data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The base of the pyramid \\( P A B C D \\) is a trapezoid \\( A B C D \\), with the base \\( A D \\) being twice as large as the base \\( B C \\). The segment \\( M N \\) is the midline of the triangle \\( A B P \\), parallel to the side \\( A B \\). Find the ratio of the volume... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{13}"
} | {
"split": "train",
"index": 25849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Person A arrives between 7:00 and 8:00, while person B arrives between 7:20 and 7:50. The one who arrives first waits for the other for 10 minutes, after which they leave. Calculate the probability that the two people will meet. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 25850,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The fraction $\\frac{1}{2015}$ has a unique \"(restricted) partial fraction decomposition\" of the form $\\frac{1}{2015}=\\frac{a}{5}+\\frac{b}{13}+\\frac{c}{31}$ where $a, b, c$ are integers with $0 \\leq a<5$ and $0 \\leq b<13$. Find $a+b$. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 25851,
"solution": "This is equivalent to $1=13 \\cdot 31 a+5 \\cdot 31 b+5 \\cdot 13 c$. Taking modulo 5 gives $1 \\equiv 3 \\cdot 1 a (\\bmod 5)$, so $a \\equiv 2(\\bmod 5)$. Taking modulo 13 gives $1 \\equiv 5 \\cdot 5 b=25 b \\equiv-b(\\bmod 13)$, so $b \\equiv 12 (\\bmod 13)$. Th... |
deepscaler | [
{
"role": "user",
"content": "If $M$ is $30 \\%$ of $Q$, $Q$ is $20 \\%$ of $P$, and $N$ is $50 \\%$ of $P$, then $\\frac {M}{N} =$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3}{25}"
} | {
"split": "train",
"index": 25852,
"solution": "1. **Express $M$ in terms of $Q$:**\n Given that $M$ is $30\\%$ of $Q$, we can write:\n \\[\n M = 0.30 \\cdot Q\n \\]\n\n2. **Express $Q$ in terms of $P$:**\n Given that $Q$ is $20\\%$ of $P$, we can write:\n \\[\n Q = 0.20 \\cdot P\n \\]\n\n3. **Su... |
deepscaler | [
{
"role": "user",
"content": "The line with equation $y=2x-6$ is translated upwards by 4 units. What is the $x$-intercept of the resulting line? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 25853,
"solution": "The line with equation $y=2 x-6$ has slope 2. When this line is translated, the slope does not change. The line with equation $y=2 x-6$ has $y$-intercept -6. When this line is translated upwards by 4 units, its $y$-intercept is translated upwards by 4 units and so ... |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of the expression \\((\\sqrt{2(1+\\cos 2x)} - \\sqrt{36 - 4\\sqrt{5}} \\sin x + 2) \\cdot (3 + 2\\sqrt{10 - \\sqrt{5}} \\cos y - \\cos 2y)\\). If the answer is not an integer, round it to the nearest whole number. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "-27"
} | {
"split": "train",
"index": 25854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(ABCD\\) be a quadrilateral inscribed in a unit circle with center \\(O\\). Suppose that \\(\\angle AOB = \\angle COD = 135^\\circ\\), and \\(BC = 1\\). Let \\(B'\\) and \\(C'\\) be the reflections of \\(A\\) across \\(BO\\) and \\(CO\\) respectively. Let \\(H_1\\) and \\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}(8-\\sqrt{6}-3\\sqrt{2})"
} | {
"split": "train",
"index": 25855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of three numbers \\(a\\), \\(b\\), and \\(c\\) is 150. If we increase \\(a\\) by 10, decrease \\(b\\) by 3, and multiply \\(c\\) by 4, the three resulting numbers are equal. What is the value of \\(b\\)? Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\frac{655}{9}"
} | {
"split": "train",
"index": 25856,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers $n \\geq 2$ are there such that whenever $z_1, z_2, \\dots, z_n$ are complex numbers such that\n\\[|z_1| = |z_2| = \\dots = |z_n| = 1 \\text{ and } z_1 + z_2 + \\dots + z_n = 0,\\]\nthen the numbers $z_1, z_2, \\dots, z_n$ are equally spaced on the unit ... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 25857,
"solution": "1. **Case $n=2$:** \n Consider $z_1$ and $z_2$ on the unit circle such that $|z_1| = |z_2| = 1$ and $z_1 + z_2 = 0$. This implies $z_2 = -z_1$. Since both $z_1$ and $-z_1$ lie on the unit circle and are diametrically opposite, they are equally spaced on the unit... |
deepscaler | [
{
"role": "user",
"content": "Find the $r$ that satisfies $\\log_{16} (r+16) = \\frac{5}{4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 25858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Select 5 people from 4 boys and 5 girls to participate in a math extracurricular group. How many different ways are there to select under the following conditions?\n\n (1) Select 2 boys and 3 girls, and girl A must be selected;\n\n (2) Select at most 4 girls, and boy A and girl... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 25859,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the product \\[\\frac{8}{4}\\cdot\\frac{12}{8}\\cdot\\frac{16}{12} \\dotsm \\frac{4n+4}{4n} \\dotsm \\frac{2008}{2004}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "502"
} | {
"split": "train",
"index": 25860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arrange the letters a, a, b, b, c, c into three rows and two columns, with the requirement that each row has different letters and each column also has different letters, and find the total number of different arrangements. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 25861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The volume of a box is 360 cubic units where $a, b,$ and $c$ are integers such that $1 < c < b < a$ with $c$ being a prime number. Determine the largest possible value of $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 25862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can 8 people be seated in a row of chairs if three of those people, Alice, Bob, and Charlie, refuse to sit next to each other in any order? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36000"
} | {
"split": "train",
"index": 25863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the plane Cartesian coordinate system \\( xOy \\), a moving line \\( l \\) is tangent to the parabola \\( \\Gamma: y^{2} = 4x \\), and intersects the hyperbola \\( \\Omega: x^{2} - y^{2} = 1 \\) at one point on each of its branches, left and right, labeled \\( A \\) and \\( ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"split": "train",
"index": 25864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function $f(x)$ be defined on $\\mathbb{R}$ and satisfy $f(2-x) = f(2+x)$ and $f(7-x) = f(7+x)$. Also, in the closed interval $[0, 7]$, only $f(1) = f(3) = 0$. Determine the number of roots of the equation $f(x) = 0$ in the closed interval $[-2005, 2005]$. Let's think s... | math | {
"style": "rule",
"ground_truth": "802"
} | {
"split": "train",
"index": 25865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular hexagon \\(ABCDEF\\) is inscribed in a circle with a radius of \\(3+\\sqrt{3}\\). Find the radius of the inscribed circle of triangle \\(ACD\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 25866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct four-digit positive integers are there such that the product of their digits equals 18? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 25867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the number of faces, edges and vertices of a triangular prism?\n\n[asy]\ndraw((0,0)--(10,0)--(5,8.7)--cycle);\ndraw((0,0)--(20,20),dashed);\ndraw((10,0)--(30,20));\ndraw((5,8.7)--(25,28.7));\ndraw((25,28.7)--(30,20)--(20,20)--cycle,dashed);\ndraw((25,28.7)--(... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 25868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular box has interior dimensions 6-inches by 5-inches by 10-inches. The box is filled with as many solid 3-inch cubes as possible, with all of the cubes entirely inside the rectangular box. What percent of the volume of the box is taken up by the cubes? Let's think ste... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 25869,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\left(2x-a\\right)^{7}=a_{0}+a_{1}(x+1)+a_{2}(x+1)^{2}+a_{3}(x+1)^{3}+\\ldots +a_{7}(x+1)^{7}$, and $a_{4}=-560$.<br/>$(1)$ Find the value of the real number $a$;<br/>$(2)$ Find the value of $|a_{1}|+|a_{2}|+|a_{3}|+\\ldots +|a_{6}|+|a_{7}|$. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "2186"
} | {
"split": "train",
"index": 25870,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the square of 1085 without using a calculator. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1177225"
} | {
"split": "train",
"index": 25871,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=-3x^2+x-4$, $g(x)=-5x^2+3x-8$, and $h(x)=5x^2+5x+1$. Express $f(x)+g(x)+h(x)$ as a single polynomial, with the terms in order by decreasing degree. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3x^2 +9x -11"
} | {
"split": "train",
"index": 25872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a point P on the curve $y = x^2 - \\ln x$, find the minimum distance from point P to the line $y = x + 2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 25873,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\(\\log_{2} a + \\log_{2} b \\geq \\gamma\\), determine the smallest positive value \\(\\delta\\) for \\(a+b\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 25874,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a clock workshop, there are several digital clocks (more than one), displaying time in a 12-hour format (the number of hours on the clock screen ranges from 1 to 12). All clocks run at the same speed but show completely different times: the number of hours on the screen of a... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 25875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let's divide a sequence of natural numbers into groups:\n\n\\((1), (2,3), (4,5,6), (7,8,9,10), \\ldots\\)\n\nLet \\( S_{n} \\) denote the sum of the \\( n \\)-th group of numbers. Find \\( S_{16} - S_{4} - S_{1} \\). Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"split": "train",
"index": 25876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ a, b, c, d,m, n \\in \\mathbb{Z}^\\plus{}$ such that \\[ a^2\\plus{}b^2\\plus{}c^2\\plus{}d^2 \\equal{} 1989,\\]\n\\[ a\\plus{}b\\plus{}c\\plus{}d \\equal{} m^2,\\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$ Let's think... | math | {
"style": "rule",
"ground_truth": "m = 9,n = 6"
} | {
"split": "train",
"index": 25877,
"solution": "\nTo solve for the values of \\( m \\) and \\( n \\), we have the given conditions:\n\n1. \\( a^2 + b^2 + c^2 + d^2 = 1989 \\)\n2. \\( a + b + c + d = m^2 \\)\n3. The largest of \\( a, b, c, d \\) is \\( n^2 \\)\n\nWe need to find positive integers \\( m \\) and \\... |
deepscaler | [
{
"role": "user",
"content": "How many positive three-digit integers are there in which each of the three digits is either prime or a perfect square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "343"
} | {
"split": "train",
"index": 25878,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( P \\) be a point inside regular pentagon \\( ABCDE \\) such that \\( \\angle PAB = 48^\\circ \\) and \\( \\angle PDC = 42^\\circ \\). Find \\( \\angle BPC \\), in degrees. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 25879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three natural numbers are written on the board: two ten-digit numbers \\( a \\) and \\( b \\), and their sum \\( a + b \\). What is the maximum number of odd digits that could be written on the board? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 25880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $8 \\times 8 \\times 8$ cube has three of its faces painted red and the other three faces painted blue (ensuring that any three faces sharing a common vertex are not painted the same color), and then it is cut into 512 $1 \\times 1 \\times 1$ smaller cubes. Among these 512 sm... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 25881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Fibonacci sequence is defined $F_1 = F_2 = 1$ and $F_n = F_{n - 1} + F_{n - 2}$ for all $n \\ge 3.$\n\nThe Fibonacci numbers $F_a,$ $F_b,$ $F_c$ form an increasing arithmetic sequence. If $a + b + c = 2000,$ compute $a.$ Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "665"
} | {
"split": "train",
"index": 25882,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = Ax - 2B^2$ and $g(x) = Bx$, where $B \\neq 0$. If $f(g(1)) = 0$, what is $A$ in terms of $B$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2B"
} | {
"split": "train",
"index": 25883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $EFGH$ has sides of length 4. Segments $EK$ and $EL$ divide the square's area into two equal parts. Calculate the length of segment $EK$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"split": "train",
"index": 25884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle \\\\(ABC\\\\), where \\\\(\\angle C = 90^{\\circ}\\\\) and \\\\(\\angle A = 30^{\\circ}\\\\), the eccentricity of the ellipse, which has \\\\(A\\\\) and \\\\(B\\\\) as its foci and passes through point \\\\(C\\\\), is \\_\\_\\_\\_\\_. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} - 1"
} | {
"split": "train",
"index": 25885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $M$ on the parabola $y=4x^{2}$ is at a distance of $1$ from the focus. The ordinate of point $M$ is __________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{16}"
} | {
"split": "train",
"index": 25886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quadrilateral \\(ABCD\\) is circumscribed around a circle with a radius of \\(1\\). Find the greatest possible value of \\(\\left| \\frac{1}{AC^2} + \\frac{1}{BD^2} \\right|\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/4"
} | {
"split": "train",
"index": 25887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integers less than $2000$ that are neither $5$-nice nor $6$-nice. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1333"
} | {
"split": "train",
"index": 25888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A palindrome is an integer that reads the same forward and backward, such as 3663. What percent of the palindromes between 100 and 500 contain at least one 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10\\%"
} | {
"split": "train",
"index": 25889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest common divisor of $654321$ and $543210$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 25890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 11 ones, 22 twos, 33 threes, and 44 fours on a blackboard. The following operation is performed: each time, erase 3 different numbers and add 2 more of the fourth number that was not erased. For example, if 1 one, 1 two, and 1 three are erased in one operation, write ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 25891,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be real numbers so that the roots of\n\\[z^2 + (10 + ai) z + (27 + bi) = 0\\]are complex conjugates. Enter the ordered pair $(a,b).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0,0)"
} | {
"split": "train",
"index": 25892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $C: \\frac{x^{2}}{4} - \\frac{y^{2}}{3} = 1$, with its right vertex at $P$.\n(1) Find the standard equation of the circle centered at $P$ and tangent to both asymptotes of the hyperbola $C$;\n(2) Let line $l$ pass through point $P$ with normal vector $\\over... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 25893,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of rectangle \\( PQRS \\) if the perimeter of rectangle \\( TVWY \\) is 60? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 25894,
"solution": "The perimeter of \\( TVWY \\) is 60, so \\( 12r = 60 \\) or \\( r = 5 \\). The area of \\( PQRS \\) is \\( 30 \\times 20 = 600 \\)."
} |
deepscaler | [
{
"role": "user",
"content": "Let $[x]$ denote the greatest integer less than or equal to the real number $x$,\n$$\n\\begin{array}{c}\nS=\\left[\\frac{1}{1}\\right]+\\left[\\frac{2}{1}\\right]+\\left[\\frac{1}{2}\\right]+\\left[\\frac{2}{2}\\right]+\\left[\\frac{3}{2}\\right]+ \\\\\n{\\left[\\frac{4}{2}\\ri... | math | {
"style": "rule",
"ground_truth": "1078"
} | {
"split": "train",
"index": 25895,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a $6 \\times 6$ grid of lights. There is a switch at the top of each column and on the left of each row. A light will only turn on if the switches corresponding to both its column and its row are in the \"on\" position. Compute the number of different configurations of... | math | {
"style": "rule",
"ground_truth": "3970"
} | {
"split": "train",
"index": 25896,
"solution": "Take any configuration of switches such that there exists at least one row and one column which are switched on. There are $\\left(2^{6}-1\\right)^{2}=3969$ such configurations. We prove that any two such configurations $A$ and $B$ lead to a different set of lights... |
deepscaler | [
{
"role": "user",
"content": "Given the binary operations $\\clubsuit$ and $\\spadesuit$ defined as $\\clubsuit$ : $a^{\\log_5(b)}$ and $\\spadesuit$ : $a^{\\frac{1}{\\log_5(b)}}$ for real numbers $a$ and $b$ where these expressions are defined, a sequence $(b_n)$ is defined recursively as $b_4 = 4 \\spades... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 25897,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $g : \\mathbb{R} \\to \\mathbb{R}$ satisfies\n\\[g(x) + 3g(1 - x) = 2x^2 + 1\\]for all $x.$ Find $g(5).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"split": "train",
"index": 25898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a function $f(x)$ defined on $R$ such that $f(2x+2)=-f(2x)$. If $f(x)=4x+3$ when $x\\in(\\frac{1}{2},\\frac{5}{4})$, then $f(2023)=\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 25899,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the quadratic polynomial, with real coefficients, which has $3 + i$ as a root, and where the coefficient of $x^2$ is 2. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2x^2 - 12x + 20"
} | {
"split": "train",
"index": 25900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two arbitrary positive integers \\( n \\) and \\( k \\), let \\( f(n, k) \\) denote the number of unit squares that one of the diagonals of an \\( n \\times k \\) grid rectangle passes through. How many such pairs \\( (n, k) \\) are there where \\( n \\geq k \\) and \\( f... | math | {
"style": "rule",
"ground_truth": "874"
} | {
"split": "train",
"index": 25901,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, a line passing through its right focus $F$ and parallel to the asymptote $y = -\\frac{b}{a}x$ intersects the right branch of the hyperbola and the other asymptote at points $A$ and $B$ respect... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 25902,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x=\\frac{\\sum\\limits_{n=1}^{44} \\cos n^\\circ}{\\sum\\limits_{n=1}^{44} \\sin n^\\circ}$. What is the greatest integer that does not exceed $100x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "241"
} | {
"split": "train",
"index": 25903,
"solution": "A slight variant of the above solution, note that\n\\begin{eqnarray*} \\sum_{n=1}^{44} \\cos n + \\sum_{n=1}^{44} \\sin n &=& \\sum_{n=1}^{44} \\sin n + \\sin(90-n)\\\\ &=& \\sqrt{2}\\sum_{n=1}^{44} \\cos(45-n) = \\sqrt{2}\\sum_{n=1}^{44} \\cos n\\\\ \\sum_{n=1}^{4... |
deepscaler | [
{
"role": "user",
"content": "Given that the first term of a geometric sequence $\\{a\\_n\\}$ is $\\frac{3}{2}$, and the sum of the first $n$ terms is $S\\_n$, where $n \\in \\mathbb{N}^*$. Also, $-2S\\_2$, $S\\_3$, and $4S\\_4$ form an arithmetic sequence.\n\n1. Find the general term formula for the sequen... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 25904,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Factor the expression $2x(x-3) + 3(x-3)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2x+3)(x-3)"
} | {
"split": "train",
"index": 25905,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A card is chosen at random from a standard deck of 52 cards, and then it is replaced and another card is chosen. What is the probability that at least one of the cards is a diamond or an ace? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{88}{169}"
} | {
"split": "train",
"index": 25906,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $17^{-1} \\equiv 26 \\pmod{53}$, find $36^{-1} \\pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 25907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $\\left(\\frac{{a}^{2}-1}{a-3}-a-1\\right) \\div \\frac{a+1}{{a}^{2}-6a+9}$, where $a=3-\\sqrt{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2\\sqrt{2}"
} | {
"split": "train",
"index": 25908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(2x)=\\frac{2}{2+x}$ for all $x>0$, then $2f(x)=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{4+x}"
} | {
"split": "train",
"index": 25909,
"solution": "Given the function $f(2x) = \\frac{2}{2+x}$ for all $x > 0$, we need to find the expression for $2f(x)$.\n\n1. **Substitute $x$ with $\\frac{x}{2}$ in the given function**: \n Since we know $f(2x) = \\frac{2}{2+x}$, we can replace $x$ with $\\frac{x}{2}$ to find... |
deepscaler | [
{
"role": "user",
"content": "Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.293"
} | {
"split": "train",
"index": 25910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin \\left(\\theta -\\frac{\\pi }{3}\\right)=\\frac{3}{5}$, then $\\sin \\left(2\\theta -\\frac{\\pi }{6}\\right)=$_______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{25}"
} | {
"split": "train",
"index": 25911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{14}{39}"
} | {
"split": "train",
"index": 25912,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of $\\sin 135^{\\circ}\\cos 15^{\\circ}-\\cos 45^{\\circ}\\sin (-15^{\\circ})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 25913,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the reciprocals of the roots of $x^2-13x+4=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{4}"
} | {
"split": "train",
"index": 25914,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In coordinate space, $A = (6,-7,7),$ $B = (16,-17,12),$ $C = (0,3,-6),$ and $D = (2,-5,10).$ Find the intersection point of lines $AB$ and $CD.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{4}{3}, -\\frac{7}{3}, \\frac{14}{3} \\right)"
} | {
"split": "train",
"index": 25915,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The set of vectors $\\left\\{ \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix}, \\begin{pmatrix} 3 \\\\ k \\end{pmatrix} \\right\\}$ is linearly dependent. Find all possible values of $k.$ Enter all the possible values, separated by commas. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 25916,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, let $D$ be the region represented by the inequality $|x| + |y| \\leq 1$, and $E$ be the region consisting of points whose distance from the origin is no greater than 1. If a point is randomly thrown into $E$, the probability that this p... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{\\pi}"
} | {
"split": "train",
"index": 25917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the period of $y = \\sin x + \\cos x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\pi"
} | {
"split": "train",
"index": 25918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of the sum $$\\frac{3}{1^{2} \\cdot 2^{2}}+\\frac{5}{2^{2} \\cdot 3^{2}}+\\frac{7}{3^{2} \\cdot 4^{2}}+\\cdots+\\frac{29}{14^{2} \\cdot 15^{2}}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{224}{225}"
} | {
"split": "train",
"index": 25919,
"solution": "The sum telescopes as $$\\left(\\frac{1}{1^{2}}-\\frac{1}{2^{2}}\\right)+\\left(\\frac{1}{2^{2}}-\\frac{1}{3^{2}}\\right)+\\cdots+\\left(\\frac{1}{14^{2}}-\\frac{1}{15^{2}}\\right)=\\frac{1}{1^{2}}-\\frac{1}{15^{2}}=\\frac{224}{225}$$"
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the greatest integer less than or equal to \\[\\frac{5^{150} + 3^{150}}{5^{147} + 3^{147}}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "124"
} | {
"split": "train",
"index": 25920,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $\\triangle DEF$ has a right angle at $F$, $\\angle D = 60^\\circ$, and $DF=12$. Find the radius of the incircle of $\\triangle DEF$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6(\\sqrt{3}-1)"
} | {
"split": "train",
"index": 25921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "With four standard six-sided dice in play, Vivian rolls all four and can choose to reroll any subset of them. To win, Vivian needs the sum of the four dice after possibly rerolling some of them to be exactly 12. Vivian plays optimally to maximize her chances of winning. What is... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 25922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The same amount of steel used to create six solid steel balls, each with a radius of 2 inches, is used to create one larger steel ball. What is the radius of the larger ball? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt[3]{2}"
} | {
"split": "train",
"index": 25923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the parallelogram shown, in square feet?\n\n[asy]\ndraw((0,0)--(15,0)--(19,3)--(4,3)--cycle,linewidth(1));\ndraw((15,0)--(15,3),dashed);\ndraw((15,2.5)--(15.5,2.5)--(15.5,3));\nlabel(\"15 ft\",(7.5,0),S);\nlabel(\"3 ft\",(15,1.5),W);\nlabel(\"5 ft\",(17,1.5)... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 25924,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{v_n\\}$ defined by $v_1 = 7$ and the relationship $v_{n+1} - v_n = 2 + 5(n-1)$ for $n=1,2,3,\\ldots$, express $v_n$ as a polynomial in $n$ and find the sum of its coefficients. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 25925,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During the New Year's Day holiday of 2018, 8 high school students from four different classes, each with 2 students, plan to carpool for a trip. Classes are denoted as (1), (2), (3), and (4). They will be divided between two cars, Car A and Car B, each with a capacity for 4 stu... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 25926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given triangle $\\triangle ABC$ with $\\cos C = \\frac{2}{3}$, $AC = 4$, and $BC = 3$, calculate the value of $\\tan B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{5}"
} | {
"split": "train",
"index": 25927,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the following 4 propositions, the correct one is: \n(1) If a solid's three views are completely identical, then the solid is a cube; \n(2) If a solid's front view and top view are both rectangles, then the solid is a cuboid; \n(3) If a solid's three views are all recta... | math | {
"style": "rule",
"ground_truth": "(3)"
} | {
"split": "train",
"index": 25928,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the number 79777 has the digit 9 crossed out, the result is the number 7777. How many different five-digit numbers exist from which 7777 can be obtained by crossing out one digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 25929,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the integer root of the polynomial \\[2x^3 + ax^2 + bx + c = 0,\\] where $a, b$, and $c$ are rational numbers. The equation has $4-2\\sqrt{3}$ as a root and another root whose sum with $4-2\\sqrt{3}$ is 8. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 25930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be a set of $2020$ distinct points in the plane. Let \n\\[M=\\{P:P\\text{ is the midpoint of }XY\\text{ for some distinct points }X,Y\\text{ in }S\\}.\\]\n\nFind the least possible value of the number of points in $M$ . Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "4037"
} | {
"split": "train",
"index": 25931,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is one-half times two-thirds times three-fourths? 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": 25932,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points \\( P \\) and \\( Q \\) are located on the sides \\( AB \\) and \\( AC \\) of triangle \\( ABC \\) such that \\( AP:PB = 1:4 \\) and \\( AQ:QC = 3:1 \\). Point \\( M \\) is chosen randomly on side \\( BC \\). Find the probability that the area of triangle \\( ABC \\) exc... | math | {
"style": "rule",
"ground_truth": "13/40"
} | {
"split": "train",
"index": 25933,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petya cut an 8x8 square along the borders of the cells into parts of equal perimeter. It turned out that not all parts are equal. What is the maximum possible number of parts he could get? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 25934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the eighth-sphere $\\left\\{(x, y, z) \\mid x, y, z \\geq 0, x^{2}+y^{2}+z^{2}=1\\right\\}$. What is the area of its projection onto the plane $x+y+z=1$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi \\sqrt{3}}{4}"
} | {
"split": "train",
"index": 25935,
"solution": "Consider the three flat faces of the eighth-ball. Each of these is a quarter-circle of radius 1, so each has area $\\frac{\\pi}{4}$. Furthermore, the projections of these faces cover the desired area without overlap. To find the projection factor one can find the c... |
deepscaler | [
{
"role": "user",
"content": "Five identical rectangles are arranged to form a larger rectangle $PQRS$, as shown. The area of $PQRS$ is $4000$. What is the length, $x$, rounded off to the nearest integer? [asy]\nreal x = 1; real w = 2/3;\n\n// Draw outer square and labels\npair s = (0, 0); pair r = (2 * x... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 25936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function \\( f(x) = 5(x+1)^{2} + \\frac{a}{(x+1)^{5}} \\) for \\( a > 0 \\), find the minimum value of \\( a \\) such that \\( f(x) \\geqslant 24 \\) when \\( x \\geqslant 0 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{\\left(\\frac{24}{7}\\right)^7}"
} | {
"split": "train",
"index": 25937,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "231361"
} | {
"split": "train",
"index": 25938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The bases \\(AB\\) and \\(CD\\) of the trapezoid \\(ABCD\\) are 367 and 6 respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors \\(\\overrightarrow{AD}\\) and \\(\\overrightarrow{BC}\\). Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "2202"
} | {
"split": "train",
"index": 25939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of three-element subsets of the set \\(\\{1, 2, 3, 4, \\ldots, 120\\}\\) for which the sum of the three elements is a multiple of 3. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "93640"
} | {
"split": "train",
"index": 25940,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the sequences $(a_n),(b_n)$ by\n\\begin{align*}\n& a_n, b_n > 0, \\forall n\\in\\mathbb{N_+} \\\\ \n& a_{n+1} = a_n - \\frac{1}{1+\\sum_{i=1}^n\\frac{1}{a_i}} \\\\ \n& b_{n+1} = b_n + \\frac{1}{1+\\sum_{i=1}^n\\frac{1}{b_i}}\n\\end{align*}\n1) If $a_{100}b_{100} = a_{101... | math | {
"style": "rule",
"ground_truth": "199"
} | {
"split": "train",
"index": 25941,
"solution": "\n\nDefine the sequences \\( (a_n) \\) and \\( (b_n) \\) by\n\\[\n\\begin{align*}\n& a_n, b_n > 0, \\forall n \\in \\mathbb{N_+}, \\\\\n& a_{n+1} = a_n - \\frac{1}{1 + \\sum_{i=1}^n \\frac{1}{a_i}}, \\\\\n& b_{n+1} = b_n + \\frac{1}{1 + \\sum_{i=1}^n \\frac{1}{b_i}... |
deepscaler | [
{
"role": "user",
"content": "Eleanor is making chocolate chip cookies for her friends. If she divides the cookies equally among $11$ of her friends, she'll have $4$ cookies left over. If she divides the cookies equally among $7$ of her friends, she'll have $1$ cookie left over. Assuming that Eleanor made f... | math | {
"style": "rule",
"ground_truth": "107"
} | {
"split": "train",
"index": 25942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integers less than or equal to $1200$ that are neither $5$-nice nor $6$-nice. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "800"
} | {
"split": "train",
"index": 25943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A car travels the 120 miles from $A$ to $B$ at 60 miles per hour, and then returns to $A$ on the same road. If the average rate of the round trip is 45 miles per hour, what is the rate, in miles per hour, of the car traveling back from $B$ to $A$? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 25944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a box containing 3 red balls and 3 white balls, which are identical in size and shape. A fair die is rolled, and the number rolled determines the number of balls drawn from the box. What is the probability that the number of red balls drawn is greater than the number o... | math | {
"style": "rule",
"ground_truth": "19/60"
} | {
"split": "train",
"index": 25945,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A deck of fifty-two cards consists of four $1$'s, four $2$'s, ..., four $13$'s. Two matching pairs (two sets of two cards with the same number) are removed from the deck. After removing these cards, find the probability, represented as a fraction $m/n$ in simplest form, where $... | math | {
"style": "rule",
"ground_truth": "299"
} | {
"split": "train",
"index": 25946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of the expression \\((\\sqrt{2(1+\\cos 2x)} - \\sqrt{3-\\sqrt{2}} \\sin x + 1) \\cdot (3 + 2\\sqrt{7-\\sqrt{2}} \\cos y - \\cos 2y)\\). If the answer is not an integer, round it to the nearest whole number. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"split": "train",
"index": 25947,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"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 put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1,099,999,999,999"
} | {
"split": "train",
"index": 25948,
"solution": ""
} |
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