data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given two fixed points $A(-2,0)$ and $B(2,0)$, a moving point $P(x,y)$ is located on the line $l:y=x+3$. An ellipse $c$ has foci at points $A$ and $B$ and passes through point $P$. Determine the maximum eccentricity of the ellipse $c$. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{26}}{13}"
} | {
"split": "train",
"index": 25749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cylindrical glass vessel is heated to a temperature of $t = 1200^{\\circ}$C, causing part of the air to be expelled. After covering the vessel with a well-sealing glass plate, it is left to cool to room temperature at $t^{\\prime}=16^{\\circ}$C. The remaining air inside becom... | math | {
"style": "rule",
"ground_truth": "19.6\\%"
} | {
"split": "train",
"index": 25750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $0.\\overline{789}-0.\\overline{456}-0.\\overline{123}?$ Express your answer as a fraction in lowest terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{70}{333}"
} | {
"split": "train",
"index": 25751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square has a side length of 4 units. Points $R$ and $S$ are chosen such that $R$ is the midpoint of the bottom side of the square, and $S$ is the midpoint of the right side of the square. What fraction of the interior of the square is not shaded if the triangle formed by $R$,... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"split": "train",
"index": 25752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $2^8=16^x$, find $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 25753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle A B C$ be a triangle inscribed in a unit circle with center $O$. Let $I$ be the incenter of $\\triangle A B C$, and let $D$ be the intersection of $B C$ and the angle bisector of $\\angle B A C$. Suppose that the circumcircle of $\\triangle A D O$ intersects $B ... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{169}"
} | {
"split": "train",
"index": 25754,
"solution": "Consider the following lemma: Lemma. $A D \\perp E O$. Proof. By the Shooting Lemma, the reflection of the midpoint $M$ of arc $B C$ not containing $A$ over $B C$ lies on $(A D O)$. Hence $\\measuredangle A D E+\\measuredangle D E O=\\measuredangle M D C+\\measured... |
deepscaler | [
{
"role": "user",
"content": "A rectangle is called cool if the number of square units in its area is equal to twice the number of units in its perimeter. A cool rectangle also must have integer side lengths. What is the sum of all the different possible areas of cool rectangles? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "236"
} | {
"split": "train",
"index": 25755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse with $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$), its left focal point is $F_1(-1, 0)$, and vertex P on the ellipse satisfies $\\angle PF_1O = 45^\\circ$ (where O is the origin).\n(1) Determine the values of $a$ and $b$;\n(2) Given that line $l_1: y... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 25756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Diana drew a rectangular grid of 2 by 1009 squares. Some of the squares were then painted black. In each white square, she wrote the number of black squares that shared an edge with it (a whole edge, not just a vertex). What is the maximum value that she could obtain as the res... | math | {
"style": "rule",
"ground_truth": "3025"
} | {
"split": "train",
"index": 25757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle with dimensions \\(24 \\times 60\\) is divided into unit squares by lines parallel to its sides. Into how many parts will this rectangle be divided if its diagonal is also drawn? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1512"
} | {
"split": "train",
"index": 25758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the center of a circular field stands a geologists' house. From it, 8 straight roads extend, dividing the field into 8 equal sectors. Two geologists embark on a journey from their house at a speed of 5 km/h, each choosing a road at random. Determine the probability that the ... | math | {
"style": "rule",
"ground_truth": "0.375"
} | {
"split": "train",
"index": 25759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α\\in\\mathbb{R}$ and $\\sin α + 2\\cos α = \\frac{\\sqrt{10}}{2}$, find the value of $\\tan α$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"split": "train",
"index": 25760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Polly has three circles cut from three pieces of colored card. She originally places them on top of each other as shown. In this configuration, the area of the visible black region is seven times the area of the white circle.\n\nPolly moves the circles to a new position, as sho... | math | {
"style": "rule",
"ground_truth": "7:6"
} | {
"split": "train",
"index": 25761,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an exam, there are 6 questions, and each question is solved by exactly 100 people. Each pair of examinees has at least one question that neither of them has solved. What is the minimum number of participants in the exam? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 25762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Robert read a book for 10 days. He read an average of 25 pages per day for the first 5 days and an average of 40 pages per day for the next 4 days, and read 30 more pages on the last day. Calculate the total number of pages in the book. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "315"
} | {
"split": "train",
"index": 25763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and it is given that $(a+b)(\\sin A-\\sin B)=c(\\sin C-\\sin B)$. \n$(1)$ Find $A$. \n$(2)$ If $a=4$, find the maximum value of the area $S$ of $\\triangle ABC$. Let's t... | math | {
"style": "rule",
"ground_truth": "4 \\sqrt {3}"
} | {
"split": "train",
"index": 25764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a sphere with a radius of $\\frac{\\sqrt{3}}{2}$, on which 4 points $A, B, C, D$ form a regular tetrahedron. What is the maximum distance from the center of the sphere to the faces of the regular tetrahedron $ABCD$? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{6}"
} | {
"split": "train",
"index": 25765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than 2019 are divisible by either 18 or 21, but not both? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "176"
} | {
"split": "train",
"index": 25766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine all positive integers $ n\\geq 2$ that satisfy the following condition: for all $ a$ and $ b$ relatively prime to $ n$ we have \\[a \\equiv b \\pmod n\\qquad\\text{if and only if}\\qquad ab\\equiv 1 \\pmod n.\\] Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "2, 3, 4, 6, 8, 12, 24"
} | {
"split": "train",
"index": 25767,
"solution": "To determine all positive integers \\( n \\geq 2 \\) that satisfy the given condition, we need to analyze when \\( a \\equiv b \\pmod{n} \\) if and only if \\( ab \\equiv 1 \\pmod{n} \\) for all \\( a \\) and \\( b \\) that are relatively prime to \\( n \\).\n\n###... |
deepscaler | [
{
"role": "user",
"content": "Two cross sections of a right octagonal pyramid are obtained by cutting the pyramid with planes parallel to the octagonal base. The areas of the cross sections are $300\\sqrt{2}$ square feet and $675\\sqrt{2}$ square feet. The two planes are $10$ feet apart. How far from the ap... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 25768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with the first term $a$ ($a \\in \\mathbb{R}, a \\neq 0$). Let the sum of the first $n$ terms of the sequence be $S_n$, and for any positive integer $n$, it holds that $$\\frac {a_{2n}}{a_{n}}= \\frac {4n-1}{2n-1}$$.\n(1) Find the genera... | math | {
"style": "rule",
"ground_truth": "k=3"
} | {
"split": "train",
"index": 25769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We know the following to be true:\n\n$\\bullet$ 1. $Z$ and $K$ are integers with $500 < Z < 1000$ and $K > 1;$\n\n$\\bullet$ 2. $Z$ = $K \\times K^2.$\n\nWhat is the value of $K$ for which $Z$ is a perfect square? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 25770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the cells of an $8 \\times 8$ chessboard, there are 8 white and 8 black pieces arranged such that no two pieces are in the same cell. Additionally, no pieces of the same color are in the same row or column. For each white piece, the distance to the black piece in the same co... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 25771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number\n$\\frac 2{\\log_4{2000^6}} + \\frac 3{\\log_5{2000^6}}$\n can be written as $\\frac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 25772,
"solution": "$\\frac 2{\\log_4{2000^6}} + \\frac 3{\\log_5{2000^6}}$\n$=\\frac{\\log_4{16}}{\\log_4{2000^6}}+\\frac{\\log_5{125}}{\\log_5{2000^6}}$\n$=\\frac{\\log{16}}{\\log{2000^6}}+\\frac{\\log{125}}{\\log{2000^6}}$\n$=\\frac{\\log{2000}}{\\log{2000^6}}$\n$=\\frac{\\log{2000... |
deepscaler | [
{
"role": "user",
"content": "A ray of light passing through the point $A = (-3,9,11),$ reflects off the plane $x + y + z = 12$ at $B,$ and then passes through the point $C = (3,5,9).$ Find the point $B.$\n\n[asy]\nimport three;\n\nsize(180);\ncurrentprojection = perspective(6,3,2);\n\ntriple A, B, C;\n\nA... | math | {
"style": "rule",
"ground_truth": "\\left( -\\frac{5}{3}, \\frac{16}{3}, \\frac{25}{3} \\right)"
} | {
"split": "train",
"index": 25773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One interior angle in a triangle measures $50^{\\circ}$. What is the angle between the bisectors of the remaining two interior angles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 25774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a circle, points $A, B, C, D, E, F, G$ are located clockwise as shown in the diagram. It is known that $AE$ is the diameter of the circle. Additionally, it is known that $\\angle ABF = 81^\\circ$ and $\\angle EDG = 76^\\circ$. How many degrees is the angle $FCG$? Let's think... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 25775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that there is a geometric sequence $\\{a_n\\}$ with a common ratio $q > 1$ and the sum of the first $n$ terms is $S_n$, $S_3 = 7$, the sequence $a_1+3$, $3a_2$, $a_3+4$ forms an arithmetic sequence. The sum of the first $n$ terms of the sequence $\\{b_n\\}$ is $T_n$, and ... | math | {
"style": "rule",
"ground_truth": "3318"
} | {
"split": "train",
"index": 25776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the squares of the solutions to the equation\n\\[\\left| x^2 - x + \\frac{1}{2010} \\right| = \\frac{1}{2010}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2008}{1005}"
} | {
"split": "train",
"index": 25777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An insect has just told me that she has laid $154_6$ eggs. In base 10, how many eggs did she lay? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 25778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each integer $n\\geq 4$, let $a_n$ denote the base-$n$ number $0.\\overline{133}_n$. The product $a_4a_5 \\dotsm a_{99}$ can be expressed as $\\frac{m}{n!}$, where $m$ and $n$ are positive integers and $n$ is as small as possible. What is the value of $m$? Let's think ste... | math | {
"style": "rule",
"ground_truth": "962"
} | {
"split": "train",
"index": 25779,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If two distinct members of the set $\\{ 4, 10, 15, 24, 30, 40, 60 \\}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 120? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{7}"
} | {
"split": "train",
"index": 25780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "10 chatterboxes sat in a circle. Initially, one of them told one joke, the next one clockwise told two jokes, the next one three jokes, and so on in a circle until one of them told 100 jokes at once. Then the chatterboxes got tired, and the next one clockwise told 99 jokes, the... | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 25781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose\n\\[\\frac{1}{x^3-x^2-21x+45}=\\frac{A}{x+5}+\\frac{B}{x-3} + \\frac{C}{(x - 3)^2}\\]where $A$, $B$, and $C$ are real constants. What is $A$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{64}"
} | {
"split": "train",
"index": 25782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest base-10 integer that can be represented as $CC_6$ and $DD_8$, where $C$ and $D$ are valid digits in their respective bases? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 25783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of sets $A$ that satisfy the three conditions: $\\star$ $A$ is a set of two positive integers $\\star$ each of the numbers in $A$ is at least $22$ percent the size of the other number $\\star$ $A$ contains the number $30.$ Let's think step by step... | math | {
"style": "rule",
"ground_truth": "129"
} | {
"split": "train",
"index": 25784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the set \\( S = \\{1, 2, 3, \\cdots, 2010, 2011\\} \\). A subset \\( T \\) of \\( S \\) is said to be a \\( k \\)-element RP-subset if \\( T \\) has exactly \\( k \\) elements and every pair of elements of \\( T \\) are relatively prime. Find the smallest positive inte... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 25785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If in an arithmetic sequence, the sum of the first three terms is 34, the sum of the last three terms is 146, and the sum of all terms is 390, then the number of terms in the sequence is __________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 25786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a tetrahedron \\(A B C D\\), where \\(B D = D C = C B = \\sqrt{2}\\), \\(A C = \\sqrt{3}\\), and \\(A B = A D = 1\\), find the cosine of the angle between line \\(B M\\) and line \\(A C\\), where \\(M\\) is the midpoint of \\(C D\\). Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{3}"
} | {
"split": "train",
"index": 25787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The difference of the logarithms of the hundreds digit and the tens digit of a three-digit number is equal to the logarithm of the difference of the same digits, and the sum of the logarithms of the hundreds digit and the tens digit is equal to the logarithm of the sum of the s... | math | {
"style": "rule",
"ground_truth": "421"
} | {
"split": "train",
"index": 25788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle is divided into six equal sections. Each section is to be coloured with a single colour so that three sections are red, one is blue, one is green, and one is yellow. Two circles have the same colouring if one can be rotated to match the other. How many different colour... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 25789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x = \\frac{5}{7}$ is a solution to the equation $56 x^2 + 27 = 89x - 8,$ what is the other value of $x$ that will solve the equation? 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{7}{8}"
} | {
"split": "train",
"index": 25790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A granary has collected 1536 shi of rice, and upon inspection, it is found that out of 224 grains, 28 are weeds. Determine the approximate amount of weeds in this batch of rice. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 25791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fluffball and Shaggy the squirrels ate a basket of berries and a pack of seeds containing between 50 and 65 seeds, starting and finishing at the same time. Initially, Fluffball ate berries while Shaggy ate seeds. Later, they swapped tasks. Shaggy ate berries six times faster th... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 25792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "2005^{2} + 2 \\times 2005 \\times 1995 + 1995^{2} divided by 800. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20000"
} | {
"split": "train",
"index": 25793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Captain Billy the Pirate looted 1010 gold doubloons and set sail on his ship to a deserted island to bury his treasure. Each evening of their voyage, he paid each of his pirates one doubloon. On the eighth day of sailing, the pirates plundered a Spanish caravel, doubling Billy'... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 25794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest natural number from which it is impossible to obtain a number divisible by 11 by deleting some of its digits. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "987654321"
} | {
"split": "train",
"index": 25795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a class has 5 students participating in the duty roster from Monday to Friday, with one student arranged each day, student A can only be arranged on Monday or Tuesday, and student B cannot be arranged on Friday, calculate the number of different duty arrangements for... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 25796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, if $a= \\sqrt {5}$, $b= \\sqrt {15}$, $A=30^{\\circ}$, then $c=$ \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {5}"
} | {
"split": "train",
"index": 25797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD$, we have $A=(6,-22)$, $B=(2006,178)$, $D=(8,y)$, for some integer $y$. What is the area of rectangle $ABCD$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40400"
} | {
"split": "train",
"index": 25798,
"solution": "\n1. **Identify the coordinates and slopes**:\n - Coordinates given: $A = (6, -22)$, $B = (2006, 178)$, $D = (8, y)$.\n - Slope of $AB$, $m_1$: \n \\[\n m_1 = \\frac{178 - (-22)}{2006 - 6} = \\frac{200}{2000} = \\frac{1}{10}\n \\]\n\n2. **Determine ... |
deepscaler | [
{
"role": "user",
"content": "The slope angle of the line $x - y + 3 = 0$ is $\\tan^{-1}(1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 25799,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of\n\\[r = -2 \\cos \\theta + 6 \\sin \\theta\\]is a circle. Find the area of the circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10 \\pi"
} | {
"split": "train",
"index": 25800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nA school has 1200 students, and each student participates in exactly \\( k \\) clubs. It is known that any group of 23 students all participate in at least one club in common, but no club includes all 1200 students. Find the minimum possible value of \\( k \\). Let's think st... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 25801,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Exactly three faces of a \\(2 \\times 2 \\times 2\\) cube are partially shaded. Calculate the fraction of the total surface area of the cube that is shaded. 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": 25802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S=\\{1,2, \\ldots, 2021\\}$, and let $\\mathcal{F}$ denote the set of functions $f: S \\rightarrow S$. For a function $f \\in \\mathcal{F}$, let $$T_{f}=\\left\\{f^{2021}(s): s \\in S\\right\\}$$ where $f^{2021}(s)$ denotes $f(f(\\cdots(f(s)) \\cdots))$ with 2021 copies of... | math | {
"style": "rule",
"ground_truth": "255"
} | {
"split": "train",
"index": 25803,
"solution": "The key idea is that $t \\in T_{f}$ if and only if $f^{k}(t)=t$ for some $k>0$. To see this, let $s \\in S$ and consider $$s, f(s), f(f(s)), \\ldots, f^{2021}(s)$$ This sequence has 2022 terms that are all in $S$, so we must have a repeat. Suppose $f^{m}(s)=f^{n}(s... |
deepscaler | [
{
"role": "user",
"content": "Two chords \\(AB\\) and \\(CD\\) of a circle with center \\(O\\) each have a length of 10. The extensions of segments \\(BA\\) and \\(CD\\) beyond points \\(A\\) and \\(D\\) respectively intersect at point \\(P\\), with \\(DP = 3\\). The line \\(PO\\) intersects segment \\(AC\\... | math | {
"style": "rule",
"ground_truth": "3/13"
} | {
"split": "train",
"index": 25804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC,$ $AB = 20$ and $BC = 15.$ Find the largest possible value of $\\tan A.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{7}}{7}"
} | {
"split": "train",
"index": 25805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T_1$ be a triangle with side lengths $2011, 2012,$ and $2013$. For $n \\ge 1$, if $T_n = \\triangle ABC$ and $D, E,$ and $F$ are the points of tangency of the incircle of $\\triangle ABC$ to the sides $AB, BC$, and $AC,$ respectively, then $T_{n+1}$ is a triangle with side... | math | {
"style": "rule",
"ground_truth": "\\frac{1509}{128}"
} | {
"split": "train",
"index": 25806,
"solution": "1. **Understanding the Problem:**\n We are given a sequence of triangles where each triangle's sides are determined by the tangency points of the incircle with the sides of the previous triangle. We need to find the perimeter of the last triangle in this sequence... |
deepscaler | [
{
"role": "user",
"content": "Susan wants to determine the average and median number of candies in a carton. She buys 9 cartons of candies, opens them, and counts the number of candies in each one. She finds that the cartons contain 5, 7, 8, 10, 12, 14, 16, 18, and 20 candies. What are the average and media... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 25807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four students participate in a knowledge contest, each student must choose one of the two questions, A or B, to answer. Correctly answering question A earns 60 points, while an incorrect answer results in -60 points. Correctly answering question B earns 180 points, while an inc... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 25808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five positive integers (not necessarily all different) are written on five cards. Boris calculates the sum of the numbers on every pair of cards. He obtains only three different totals: 57, 70, and 83. What is the largest integer on any card?\n- A) 35\n- B) 42\n- C) 48\n- D) 53... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 25809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of \n\\[x^3 + 12x + \\frac{81}{x^4}\\]\nfor $x > 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 25810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what value of $n$ is $3^3-5=4^2+n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 25811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Year 0 of Cambridge there is one squirrel and one rabbit. Both animals multiply in numbers quickly. In particular, if there are $m$ squirrels and $n$ rabbits in Year $k$, then there will be $2 m+2019$ squirrels and $4 n-2$ rabbits in Year $k+1$. What is the first year in... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 25812,
"solution": "In year $k$, the number of squirrels is $$2(2(\\cdots(2 \\cdot 1+2019)+2019)+\\cdots)+2019=2^{k}+2019 \\cdot\\left(2^{k-1}+2^{k-2}+\\cdots+1\\right)=2020 \\cdot 2^{k}-2019$$ and the number of rabbits is $$4(4(\\cdots(4 \\cdot 1-2)-2)-\\cdots)-2=4^{k}-2 \\cdot\\left... |
deepscaler | [
{
"role": "user",
"content": "What is the smallest base-10 integer that can be represented as $CC_6$ and $DD_8$, where $C$ and $D$ are valid digits in their respective bases? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63_{10}"
} | {
"split": "train",
"index": 25813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of the expression \\((\\sqrt{8-4 \\sqrt{3}} \\sin x - 3 \\sqrt{2(1+\\cos 2x)} - 2) \\cdot (3 + 2 \\sqrt{11 - \\sqrt{3}} \\cos y - \\cos 2y)\\). If the answer is a non-integer, round it to the nearest whole number. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 25814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 10 boys, each with a unique weight and height. For any two boys, $\\mathbf{A}$ and $\\mathbf{B}$, if $\\mathbf{A}$ is heavier than $\\mathbf{B}$, or if $\\mathbf{A}$ is taller than $\\mathbf{B}$, then $\\mathbf{A}$ is not considered worse than $\\mathbf{B}$. A boy who... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 25815,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The increasing sequence $1, 3, 4, 9, 10, 12, 13, \\dots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $150^{\\text{th}}$ term of this sequence. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "2280"
} | {
"split": "train",
"index": 25816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find \\( k \\) such that, for all \\( n \\), the following expression is a perfect square:\n\n$$\n4 n^{2} + k n + 9\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 25817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $f'(x_0)=2$, find the value of $\\lim_{k\\rightarrow 0} \\frac{f(x_0-k)-f(x_0)}{2k}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 25818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the sides $AB$ and $AC$ of triangle $ABC$ lie points $K$ and $L$, respectively, such that $AK: KB = 4:7$ and $AL: LC = 3:2$. The line $KL$ intersects the extension of side $BC$ at point $M$. Find the ratio $CM: BC$. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "8 : 13"
} | {
"split": "train",
"index": 25819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an isosceles triangle \\(ABC\\) with \\(\\angle B\\) equal to \\(30^{\\circ}\\) and \\(AB = BC = 6\\), the altitude \\(CD\\) of triangle \\(ABC\\) and the altitude \\(DE\\) of triangle \\(BDC\\) are drawn.\n\nFind \\(BE\\). Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 25820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\(PRTY\\) and \\(WRSU\\) are squares. Point \\(Q\\) is on \\(PR\\) and point \\(X\\) is on \\(TY\\) so that \\(PQXY\\) is a rectangle. Also, point \\(T\\) is on \\(SU\\), point \\(W\\) is on \\(QX\\), and point \\(V\\) is the point of intersection of \\(UW\\) a... | math | {
"style": "rule",
"ground_truth": "5.5"
} | {
"split": "train",
"index": 25821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$n$ mushroom gatherers went to the forest and brought back a total of 338 mushrooms (it is possible that some of the gatherers did not bring back any mushrooms). A boy named Petya, upon learning this, stated, \"Some two of them must have brought an identical number of mushrooms... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 25822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A straight one-mile stretch of highway, 40 feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at 5 miles per hour, how many hours will it take to cover the one-mile stretch?\n[asy]\nsize(10cm); pathpen=black; pointpen=black; D(ar... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{10}"
} | {
"split": "train",
"index": 25823,
"solution": "\n#### Step-by-step Analysis:\n\n1. **Understanding the Path**: Robert rides his bike on a path composed of semicircles along a one-mile stretch of highway that is 40 feet wide. We need to determine the radius of these semicircles. Since the highway is 40 feet wide... |
deepscaler | [
{
"role": "user",
"content": "On a $300 \\times 300$ board, several rooks are placed that beat the entire board. Within this case, each rook beats no more than one other rook. At what least $k$ , it is possible to state that there is at least one rook in each $k\\times k$ square ? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "201"
} | {
"split": "train",
"index": 25824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each integer $n\\geq3$, let $f(n)$ be the number of $3$-element subsets of the vertices of the regular $n$-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$. Let's think step by... | math | {
"style": "rule",
"ground_truth": "245"
} | {
"split": "train",
"index": 25825,
"solution": "Considering $n \\pmod{6}$, we have the following formulas:\n Even and a multiple of 3: $\\frac{n(n-4)}{2} + \\frac{n}{3}$\n Even and not a multiple of 3: $\\frac{n(n-2)}{2}$\n Odd and a multiple of 3: $\\frac{n(n-3)}{2} + \\frac{n}{3}$\n Odd and not a multiple of 3... |
deepscaler | [
{
"role": "user",
"content": "Around the circle, a pentagon is described, the lengths of the sides of which are integers, and the first and third sides are equal to 1. Into which segments does the point of tangency divide the second side? 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": 25826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $(2-3z) - (3+4z)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1-7z"
} | {
"split": "train",
"index": 25827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ten-letter code $\\text{BEST OF LUCK}$ represents the ten digits $0-9$, in order. What 4-digit number is represented by the code word $\\text{CLUE}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8671"
} | {
"split": "train",
"index": 25828,
"solution": "The problem states that the ten-letter code $\\text{BEST OF LUCK}$ represents the ten digits $0-9$ in order. We can assign each letter a corresponding digit based on its position in the sequence:\n\n- B = 0\n- E = 1\n- S = 2\n- T = 3\n- O = 4\n- F = 5\n- L = 6\n- U... |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, $a+b=11$. Choose one of the following two conditions as known, and find:<br/>$(Ⅰ)$ the value of $a$;<br/>$(Ⅱ)$ $\\sin C$ and the area of $\\triangle ABC$.<br/>Condition 1: $c=7$, $\\cos A=-\\frac{1}{7}$;<br/>Condition 2: $\\cos A=\\frac{1}{8}$, $\\... | math | {
"style": "rule",
"ground_truth": "\\frac{15\\sqrt{7}}{4}"
} | {
"split": "train",
"index": 25829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two balls, a red one and a green one, are randomly and independently tossed into bins numbered with positive integers. For each ball, the probability that it is tossed into bin $k$ is given by $p_k = \\frac{1}{k(k+1)}$ for $k = 1, 2, 3, ...$. Calculate the probability that the ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 25830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 4 willow trees and 4 poplar trees planted in a row. How many ways can they be planted alternately? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1152"
} | {
"split": "train",
"index": 25831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\tan(\\omega x + \\phi)$ $(\\omega \\neq 0, \\left|\\phi\\right| < \\frac{\\pi}{2})$, points $\\left(\\frac{2\\pi}{3}, 0\\right)$ and $\\left(\\frac{7\\pi}{6}, 0\\right)$ are two adjacent centers of symmetry for $f(x)$, and the function is monotonicall... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\pi}{6}"
} | {
"split": "train",
"index": 25832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a bag are all natural numbers less than or equal to $999$ whose digits sum to $6$ . What is the probability of drawing a number from the bag that is divisible by $11$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/7"
} | {
"split": "train",
"index": 25833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain warehouse, there are $1335$ boxes, each containing $39$ books.\n\nMelvin's boss orders him to unpack all the books and repack them so that there are $40$ books in each box. After packing as many such boxes as possible, Melvin has how many books left over? Let's thi... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 25834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $f(x)$ has a derivative and satisfies $\\lim_{\\Delta x \\to 0} \\, \\frac{f(1)-f(1-2\\Delta x)}{2\\Delta x}=-1$, find the slope of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 25835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two distinct similar rhombi share a diagonal. The smaller rhombus has area 1, and the larger rhombus has area 9. Compute the side length of the larger rhombus. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{15}"
} | {
"split": "train",
"index": 25836,
"solution": "Let $d$ be the length of the smaller diagonal of the smaller rhombus. Since the ratio of the areas is $9: 1$, the ratio of the lengths is $3: 1$. This means that the smaller diagonal of the larger rhombus (which is also the longer diagonal of the smaller rhombus) h... |
deepscaler | [
{
"role": "user",
"content": "Given the positive numbers \\( a, b, c, x, y, z \\) that satisfy the equations \\( cy + bz = a \\), \\( az + cx = b \\), and \\( bx + ay = c \\), find the minimum value of the function\n\\[ \nf(x, y, z) = \\frac{x^{2}}{1+x} + \\frac{y^{2}}{1+y} + \\frac{z^{2}}{1+z}.\n\\] Let's ... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 25837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 30 students in Mrs. Taylor's kindergarten class. If there are twice as many students with blond hair as with blue eyes, 6 students with blond hair and blue eyes, and 3 students with neither blond hair nor blue eyes, how many students have blue eyes? Let's think step ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 25838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function \\( f(x) = x^3 + ax^2 + bx + c \\) where \\( a, b, \\) and \\( c \\) are nonzero integers, if \\( f(a) = a^3 \\) and \\( f(b) = b^3 \\), what is the value of \\( c \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 25839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f^{-1}(g(x))=x^4-4$ and $g$ has an inverse, find $g^{-1}(f(15))$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[4]{19}"
} | {
"split": "train",
"index": 25840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We have a calculator with two buttons that displays an integer $x$. Pressing the first button replaces $x$ by $\\left\\lfloor\\frac{x}{2}\\right\\rfloor$, and pressing the second button replaces $x$ by $4 x+1$. Initially, the calculator displays 0. How many integers less than o... | math | {
"style": "rule",
"ground_truth": "233"
} | {
"split": "train",
"index": 25841,
"solution": "We consider the integers from this process written in binary. The first operation truncates the rightmost digit, while the second operation appends 01 to the right. We cannot have a number with a substring 11. For simplicity, call a string valid if it has no consec... |
deepscaler | [
{
"role": "user",
"content": "The sequence 12, 15, 18, 21, 51, 81, $\\ldots$ consists of all positive multiples of 3 that contain at least one digit that is a 1. What is the $50^{\\mathrm{th}}$ term of the sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "318"
} | {
"split": "train",
"index": 25842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ and $c$ be positive real numbers such that $abc = 8.$ Find the minimum value of\n\\[(3a + b)(a + 3c)(2bc + 4).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "384"
} | {
"split": "train",
"index": 25843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle{ABC}$ with $AB = 12$, $BC = 13$, and $AC = 15$, let $M$ be a point on $\\overline{AC}$ such that the incircles of $\\triangle{ABM}$ and $\\triangle{BCM}$ have equal radii. Then $\\frac{AM}{CM} = \\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integ... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 25844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The edge of cube \\( ABCD A_1 B_1 C_1 D_1 \\) is 12. Point \\( K \\) lies on the extension of edge \\( BC \\) at a distance of 9 from vertex \\( C \\). Point \\( L \\) on edge \\( AB \\) is at a distance of 5 from \\( A \\). Point \\( M \\) divides segment \\( A_1 C_1 \\) in a ... | math | {
"style": "rule",
"ground_truth": "156"
} | {
"split": "train",
"index": 25845,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a park, 10,000 trees are planted in a square grid pattern (100 rows of 100 trees). What is the maximum number of trees that can be cut down such that if one stands on any stump, no other stumps are visible? (Trees can be considered thin enough for this condition.)\n Let's th... | math | {
"style": "rule",
"ground_truth": "2500"
} | {
"split": "train",
"index": 25846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Wanda is trying to locate the Fermat point $P$ of $\\triangle ABC$, where $A$ is at the origin, $B$ is at $(10,0)$, and $C$ is at $(3,5)$ (the Fermat point is the point such that the sum of its distances from the vertices of a triangle is minimized). She guesses that the point ... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 25847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Square $ABCD$ has side length $5$, point $M$ is chosen on side $AB$ so that $\\angle AMD = \\angle CMD$, calculate the degree measure of $\\angle AMD$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 25848,
"solution": ""
} |
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