve to definitely have a higher score than any ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 17772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many $n$ in $\\{1, 2, 3, ..., 100 \\}$ is the tens digit of $n^2$ odd? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 17773,
"solution": "1. **Express $n$ in terms of its digits**: Let $n = 10a + b$, where $a$ is the tens digit and $b$ is the units digit of $n$. Then $n^2 = (10a + b)^2$.\n\n2. **Expand $n^2$**: \n \\[\n n^2 = (10a + b)^2 = 100a^2 + 20ab + b^2\n \\]\n - The term $100a^2$ contr... |
deepscaler | [
{
"role": "user",
"content": "What is the next term in the geometric sequence $$2, 6x, 18x^2, 54x^3, \\ldots ?$$ Express your answer in terms of $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "162x^4"
} | {
"split": "train",
"index": 17774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the determinant of the following matrix:\n\\[\n\\begin{vmatrix} 3 & 1 & 0 \\\\ 8 & 5 & -2 \\\\ 3 & -1 & 6 \\end{vmatrix}.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "138"
} | {
"split": "train",
"index": 17775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $F$ is taken in side $AD$ of square $ABCD$. At $C$ a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. The area of $ABCD$ is $256$ square inches and the area of $\\triangle CEF$ is $200$ square inches. Then the number of inches in $BE$ is: Let's think step by ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 17776,
"solution": "1. **Identify the properties of the square**: Given that $ABCD$ is a square with an area of $256$ square inches, we can find the side length of the square:\n \\[\n s^2 = 256 \\implies s = 16 \\text{ inches}\n \\]\n Therefore, $AB = BC = CD = DA = 16$ inches... |
deepscaler | [
{
"role": "user",
"content": "The diagonal lengths of a rhombus are 18 units and 26 units. Calculate both the area and the perimeter of the rhombus. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20\\sqrt{10}"
} | {
"split": "train",
"index": 17777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( P_{1} \\) and \\( P_{2} \\) be any two different points on the ellipse \\(\\frac{x^{2}}{9}+\\frac{y^{2}}{4}=1\\), and let \\( P \\) be a variable point on the circle with diameter \\( P_{1} P_{2} \\). Find the maximum area of the circle with radius \\( OP \\). Let's thi... | math | {
"style": "rule",
"ground_truth": "13 \\pi"
} | {
"split": "train",
"index": 17778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $a_1$, $a_2$, $a_3$ are not all zero, and positive numbers $x$, $y$ satisfy $x+y=2$. Let the maximum value of $$\\frac {xa_{1}a_{2}+ya_{2}a_{3}}{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$$ be $M=f(x,y)$, then the minimum value of $M$ is \\_\\_\\_\\_\\_\\_. Let's think s... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {2}}{2}"
} | {
"split": "train",
"index": 17779,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given triangle $ABC$ with sides $AB = 7$, $AC = 8$, and $BC = 5$, find the value of \n\\[\\frac{\\cos \\frac{A - B}{2}}{\\sin \\frac{C}{2}} - \\frac{\\sin \\frac{A - B}{2}}{\\cos \\frac{C}{2}}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{7}"
} | {
"split": "train",
"index": 17780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The three-tiered \"pyramid\" shown in the image is built from $1 \\mathrm{~cm}^{3}$ cubes and has a surface area of $42 \\mathrm{~cm}^{2}$. We made a larger \"pyramid\" based on this model, which has a surface area of $2352 \\mathrm{~cm}^{2}$. How many tiers does it have? Let's... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 17781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The congruent sides of an isosceles triangle are each 5 cm long, and the perimeter is 17 cm. In centimeters, what is the length of the base? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 17782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S_{1}, S_{2}, \\ldots, S_{10}$ be the first ten terms of an arithmetic progression (A.P.) of positive integers. If $S_{1} + S_{2} + \\ldots + S_{10} = 55$ and $\\left(S_{10} - S_{8}\\right) + \\left(S_{9} - S_{7}\\right) + \\ldots + \\left(S_{3} - S_{1}\\right) = d$, find ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 17783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The MathMatters competition consists of 10 players $P_1$ , $P_2$ , $\\dots$ , $P_{10}$ competing in a ladder-style tournament. Player $P_{10}$ plays a game with $P_9$ : the loser is ranked 10th, while the winner plays $P_8$ . The loser of that game is ranked 9th, whi... | math | {
"style": "rule",
"ground_truth": "512"
} | {
"split": "train",
"index": 17784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of the areas of two squares is $\\frac{50}{98}$. After rationalizing the denominator, express the simplified form of the ratio of their side lengths in the form $\\frac{a \\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. Find the sum $a+b+c$. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 17785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ordered triples of integers $(a,b,c)$, with $a \\ge 2$, $b\\ge 1$, and $c \\ge 0$, satisfy both $\\log_a b = c^{2005}$ and $a + b + c = 2005$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 17786,
"solution": "We are given two conditions:\n1. $\\log_a b = c^{2005}$\n2. $a + b + c = 2005$\n\nFrom the first condition, we can rewrite it using the definition of logarithms:\n\\[ a^{c^{2005}} = b \\]\n\nWe will analyze the possible values of $c$ and their implications on $a$ a... |
deepscaler | [
{
"role": "user",
"content": "\\(ABC\\) is a triangle with \\(AB = 15\\), \\(BC = 14\\), and \\(CA = 13\\). The altitude from \\(A\\) to \\(BC\\) is extended to meet the circumcircle of \\(ABC\\) at \\(D\\). Find \\(AD\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{63}{4}"
} | {
"split": "train",
"index": 17787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\ln (ax+1)+ \\frac {x^{3}}{3}-x^{2}-ax(a∈R)$,\n(1) Find the range of values for the real number $a$ such that $y=f(x)$ is an increasing function on $[4,+∞)$;\n(2) When $a\\geqslant \\frac {3 \\sqrt {2}}{2}$, let $g(x)=\\ln [x^{2}(ax+1)]+ \\frac {x^{3}}... | math | {
"style": "rule",
"ground_truth": "\\ln 2- \\frac {2}{3}"
} | {
"split": "train",
"index": 17788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alli rolls a standard 8-sided die twice. What is the probability of rolling integers that differ by 3 on her first two rolls? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{32}"
} | {
"split": "train",
"index": 17789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We can view these conditions as a geometry diagram as seen below. So, we know that $\\frac{e}{f}=\\frac{3}{4}$ (since $e=a-b=\\frac{3}{4} c-\\frac{3}{4} d=\\frac{3}{4} f$ and we know that $\\sqrt{e^{2}+f^{2}}=15$ (since this is $\\left.\\sqrt{a^{2}+c^{2}}-\\sqrt{b^{2}+d^{2}}\\r... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 17790,
"solution": "The value of $ef$ is 108."
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can 10 people be seated in a row of chairs if four of the people, Alice, Bob, Charlie, and Dana, refuse to sit in four consecutive seats? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3507840"
} | {
"split": "train",
"index": 17791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The organizers of a ping-pong tournament have only one table. They call two participants to play, who have not yet played against each other. If after the game the losing participant suffers their second defeat, they are eliminated from the tournament (since there are no ties i... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 17792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, $XY = 5$, $XZ = 7$, and $YZ = 8$. The medians $XM$, $YN$, and $ZO$ of triangle $XYZ$ intersect at the centroid $G$. Let the projections of $G$ onto $YZ$, $XZ$, and $XY$ be $P$, $Q$, and $R$ respectively. Find $GP + GQ + GR$. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "\\frac{131\\sqrt{3}}{42}"
} | {
"split": "train",
"index": 17793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A Ferris wheel rotates at a constant speed, completing one revolution every 12 minutes. The lowest point of the Ferris wheel is 2 meters above the ground, and the highest point is 18 meters above the ground. Let P be a fixed point on the circumference of the Ferris wheel. Start... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 17794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For which integers $n \\in\\{1,2, \\ldots, 15\\}$ is $n^{n}+1$ a prime number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1, 2, 4"
} | {
"split": "train",
"index": 17795,
"solution": "$n=1$ works. If $n$ has an odd prime factor, you can factor, and this is simulated also by $n=8$: $$a^{2 k+1}+1=(a+1)\\left(\\sum_{i=0}^{2 k}(-a)^{i}\\right)$$ with both parts larger than one when $a>1$ and $k>0$. So it remains to check 2 and 4, which work. Thus th... |
deepscaler | [
{
"role": "user",
"content": "Four points \\( B, A, E, L \\) are on a straight line. \\( G \\) is a point off the line such that \\(\\angle B A G = 120^\\circ\\) and \\(\\angle G E L = 80^\\circ\\). If the reflex angle at \\( G \\) is \\( x^\\circ \\), then \\( x \\) equals: Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "340"
} | {
"split": "train",
"index": 17796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mr. Fat needs 20 minutes to eat a pound of cereal, while Mr. Thin needs 30 minutes. If they eat together, how long does it take for them to finish off three pounds of cereal? Express your answer in minutes. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 17797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $\\left(\\frac{1}{2}\\right)^{4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"split": "train",
"index": 17798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $\\alpha$ is inversely proportional to $\\beta$. If $\\alpha = -3$ when $\\beta = -6$, find $\\alpha$ when $\\beta = 8$. Express your answer as a fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"split": "train",
"index": 17799,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given three integers \\( x, y, z \\) satisfying \\( x + y + z = 100 \\) and \\( x < y < 2z \\), what is the minimum value of \\( z \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 17800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the markings on the number line are equally spaced, what is the number $\\text{y}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 17801,
"solution": "1. **Identify the total distance and number of steps**: The problem states that five steps are taken to get from $0$ to $20$ on a number line where the markings are equally spaced.\n\n2. **Calculate the length of each step**: Since there are five equal steps coveri... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2\\cos^2x+2\\sqrt{3}\\sin x\\cos x+a$, and when $x\\in\\left[0, \\frac{\\pi}{2}\\right]$, the minimum value of $f(x)$ is $2$,\n\n$(1)$ Find the value of $a$, and determine the intervals where $f(x)$ is monotonically increasing;\n\n$(2)$ First, transform... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 17802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three $1 \\times 1 \\times 1$ cubes are joined face to face in a single row and placed on a table, and have a total of 11 exposed $1 \\times 1$ faces. Determine the number of exposed $1 \\times 1$ faces when sixty $1 \\times 1 \\times 1$ cubes are joined face to face in a singl... | math | {
"style": "rule",
"ground_truth": "182"
} | {
"split": "train",
"index": 17803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given sets $M=\\{1, 2, a^2 - 3a - 1 \\}$ and $N=\\{-1, a, 3\\}$, and the intersection of $M$ and $N$ is $M \\cap N = \\{3\\}$, find the set of all possible real values for $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\{4\\}"
} | {
"split": "train",
"index": 17804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $M \\Theta N$ represent the remainder when the larger of $M$ and $N$ is divided by the smaller one. For example, $3 \\Theta 10 = 1$. For non-zero natural numbers $A$ less than 40, given that $20 \\Theta(A \\bigodot 20) = 7$, find $A$. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 17805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "**p22.** Consider the series $\\{A_n\\}^{\\infty}_{n=0}$ , where $A_0 = 1$ and for every $n > 0$ , $$ A_n = A_{\\left[ \\frac{n}{2023}\\right]} + A_{\\left[ \\frac{n}{2023^2}\\right]}+A_{\\left[ \\frac{n}{2023^3}\\right]}, $$ where $[x]$ denotes the largest integer v... | math | {
"style": "rule",
"ground_truth": "653"
} | {
"split": "train",
"index": 17806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively, satisfying $2a\\sin A = (2\\sin B - \\sqrt{3}\\sin C)b + (2\\sin C - \\sqrt{3}\\sin B)c$.\n\n(1) Find the measure of angle $A$.\n\n(2) If $a=2$ and $b=2\\sqrt{3}$, find the are... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 17807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sqrt {3}\\sin x+\\cos x= \\frac {2}{3}$, find the value of $\\tan (x+ \\frac {7\\pi}{6})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{4}"
} | {
"split": "train",
"index": 17808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the integers \\(1, 2, \\ldots, 2004\\) can be represented as \\(\\frac{mn+1}{m+n}\\) for positive integers \\(m\\) and \\(n\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"split": "train",
"index": 17809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a certain square, two vertices lie on the line $y = 2x - 17,$ and the other two vertices lie on the parabola $y = x^2.$ Find the smallest possible area of the square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 17810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $\\frac{~\\frac{2}{5}~}{\\frac{3}{7}}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{14}{15}"
} | {
"split": "train",
"index": 17811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum of the $2023$ roots of $(x-1)^{2023} + 2(x-2)^{2022} + 3(x-3)^{2021} + \\cdots + 2022(x-2022)^2 + 2023(x-2023)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"split": "train",
"index": 17812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Simplify: $\\dfrac {\\tan (3\\pi-\\alpha)\\cos (2\\pi-\\alpha)\\sin (-\\alpha+ \\dfrac {3\\pi}{2})}{\\cos (-\\alpha-\\pi)\\sin (-\\pi+\\alpha)\\cos (\\alpha+ \\dfrac {5\\pi}{2})}$; \n(2) Given $\\tan \\alpha= \\dfrac {1}{4}$, find the value of $\\dfrac {1}{2\\cos ^{2}\\alp... | math | {
"style": "rule",
"ground_truth": "\\dfrac {17}{20}"
} | {
"split": "train",
"index": 17813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = -x^3 + ax^2 - 4$ has an extremum at $x = 2$, and $m, n \\in [-1, 1]$, then the minimum value of $f(m) + f'(n)$ is \\_\\_\\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-13"
} | {
"split": "train",
"index": 17814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ABCD$, a rectangle with $AB = 12$ and $BC = 16$, is the base of pyramid $P$, which has a height of $24$. A plane parallel to $ABCD$ is passed through $P$, dividing $P$ into a frustum $F$ and a smaller pyramid $P'$. Let $X$ denote the center of the circumsphere of $F$, and let ... | math | {
"style": "rule",
"ground_truth": "177"
} | {
"split": "train",
"index": 17815,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\frac{11! - 10!}{9!}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 17816,
"solution": "1. Start by rewriting the expression $\\dfrac{11!-10!}{9!}$ using the definition of factorial:\n \\[\n \\frac{11! - 10!}{9!} = \\frac{11 \\times 10! - 10!}{9!}\n \\]\n\n2. Factor out $10!$ from the numerator:\n \\[\n \\frac{11 \\times 10! - 10!}{9!} = \\f... |
deepscaler | [
{
"role": "user",
"content": "In rectangle \\(ABCD\\), a point \\(E\\) is marked on the extension of side \\(CD\\) beyond point \\(D\\). The bisector of angle \\(ABC\\) intersects side \\(AD\\) at point \\(K\\), and the bisector of angle \\(ADE\\) intersects the extension of side \\(AB\\) at point \\(M\\). ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{55}"
} | {
"split": "train",
"index": 17817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(ABCD\\) be an isosceles trapezoid such that \\(AD = BC\\), \\(AB = 3\\), and \\(CD = 8\\). Let \\(E\\) be a point in the plane such that \\(BC = EC\\) and \\(AE \\perp EC\\). Compute \\(AE\\). Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{6}"
} | {
"split": "train",
"index": 17818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F(c,0)$ is the right focus of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > b > 0)$, a line perpendicular to one asymptote of the hyperbola passes through $F$ and intersects the two asymptotes at points $A$ and $B$. $O$ is the coordinate origin, and th... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"split": "train",
"index": 17819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 17820,
"solution": "1. **Understanding the Problem:**\n We are given a line of balls consisting of one red ball and $N$ green balls, where $N$ is a multiple of 5. We need to find the probability $P(N)$ that at least $\\frac{3}{5}$ of the green balls are on the same side of the red b... |
deepscaler | [
{
"role": "user",
"content": "Six soccer teams are competing in a tournament in Waterloo. Every team is to play three games, each against a different team. How many different schedules are possible? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 17821,
"solution": "Before we answer the given question, we determine the number of ways of choosing 3 objects from 5 objects and the number of ways of choosing 2 objects from 5 objects. Consider 5 objects labelled B, C, D, E, F. The possible pairs are: BC, BD, BE, BF, CD, CE, CF, DE,... |
deepscaler | [
{
"role": "user",
"content": "The graph of the equation $10x + 270y = 2700$ is drawn on graph paper where each square represents one unit in each direction. A second line defined by $x + y = 10$ also passes through the graph. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 17822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le x \\le 2, \\\\\n2x-b &\\text{ if } x <-2.\n\\end{array}\n\\right.\\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting yo... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 17823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular table $ 9$ rows $ \\times$ $ 2008$ columns is fulfilled with numbers $ 1$ , $ 2$ , ..., $ 2008$ in a such way that each number appears exactly $ 9$ times in table and difference between any two numbers from same column is not greater than $ 3$ . What i... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 17824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when the sum $1 + 7 + 13 + 19 + \\cdots + 253 + 259$ is divided by $6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 17825,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Olga purchases a rectangular mirror (the shaded region) that fits exactly inside a frame. The outer perimeter of the frame measures 60 cm by 80 cm. The width of each side of the frame is 10 cm. What is the area of the mirror?\n\n[asy]\nunitsize(0.15inch);\ndefaultpen(black);\nd... | math | {
"style": "rule",
"ground_truth": "2400 \\mbox{ cm}^2"
} | {
"split": "train",
"index": 17826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The square with vertices $(-a, -a), (a, -a), (-a, a), (a, a)$ is cut by the line $y = x/2$ into congruent quadrilaterals. The perimeter of one of these congruent quadrilaterals divided by $a$ equals what? Express your answer in simplified radical form. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "4+\\sqrt{5}"
} | {
"split": "train",
"index": 17827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can I choose 3 captains from a team of 11 people? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 17828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $23... | math | {
"style": "rule",
"ground_truth": "91"
} | {
"split": "train",
"index": 17829,
"solution": "1. **Calculate the cost of one T-shirt**: \n Given that each T-shirt costs $5 more than a pair of socks, and a pair of socks costs $4, the cost of one T-shirt is:\n \\[\n 4 + 5 = 9 \\text{ dollars}\n \\]\n\n2. **Calculate the total cost for one member**: ... |
deepscaler | [
{
"role": "user",
"content": "Write 1 as a sum of 4 distinct unit fractions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{7}+\\frac{1}{42}"
} | {
"split": "train",
"index": 17830,
"solution": "$\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{7}+\\frac{1}{42}$"
} |
deepscaler | [
{
"role": "user",
"content": "A woman invests in a property for $12,000 with the aim of receiving a $6\\%$ return on her investment after covering all expenses including taxes and insurance. She pays $360 annually in taxes and $240 annually for insurance. She also keeps aside $10\\%$ of each month's rent fo... | math | {
"style": "rule",
"ground_truth": "122.22"
} | {
"split": "train",
"index": 17831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the graph of the function $f(x)=ax^3+bx^2+c$ passes through the point $(0,1)$, and the equation of the tangent line at $x=1$ is $y=x$.\n\n(1) Find the analytic expression of $y=f(x)$;\n\n(2) Find the extreme values of $y=f(x)$. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "\\frac{23}{27}"
} | {
"split": "train",
"index": 17832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(\\sin x,\\cos x),\\overrightarrow{b}=(2\\sqrt{3}\\cos x-\\sin x,\\cos x)$, and $f(x)=\\overrightarrow{a}\\cdot\\overrightarrow{b}$. \n$(1)$ Find the interval where the function $f(x)$ is monotonically decreasing. \n$(2)$ If $f(x_0)=\\frac{2... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}-3\\sqrt{2}}{6}"
} | {
"split": "train",
"index": 17833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( D \\) be a point inside \\( \\triangle ABC \\) such that \\( \\angle BAD = \\angle BCD \\) and \\( \\angle BDC = 90^\\circ \\). If \\( AB = 5 \\), \\( BC = 6 \\), and \\( M \\) is the midpoint of \\( AC \\), find the length of \\( DM \\). Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{11}}{2}"
} | {
"split": "train",
"index": 17834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the xy-plane with a rectangular coordinate system, let vector $\\overrightarrow {a}$ = (cosα, sinα) and vector $\\overrightarrow {b}$ = (sin(α + π/6), cos(α + π/6)), where 0 < α < π/2.\n(1) If $\\overrightarrow {a}$ is parallel to $\\overrightarrow {b}$, find the value of α.... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6} - 7\\sqrt{2}}{20}"
} | {
"split": "train",
"index": 17835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Semicircles of diameter 2'' are lined up as shown. What is the area, in square inches, of the shaded region in a 1-foot length of this pattern? Express your answer in terms of $\\pi$.\n\n[asy]import graph;\nsize(101);\n\npath tophalf = Arc((0,0),1,180,0) -- Arc((2,0),1,180,0) -... | math | {
"style": "rule",
"ground_truth": "6\\pi"
} | {
"split": "train",
"index": 17836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fast train with a weight of $P=150$ tons travels at a maximum speed of $v=72 \\frac{\\text{km}}{\\text{hour}}$ on a horizontal track with a friction coefficient of $\\rho=0,005$. What speed can it reach on a track with the same friction conditions but with an incline having $... | math | {
"style": "rule",
"ground_truth": "10.3"
} | {
"split": "train",
"index": 17837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is the exact square.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1972"
} | {
"split": "train",
"index": 17838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a heart and the second card is a 10? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{52}"
} | {
"split": "train",
"index": 17839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when \\(2,468,135,790\\) is divided by \\(99\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 17840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mickey's age is 4 years less than $300\\%$ of Jerry's age. If Mickey is 14 years old, how old is Jerry? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 17841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the terminating decimal expansion of $\\frac{13}{200}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.052"
} | {
"split": "train",
"index": 17842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer with exactly 12 positive integer divisors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 17843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $10$ seats in each of $10$ rows of a theatre and all the seats are numbered. What is the probablity that two friends buying tickets independently will occupy adjacent seats? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{55}"
} | {
"split": "train",
"index": 17844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$A B C$ is an acute triangle with incircle $\\omega$. $\\omega$ is tangent to sides $\\overline{B C}, \\overline{C A}$, and $\\overline{A B}$ at $D, E$, and $F$ respectively. $P$ is a point on the altitude from $A$ such that $\\Gamma$, the circle with diameter $\\overline{A P}$... | math | {
"style": "rule",
"ground_truth": "\\frac{675}{4}"
} | {
"split": "train",
"index": 17845,
"solution": "By the Law of Sines we have $\\sin \\angle A=\\frac{X Y}{A P}=\\frac{4}{5}$. Let $I, T$, and $Q$ denote the center of $\\omega$, the point of tangency between $\\omega$ and $\\Gamma$, and the center of $\\Gamma$ respectively. Since we are told $A B C$ is acute, we ... |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be five-digit palindromes (without leading zeroes) such that $a<b$ and there are no other five-digit palindromes strictly between $a$ and $b$. What are all possible values of $b-a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100, 110, 11"
} | {
"split": "train",
"index": 17846,
"solution": "Let $\\overline{x y z y x}$ be the digits of the palindrome $a$. There are three cases. If $z<9$, then the next palindrome greater than $\\overline{x y z y x}$ is $\\overline{x y(z+1) y x}$, which differs by 100. If $z=9$ but $y<9$, then the next palindrome up is $... |
deepscaler | [
{
"role": "user",
"content": "For how many ordered pairs of positive integers $(x,y),$ with $y<x\\le 100,$ are both $\\frac xy$ and $\\frac{x+1}{y+1}$ integers?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 17847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The figures $F_1$, $F_2$, $F_3$, and $F_4$ shown are the first in a sequence of figures. For $n\\ge3$, $F_n$ is constructed from $F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n - 1}$ had on each side of its outs... | math | {
"style": "rule",
"ground_truth": "761"
} | {
"split": "train",
"index": 17848,
"solution": "1. **Understanding the Construction of Figures:**\n Each figure $F_n$ in the sequence is constructed by surrounding the previous figure $F_{n-1}$ with a square and adding diamonds. The number of diamonds added on each side of the new square is one more than the n... |
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