reward_model dict | prompt listlengths 1 1 |
|---|---|
{
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | [
{
"content": "Regular octagon CHILDREN has area 1. Determine the area of quadrilateral LINE. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "B",
"style": "rule"
} | [
{
"content": "Which letter will go in the square marked with $*$ in the grid where each of the letters A, B, C, D, and E appears exactly once in each row and column? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "\\dfrac{1}{3}",
"style": "rule"
} | [
{
"content": "Calculate the probability that the line $y=kx+k$ intersects with the circle ${{\\left( x-1 \\right)}^{2}}+{{y}^{2}}=1$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "27",
"style": "rule"
} | [
{
"content": "Find the smallest natural number \\( n \\) such that both \\( n^2 \\) and \\( (n+1)^2 \\) contain the digit 7. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "28",
"style": "rule"
} | [
{
"content": "Kelvin the Frog and 10 of his relatives are at a party. Every pair of frogs is either friendly or unfriendly. When 3 pairwise friendly frogs meet up, they will gossip about one another and end up in a fight (but stay friendly anyway). When 3 pairwise unfriendly frogs meet up, they will also end up... |
{
"ground_truth": "123",
"style": "rule"
} | [
{
"content": "Sasha records the numbers 1, 2, 3, 4, and 5 in some order, places arithmetic operation signs \"+\", \"-\", \"x\" and parentheses, and looks at the result of the expression obtained. For example, he can get the number 8 using the expression \\((4-3) \\times (2+5) + 1\\). Can he get the number 123?\... |
{
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | [
{
"content": "Given a circle \\\\(O: x^2 + y^2 = 2\\\\) and a line \\\\(l: y = kx - 2\\\\).\n\n\\\\((1)\\\\) If line \\\\(l\\\\) intersects circle \\\\(O\\\\) at two distinct points \\\\(A\\\\) and \\\\(B\\\\), and \\\\(\\angle AOB = \\frac{\\pi}{2}\\\\), find the value of \\\\(k\\\\).\n\n\\\\((2)\\\\) If \\\\(... |
{
"ground_truth": "20",
"style": "rule"
} | [
{
"content": "A polynomial with integer coefficients is of the form\n\\[12x^3 - 4x^2 + a_1x + 18 = 0.\\]\nDetermine the number of different possible rational roots of this polynomial. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "2",
"style": "rule"
} | [
{
"content": "The figure below shows line $\\ell$ with a regular, infinite, recurring pattern of squares and line segments.\n\nHow many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into ... |
{
"ground_truth": "3 : 2 : 1",
"style": "rule"
} | [
{
"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 output the final answer withi... |
{
"ground_truth": "13",
"style": "rule"
} | [
{
"content": "Two identical rulers are placed together. Each ruler is exactly 10 cm long and is marked in centimeters from 0 to 10. The 3 cm mark on each ruler is aligned with the 4 cm mark on the other. The overall length is \\( L \\) cm. What is the value of \\( L \\)? Let's think step by step and output the ... |
{
"ground_truth": "1346",
"style": "rule"
} | [
{
"content": "On an island of Liars and Knights, a circular arrangement is called correct if each person in the circle can say that among their two neighbors, there is at least one member of their tribe. One day, 2019 natives formed a correct circle. A liar approached them and said, \"Now we too can form a corr... |
{
"ground_truth": "21.6",
"style": "rule"
} | [
{
"content": "At night, there was a heavy snowfall. In the morning, Xiao Long and his father measured the length of a circular path in the garden by walking. They started from the same point and walked in the same direction. Xiao Long's step length is 54 cm, and his father's step length is 72 cm. Each of them w... |
{
"ground_truth": "107\\%",
"style": "rule"
} | [
{
"content": "A store owner purchases merchandise at a discount of 30% off the original list price. To ensure a profit, the owner wants to mark up the goods such that after offering a 15% discount on the new marked price, the final selling price still yields a 30% profit compared to the cost price. What percent... |
{
"ground_truth": "1,099,999,999,999",
"style": "rule"
} | [
{
"content": "Add the following two numbers: $800,000,000,000 + 299,999,999,999$.\nA) $1,000,000,000,000$\nB) $1,099,999,999,999$\nC) $1,100,000,000,000$\nD) $900,000,000,000$\nE) $2,099,999,999,999$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "2\\sqrt{13}",
"style": "rule"
} | [
{
"content": "Given triangle ABC, where sides $a$, $b$, and $c$ correspond to angles A, B, and C respectively, and $a=4$, $\\cos{B}=\\frac{4}{5}$.\n(1) If $b=6$, find the value of $\\sin{A}$;\n(2) If the area of triangle ABC, $S=12$, find the values of $b$ and $c$. Let's think step by step and output the final ... |
{
"ground_truth": "22",
"style": "rule"
} | [
{
"content": "In an opaque bag, there are $10$ balls each of red, white, and yellow colors, all identical except for the color. At least how many balls must be drawn to ensure that two balls of different colors are drawn? At least how many balls must be drawn to ensure that two yellow balls are drawn? Let's thi... |
{
"ground_truth": "3 : 2",
"style": "rule"
} | [
{
"content": "The sum of two natural numbers and their greatest common divisor is equal to their least common multiple. Determine the ratio of the two numbers. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | [
{
"content": "Three people, including one girl, are to be selected from a group of $3$ boys and $2$ girls, determine the probability that the remaining two selected individuals are boys. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "19",
"style": "rule"
} | [
{
"content": "In a geometric sequence $\\\\{a\\_n\\\\}$, $a\\_n > 0 (n \\in \\mathbb{N}^*)$, the common ratio $q \\in (0, 1)$, and $a\\_1a\\_5 + 2aa\\_5 + a\\_2a\\_8 = 25$, and the geometric mean of $a\\_3$ and $a\\_5$ is $2$.\n\n(1) Find the general term formula of the sequence $\\\\{a\\_n\\\\}$;\n\n(2) If $b\... |
{
"ground_truth": "9.49",
"style": "rule"
} | [
{
"content": "A 30 foot ladder is placed against a vertical wall of a building. The foot of the ladder is 11 feet from the base of the building. If the top of the ladder slips 6 feet, then the foot of the ladder will slide how many feet? Let's think step by step and output the final answer within \\boxed{}.",
... |
{
"ground_truth": "10",
"style": "rule"
} | [
{
"content": "The cost of purchasing a car is 150,000 yuan, and the annual expenses for insurance, tolls, and gasoline are about 15,000 yuan. The maintenance cost for the first year is 3,000 yuan, which increases by 3,000 yuan each year thereafter. Determine the best scrap year limit for this car. Let's think s... |
{
"ground_truth": "186",
"style": "rule"
} | [
{
"content": "A student, Theo, needs to earn a total of 30 homework points. For the first six homework points, he has to do one assignment each; for the next six points, he needs to do two assignments each; and so on, such that for every subsequent set of six points, the number of assignments he needs to comple... |
{
"ground_truth": "70",
"style": "rule"
} | [
{
"content": "Given that the ceiling is 3 meters above the floor, the light bulb is 15 centimeters below the ceiling, Alice is 1.6 meters tall and can reach 50 centimeters above the top of her head, and a 5 centimeter thick book is placed on top of a stool to reach the light bulb, find the height of the stool i... |
{
"ground_truth": "11",
"style": "rule"
} | [
{
"content": "There are 8 books on a shelf, which include a trilogy that must be selected together. In how many ways can 5 books be selected from this shelf if the order in which the books are selected does not matter? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
... |
{
"ground_truth": "123.53",
"style": "rule"
} | [
{
"content": "A woman buys a property for $15,000, aiming for a $6\\%$ return on her investment annually. She sets aside $15\\%$ of the rent each month for maintenance, and pays $360 annually in taxes. What must be the monthly rent to meet her financial goals?\nA) $110.00$\nB) $123.53$\nC) $130.45$\nD) $142.86$... |
{
"ground_truth": "45",
"style": "rule"
} | [
{
"content": "John is cycling east at a speed of 8 miles per hour, while Bob is also cycling east at a speed of 12 miles per hour. If Bob starts 3 miles west of John, determine the time it will take for Bob to catch up to John. Let's think step by step and output the final answer within \\boxed{}.",
"role":... |
{
"ground_truth": "67",
"style": "rule"
} | [
{
"content": "The café has enough chairs to seat $312_8$ people. If $3$ people are supposed to sit at one table, how many tables does the café have? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "11",
"style": "rule"
} | [
{
"content": "How many distinct trees with exactly 7 vertices exist? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "12.24",
"style": "rule"
} | [
{
"content": "We are laying railway tracks that are $15 \\text{ meters }$ long at a temperature of $t = -8^{\\circ} \\text{C}$. What gap should we leave between each rail if the maximum expected temperature is $t = 60^{\\circ} \\text{C}$? The coefficient of expansion for the rail is $\\lambda = 0.000012$. Let's... |
{
"ground_truth": "4.7667",
"style": "rule"
} | [
{
"content": "George now has an unfair eight-sided die. The probabilities of rolling each number from 1 to 5 are each $\\frac{1}{15}$, the probability of rolling a 6 or a 7 is $\\frac{1}{6}$ each, and the probability of rolling an 8 is $\\frac{1}{5}$. What is the expected value of the number shown when this die... |
{
"ground_truth": "153, 370, 371, 407",
"style": "rule"
} | [
{
"content": "Find the numbers $\\mathbf{1 5 3 , 3 7 0 , 3 7 1 , 4 0 7}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "18 \\pi",
"style": "rule"
} | [
{
"content": "Alice is sitting in a teacup ride with infinitely many layers of spinning disks. The largest disk has radius 5. Each succeeding disk has its center attached to a point on the circumference of the previous disk and has a radius equal to $2 / 3$ of the previous disk. Each disk spins around its cente... |
{
"ground_truth": "\\dfrac{8}{5}",
"style": "rule"
} | [
{
"content": "Given an ellipse $C$: $\\dfrac{x^{2}}{a^{2}} + \\dfrac{y^{2}}{b^{2}} = 1$ passes through points $M(2,0)$ and $N(0,1)$.\n\n$(1)$ Find the equation of ellipse $C$ and its eccentricity;\n\n$(2)$ A line $y=kx (k \\in \\mathbb{R}, k \\neq 0)$ intersects ellipse $C$ at points $A$ and $B$, point $D$ is a... |
{
"ground_truth": "120",
"style": "rule"
} | [
{
"content": "Let $T$ be a trapezoid with two right angles and side lengths $4,4,5$, and $\\sqrt{17}$. Two line segments are drawn, connecting the midpoints of opposite sides of $T$ and dividing $T$ into 4 regions. If the difference between the areas of the largest and smallest of these regions is $d$, compute ... |
{
"ground_truth": "198",
"style": "rule"
} | [
{
"content": "Using the distinct digits \\( a, b, \\) and \\( c \\), Araceli wrote the number \\( abc \\), and Luana wrote the numbers \\( ab, bc, \\) and \\( ca \\). Find the digits \\( a, b, \\) and \\( c \\), knowing that the sum of the numbers written by Luana is equal to the number written by Araceli. Let'... |
{
"ground_truth": "-\\frac{2}{5}",
"style": "rule"
} | [
{
"content": "If circle $$C_{1}: x^{2}+y^{2}+ax=0$$ and circle $$C_{2}: x^{2}+y^{2}+2ax+ytanθ=0$$ are both symmetric about the line $2x-y-1=0$, then $sinθcosθ=$ \\_\\_\\_\\_\\_\\_ . Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "10 + 2\\sqrt{10}",
"style": "rule"
} | [
{
"content": "Given point \\( A(4,0) \\) and \\( B(2,2) \\), while \\( M \\) is a moving point on the ellipse \\(\\frac{x^{2}}{25} + \\frac{y^{2}}{9} = 1\\), the maximum value of \\( |MA| + |MB| \\) is ______. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "59",
"style": "rule"
} | [
{
"content": "The houses on the south side of Crazy Street are numbered in increasing order starting at 1 and using consecutive odd numbers, except that odd numbers that contain the digit 3 are missed out. What is the number of the 20th house on the south side of Crazy Street? \n\nA) 41 \nB) 49 \nC) 51 \nD) ... |
{
"ground_truth": "0.973",
"style": "rule"
} | [
{
"content": "Calculate the sum of the series $\\sum_{n=1}^{\\infty} \\frac{(-1)^{n+1}}{n^{5}}$ with an accuracy of $10^{-3}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "9",
"style": "rule"
} | [
{
"content": "Given that $x, y,$ and $z$ are real numbers that satisfy: \\begin{align*} x &= \\sqrt{y^2-\\frac{1}{16}}+\\sqrt{z^2-\\frac{1}{16}}, \\\\ y &= \\sqrt{z^2-\\frac{1}{25}}+\\sqrt{x^2-\\frac{1}{25}}, \\\\ z &= \\sqrt{x^2 - \\frac 1{36}}+\\sqrt{y^2-\\frac 1{36}}, \\end{align*} and that $x+y+z = \\frac{m... |
{
"ground_truth": "13",
"style": "rule"
} | [
{
"content": "Let \\( \\triangle ABC \\) be a triangle such that \\( AB = 7 \\), and let the angle bisector of \\( \\angle BAC \\) intersect line \\( BC \\) at \\( D \\). If there exist points \\( E \\) and \\( F \\) on sides \\( AC \\) and \\( BC \\), respectively, such that lines \\( AD \\) and \\( EF \\) are... |
{
"ground_truth": "\\dfrac{1}{5}",
"style": "rule"
} | [
{
"content": "When someone tries to withdraw money from an ATM with a forgotten last digit of the password, calculate the probability that they will press the correct digit in no more than 2 attempts. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "35",
"style": "rule"
} | [
{
"content": "Amy writes down four integers \\(a > b > c > d\\) whose sum is 50. The pairwise positive differences of these numbers are \\(2, 3, 4, 5, 7,\\) and \\(10\\). What is the sum of the possible values for \\(a\\)? Let's think step by step and output the final answer within \\boxed{}.",
"role": "use... |
{
"ground_truth": "26",
"style": "rule"
} | [
{
"content": "The café has enough chairs to seat $310_5$ people. If $3$ people are supposed to sit at one table, how many tables does the café have? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "506",
"style": "rule"
} | [
{
"content": "Let $p$, $q$, $r$, $s$, and $t$ be positive integers such that $p+q+r+s+t=2022$. Let $N$ be the largest of the sum $p+q$, $q+r$, $r+s$, and $s+t$. What is the smallest possible value of $N$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "900",
"style": "rule"
} | [
{
"content": "In a factor tree, each value is the product of the two values below it, unless a value is a prime number already. What is the value of $A$ on the factor tree shown?\n\n[asy]\ndraw((-1,-.3)--(0,0)--(1,-.3),linewidth(1));\ndraw((-2,-1.3)--(-1.5,-.8)--(-1,-1.3),linewidth(1));\ndraw((1,-1.3)--(1.5,-.8... |
{
"ground_truth": "37.5\\%",
"style": "rule"
} | [
{
"content": "The price of an item is decreased by 20%. To bring it back to its original value and then increase it by an additional 10%, the price after restoration must be increased by what percentage. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "15",
"style": "rule"
} | [
{
"content": "Each of the integers \\(1, 2, 3, \\ldots, 9\\) is assigned to a vertex of a regular 9-sided polygon (every vertex receives exactly one integer from \\(\\{1, 2, \\ldots, 9\\}\\), and no two vertices receive the same integer) so that the sum of the integers assigned to any three consecutive vertices... |
{
"ground_truth": "2.1",
"style": "rule"
} | [
{
"content": "Given a stalk of bamboo with nine sections, with three sections from the bottom holding 3.9 liters, and the four sections from the top holding three liters, determine the combined volume of the middle two sections. Let's think step by step and output the final answer within \\boxed{}.",
"role"... |
{
"ground_truth": "\\frac{8}{5}",
"style": "rule"
} | [
{
"content": "As shown in the diagram, there is a sequence of curves \\(P_{0}, P_{1}, P_{2}, \\cdots\\). It is given that \\(P_{0}\\) is an equilateral triangle with an area of 1. Each \\(P_{k+1}\\) is obtained from \\(P_{k}\\) by performing the following operations: each side of \\(P_{k}\\) is divided into thr... |
{
"ground_truth": "\\frac{8}{3}",
"style": "rule"
} | [
{
"content": "Given that $x$ is a root of the equation $x^{2}+x-6=0$, simplify $\\frac{x-1}{\\frac{2}{{x-1}}-1}$ and find its value. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "3 - 2\\sqrt{2}",
"style": "rule"
} | [
{
"content": "The function $y = \\frac{2}{x}$ is defined on the interval $[1, 2]$, and its graph has endpoints $A(1, 2)$ and $B(2, 1)$. Find the linear approximation threshold of the function. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "56",
"style": "rule"
} | [
{
"content": "Given 8 planes in space, for each pair of planes, the line of their intersection is noted. For each pair of these noted lines, the point of their intersection is noted (if the lines intersect). What is the maximum number of noted points that could be obtained? Let's think step by step and output t... |
{
"ground_truth": "33.0876",
"style": "rule"
} | [
{
"content": "If the total sum of squared deviations of a set of data is 100, and the correlation coefficient is 0.818, then the sum of squared residuals is. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "96",
"style": "rule"
} | [
{
"content": "Let $n$ be the answer to this problem. $a$ and $b$ are positive integers satisfying $$\\begin{aligned} & 3a+5b \\equiv 19 \\quad(\\bmod n+1) \\\\ & 4a+2b \\equiv 25 \\quad(\\bmod n+1) \\end{aligned}$$ Find $2a+6b$. Let's think step by step and output the final answer within \\boxed{}.",
"role"... |
{
"ground_truth": "7",
"style": "rule"
} | [
{
"content": "The graph below shows the number of home runs in April for the top hitters in the league. What is the mean (average) number of home runs hit by these players?\n\n[asy]\ndraw((0,0)--(0,7)--(24,7)--(24,0)--cycle);\nlabel(\"KEY:\",(3,5));\nfill((3,2.5)..(3.5,2)..(3,1.5)..(2.5,2)..cycle);\nlabel(\"- o... |
{
"ground_truth": "30",
"style": "rule"
} | [
{
"content": "A chocolate bar weighed 250 g and cost 50 rubles. Recently, for cost-saving purposes, the manufacturer reduced the weight of the bar to 200 g and increased its price to 52 rubles. By what percentage did the manufacturer's income increase? Let's think step by step and output the final answer within... |
{
"ground_truth": "101",
"style": "rule"
} | [
{
"content": "Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra $P_{i}$ is defined recursively as follows: $P_{0}$ is a regular tetrahedron whose volume is 1. To obtain $P_{i + 1}$, replace the midpoint triangle of every face of $P_{i}$ by an outw... |
{
"ground_truth": "360",
"style": "rule"
} | [
{
"content": "A permutation $a_1,a_2,\\cdots ,a_6$ of numbers $1,2,\\cdots ,6$ can be transformed to $1,2,\\cdots,6$ by transposing two numbers exactly four times. Find the number of such permutations. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "225",
"style": "rule"
} | [
{
"content": "Find the number of ordered pairs of integers $(a, b) \\in\\{1,2, \\ldots, 35\\}^{2}$ (not necessarily distinct) such that $a x+b$ is a \"quadratic residue modulo $x^{2}+1$ and 35 \", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following equivalent condit... |
{
"ground_truth": "49.27",
"style": "rule"
} | [
{
"content": "At a family outing to a theme park, the Thomas family, comprising three generations, plans to purchase tickets. The two youngest members, categorized as children, get a 40% discount. The two oldest members, recognized as seniors, enjoy a 30% discount. The middle generation no longer enjoys any dis... |
{
"ground_truth": "50",
"style": "rule"
} | [
{
"content": "A town's reservoir has a usable water volume of 120 million cubic meters. Assuming the annual precipitation remains unchanged, it can sustain a water supply for 160,000 people for 20 years. After urbanization and the migration of 40,000 new residents, the reservoir will only be able to maintain th... |
{
"ground_truth": "\\frac{7}{25}",
"style": "rule"
} | [
{
"content": "There are three video game systems: the Paystation, the WHAT, and the ZBoz2 \\pi, and none of these systems will play games for the other systems. Uncle Riemann has three nephews: Bernoulli, Galois, and Dirac. Bernoulli owns a Paystation and a WHAT, Galois owns a WHAT and a ZBoz2 \\pi, and Dirac o... |
{
"ground_truth": "64",
"style": "rule"
} | [
{
"content": "The total number of matches played in the 2006 World Cup competition can be calculated by summing the number of matches determined at each stage of the competition. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "66",
"style": "rule"
} | [
{
"content": "Consider a decreasing arithmetic sequence $\\{a\\_n\\}$ with the sum of its first $n$ terms denoted as $S\\_n$. If $a\\_3a\\_5=63$ and $a\\_2+{a}\\_{6} =16$,\n\n(1) Find the general term formula of the sequence.\n(2) For what value of $n$ does $S\\_n$ reach its maximum value? Also, find the maximu... |
{
"ground_truth": "125",
"style": "rule"
} | [
{
"content": "For any positive integer $a, \\sigma(a)$ denotes the sum of the positive integer divisors of $a$. Let $n$ be the least positive integer such that $\\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a$. Find the sum of the prime factors in the prime factorization of $n$. Let's think ... |
{
"ground_truth": "67.\\overline{3}",
"style": "rule"
} | [
{
"content": "The average of the seven numbers in a list is 62. The average of the first four numbers is 58. What is the average of the last three numbers? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "52",
"style": "rule"
} | [
{
"content": "Marcus and four of his relatives are at a party. Each pair of the five people are either friends or enemies. For any two enemies, there is no person that they are both friends with. In how many ways is this possible? Let's think step by step and output the final answer within \\boxed{}.",
"rol... |
{
"ground_truth": "2+6\\ln 3",
"style": "rule"
} | [
{
"content": "Let \\\\(f(x)=a(x-5)^{2}+6\\ln x\\\\), where \\\\(a\\in\\mathbb{R}\\\\), the tangent line of the curve \\\\(y=f(x)\\\\) at point \\\\((1,f(1))\\\\) intersects the \\\\(y\\\\)-axis at point \\\\((0,6)\\\\). \n\\\\((1)\\\\) Determine the value of \\\\(a\\\\); \n\\\\((2)\\\\) Find the intervals of ... |
{
"ground_truth": "89",
"style": "rule"
} | [
{
"content": "The sequence $\\{a\\_n\\}$ satisfies $a\\_1=1$, $a\\_2=3$, and $a_{n+2}=|a_{n+1}|-a_{n}$, where $n∈N^{*}$. Let $S_{n}$ denote the sum of the first $n$ terms of the sequence. Find $S_{100}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "16",
"style": "rule"
} | [
{
"content": "At a meeting of cactus enthusiasts, 80 cactophiles presented their collections, each consisting of cacti of different species. It turned out that no single species of cactus is found in all collections simultaneously, but any 15 people have cacti of the same species. What is the minimum total numb... |
{
"ground_truth": "20",
"style": "rule"
} | [
{
"content": "A dormitory is installing a shower room for 100 students. How many shower heads are economical if the boiler preheating takes 3 minutes per shower head, and it also needs to be heated during the shower? Each group is allocated 12 minutes for showering. Let's think step by step and output the final... |
{
"ground_truth": "132",
"style": "rule"
} | [
{
"content": "Find the smallest natural number $n$ such that the expression $n^2 - n + 11$ has exactly four prime factors, which may not be distinct. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "100902018",
"style": "rule"
} | [
{
"content": "Determine the smallest positive integer $A$ with an odd number of digits such that both $A$ and the number $B$, created by removing the middle digit of $A$, are divisible by $2018$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "243",
"style": "rule"
} | [
{
"content": "Find the smallest integer $n > 3$ such that, for each partition of $\\{3, 4, \\ldots, n\\}$ into two sets, at least one of these sets contains three (not necessarily distinct) numbers $a, b, c$ for which $ab = c$. Let's think step by step and output the final answer within \\boxed{}.",
"role":... |
{
"ground_truth": "1",
"style": "rule"
} | [
{
"content": "Find all the ways of placing the integers $1, 2, 3, \\ldots, 16$ in the boxes below, such that each integer appears in exactly one box, and the sum of every pair of neighboring integers is a perfect square. \n\n\\begin{asy}\nimport graph;\nreal r=10;\nsize(r*cm);\npicture square1;\ndraw(square1, (... |
{
"ground_truth": "8128",
"style": "rule"
} | [
{
"content": "Compute the smallest integer $n \\geq 4$ such that $\\binom{n}{4}$ ends in 4 or more zeroes (i.e., the rightmost four digits of $\\binom{n}{4}$ are $0000$). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "62208",
"style": "rule"
} | [
{
"content": "Let $v$ and $w$ be real numbers such that, for all real numbers $a$ and $b$, the inequality \\[(2^{a+b}+8)(3^a+3^b) \\leq v(12^{a-1}+12^{b-1}-2^{a+b-1})+w\\] holds. Compute the smallest possible value of $128v^2+w^2$. Let's think step by step and output the final answer within \\boxed{}.",
"ro... |
{
"ground_truth": "10053",
"style": "rule"
} | [
{
"content": "Define a *growing spiral* in the plane as a sequence of points with integer coordinates $P_0=(0,0), P_1, \\dots, P_n$ such that $n \\ge 2$ and:\n\n- The directed line segments $P_0P_1, P_1P_2, \\dots, P_{n-1}P_n$ follow successive coordinate directions: east (for $P_0P_1$), north, west, south, eas... |
{
"ground_truth": "14",
"style": "rule"
} | [
{
"content": "What is the maximal number of crosses that can fit in a $10 \\times 11$ board without overlapping? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "18",
"style": "rule"
} | [
{
"content": "A large number of rocks are placed on a table. On each turn, one may remove some rocks from the table following these rules: on the first turn, only one rock may be removed, and on every subsequent turn, one may remove either twice as many rocks or the same number of rocks as they have discarded o... |
{
"ground_truth": "126",
"style": "rule"
} | [
{
"content": "Let $n$ be a positive integer. When the leftmost digit of the standard base 10 representation of $n$ is shifted to the rightmost position (the units position), the result is $\\frac{n}{3}$. Find the smallest possible value of the sum of the digits of $n$. Let's think step by step and output the fi... |
{
"ground_truth": "1234",
"style": "rule"
} | [
{
"content": "Does there exist a four-digit positive integer with different non-zero digits, which has the following property: if we add the same number written in the reverse order, then we get a number divisible by $101$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "us... |
{
"ground_truth": "114",
"style": "rule"
} | [
{
"content": "How many ways are there to color the vertices of a cube using the colors red, blue, or green such that no edge connects two vertices of the same color? Note that rotations and reflections are considered distinct colorings. Let's think step by step and output the final answer within \\boxed{}.",
... |
{
"ground_truth": "4250",
"style": "rule"
} | [
{
"content": "Compute the smallest positive integer $M$ such that there exists a positive integer $n$ for which:\n\n- $M$ is the sum of the squares of some $n$ consecutive positive integers.\n- $2M$ is the sum of the squares of some $2n$ consecutive positive integers. Let's think step by step and output the fin... |
{
"ground_truth": "60",
"style": "rule"
} | [
{
"content": "The target below is made up of concentric circles with diameters $4$, $8$, $12$, $16$, and $20$. The area of the dark region is $n\\pi$. Find $n$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "1347",
"style": "rule"
} | [
{
"content": "A sequence $\\{a_n\\}$ is defined such that $a_i = i$ for $i = 1, 2, 3, \\ldots, 2020$ and for $i > 2020$, $a_i$ is the average of the previous 2020 terms. What is the largest integer less than or equal to $\\displaystyle\\lim_{n \\to \\infty} a_n$? Let's think step by step and output the final an... |
{
"ground_truth": "457",
"style": "rule"
} | [
{
"content": "Find the sum of the decimal digits of \\[ \\left\\lfloor \\frac{51525354555657\\dots979899}{50} \\right\\rfloor. \\] Here $\\left\\lfloor x \\right\\rfloor$ is the greatest integer not exceeding $x$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
{
"ground_truth": "167",
"style": "rule"
} | [
{
"content": "Let $a_0$, $a_1$, $a_2$, $\\dots$ be an infinite sequence of real numbers such that $a_0 = \\frac{4}{5}$ and \n\\[ \na_{n} = 2 a_{n-1}^2 - 1 \n\\] \nfor every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satis... |
{
"ground_truth": "1900",
"style": "rule"
} | [
{
"content": "We color some unit squares in a $99 \\times 99$ square grid with one of $5$ given distinct colors, such that each color appears the same number of times. On each row and on each column, there are no differently colored unit squares. Find the maximum possible number of colored unit squares. Let's t... |
{
"ground_truth": "2016",
"style": "rule"
} | [
{
"content": "Let \\( f \\) be a function from \\( \\{1, 2, \\ldots, 22\\} \\) to the positive integers such that \\( mn \\mid f(m) + f(n) \\) for all \\( m, n \\in \\{1, 2, \\ldots, 22\\} \\). If \\( d \\) is the number of positive divisors of \\( f(20) \\), compute the minimum possible value of \\( d \\). Let... |
{
"ground_truth": "40",
"style": "rule"
} | [
{
"content": "Let $\\triangle ABC$ be a triangle with $BA<AC$, $BC=10$, and $BA=8$. Let $H$ be the orthocenter of $\\triangle ABC$. Let $F$ be the point on segment $AC$ such that $BF=8$. Let $T$ be the point of intersection of $FH$ and the extension of line $BC$. Suppose that $BT=8$. Find the area of $\\triangl... |
{
"ground_truth": "50",
"style": "rule"
} | [
{
"content": "A $8 \\times 8$ board is given, with sides directed north-south and east-west. It is divided into $1 \\times 1$ cells in the usual manner. In each cell, there is at most one house. A house occupies only one cell.\n\nA house is considered \"in the shade\" if there is a house in each of the cells di... |
{
"ground_truth": "60",
"style": "rule"
} | [
{
"content": "A $3 \\times 3 \\times 3$ cube composed of $27$ unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a $3 \\times 3 \\times 1$ block (the order is irrelevant) with the property that the line joining the centers of the two cubes makes a $45^... |
{
"ground_truth": "539",
"style": "rule"
} | [
{
"content": "Compute the sum of all positive integers $n < 200$ such that $\\text{gcd}(n, k) \\neq 1$ for every $k \\in \\{2 \\cdot 11 \\cdot 19, 3 \\cdot 13 \\cdot 17, 5 \\cdot 11 \\cdot 13, 7 \\cdot 17 \\cdot 19\\}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
... |
{
"ground_truth": "1183",
"style": "rule"
} | [
{
"content": "Consider an infinite sequence $x_1, x_2, \\dots$ of positive integers such that, for every integer $n \\geq 1$: \n\n- If $x_n$ is even, then $x_{n+1} = \\frac{x_n}{2}$.\n- If $x_n$ is odd, then $x_{n+1} = \\frac{x_n - 1}{2} + 2^{k-1}$, where $2^{k-1} \\leq x_n < 2^k$.\n\nDetermine the smallest pos... |
{
"ground_truth": "32",
"style": "rule"
} | [
{
"content": "Find the smallest integer $k \\geq 2$ such that for every partition of the set $\\{2, 3, \\ldots, k\\}$ 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 output the final answer within \\boxed{}.",
... |
{
"ground_truth": "60",
"style": "rule"
} | [
{
"content": "A $3 \\times 3 \\times 3$ cube composed of $27$ unit cubes rests on a horizontal plane. Determine the number of ways to select two distinct unit cubes from a $3 \\times 3 \\times 1$ block (the order is irrelevant) such that the line joining the centers of the two cubes forms a $45^\\circ$ angle wi... |
{
"ground_truth": "31",
"style": "rule"
} | [
{
"content": "In circle $\\theta_1$ with radius $1$, circles $\\phi_1, \\phi_2, \\dots, \\phi_8$, with equal radii, are drawn such that for $1 \\le i \\le 8$, $\\phi_i$ is tangent to $\\omega_1$, $\\phi_{i-1}$, and $\\phi_{i+1}$, where $\\phi_0 = \\phi_8$ and $\\phi_1 = \\phi_9$. There exists a circle $\\omega_... |
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