data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given the parabola \\( C: y^2 = 2x \\) whose directrix intersects the x-axis at point \\( A \\), a line \\( l \\) passing through point \\( B(-1,0) \\) is tangent to the parabola \\( C \\) at point \\( K \\). A line parallel to \\( l \\) is drawn through point \\( A \\) and intersects the parabola... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "5900",
"split": "train"
} |
math | [
{
"content": "Evaluate the product \\[ (n-1) \\cdot n \\cdot (n+1) \\cdot (n+2) \\cdot (n+3), \\] where $n=2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "5901",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system \\( xOy \\), find the area of the graph formed by all points \\( (x, y) \\) that satisfy \\( \\lfloor x \\rfloor \\cdot \\lfloor y \\rfloor = 2013 \\), where \\( \\lfloor x \\rfloor \\) represents the greatest integer less than or equal to the real number \\( x... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "5902",
"split": "train"
} |
math | [
{
"content": "An acronym XYZ is drawn within a 2x4 rectangular grid with grid lines spaced 1 unit apart. The letter X is formed by two diagonals crossing in a $1 \\times 1$ square. Y consists of a vertical line segment and two slanted segments each forming 45° with the vertical line, making up a symmetric lette... | {
"ground_truth": "5 + 4\\sqrt{2} + \\sqrt{5}",
"style": "rule"
} | {
"index": "5903",
"split": "train"
} |
math | [
{
"content": "Let $\\triangle ABC$ have side lengths $AB = 12$, $AC = 16$, and $BC = 20$. Inside $\\angle BAC$, two circles are positioned, each tangent to rays $\\overline{AB}$ and $\\overline{AC}$, and the segment $\\overline{BC}$. Compute the distance between the centers of these two circles. Please output t... | {
"ground_truth": "20\\sqrt{2}",
"style": "rule"
} | {
"index": "5904",
"split": "train"
} |
math | [
{
"content": "In the diagram, two pairs of identical isosceles triangles are cut off of square $ABCD$, leaving rectangle $PQRS$. The total area cut off is $200 \\text{ m}^2$. What is the length of $PR$, in meters? [asy]\nsize(5cm);\n\npair a = (0, 1); pair b = (1, 1); pair c = (1, 0); pair d = (0, 0);\npair s =... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "5905",
"split": "train"
} |
math | [
{
"content": "Given the quadratic function \\( y = x^{2} - \\frac{2n+1}{n(n+2)} x + \\frac{n+1}{n(n+2)^{2}} \\), the length of the segment intercepted on the \\( x \\)-axis is \\( d_n \\). Find the value of \\( \\sum_{n=1}^{100} d_n \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "\\frac{7625}{10302}",
"style": "rule"
} | {
"index": "5906",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, $AX = XY = YB = BC$ and the measure of angle $ABC$ is 120 degrees. What is the number of degrees in the measure of angle $BAC$?\n\n[asy]\npair A,X,Y,B,C;\nX = A + dir(30); Y = X + dir(0); B = Y + dir(60); C = B + dir(-30);\ndraw(B--Y--X--B--C--A--X);\nlabel(\"$A$\",A,W); label(\... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "5907",
"split": "train"
} |
math | [
{
"content": "The two lines $y = 4x - 19$ and $2x+y = 95$ intersect. What is the value of $x$ at the point of intersection? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "5908",
"split": "train"
} |
math | [
{
"content": "Points $A$ and $B$ are 600 kilometers apart. Two people, A and B, start bicycling from point $A$ to point $B$ simultaneously. Person A bicycles 40 kilometers every day, while person B bicycles 60 kilometers every other day, resting every other day. At the end of a certain day, the distance person ... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "5909",
"split": "train"
} |
math | [
{
"content": "Consider all triangles $ABC$ satisfying in the following conditions: $AB = AC$, $D$ is a point on $AC$ for which $BD \\perp AC$, $AC$ and $CD$ are integers, and $BD^{2} = 57$. Among all such triangles, the smallest possible value of $AC$ is Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "5910",
"split": "train"
} |
math | [
{
"content": "The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than ... | {
"ground_truth": "294",
"style": "rule"
} | {
"index": "5911",
"split": "train"
} |
math | [
{
"content": "Lesha's summer cottage has the shape of a nonagon with three pairs of equal and parallel sides. Lesha knows that the area of the triangle with vertices at the midpoints of the remaining sides of the nonagon is 12 sotkas. Help him find the area of the entire summer cottage. Please output the final ... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "5912",
"split": "train"
} |
math | [
{
"content": "Placing no more than one X in each small square, what is the greatest number of X's that can be put on the grid shown without getting three X's in a row vertically, horizontally, or diagonally?\n\n[asy] for(int a=0; a<4; ++a) { draw((a,0)--(a,3)); } for(int b=0; b<4; ++b) { draw((0,b)--(3,b... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "5913",
"split": "train"
} |
math | [
{
"content": "Two chords, $AB$ and $CD,$ meet inside a circle at $P.$ If $AP = 3$ and $CP = 8,$ then what is $\\frac{BP}{DP}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8}{3}",
"style": "rule"
} | {
"index": "5914",
"split": "train"
} |
math | [
{
"content": "There are integers $x$ that satisfy the inequality $|x-2000|+|x| \\leq 9999$. Find the number of such integers $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9999",
"style": "rule"
} | {
"index": "5915",
"split": "train"
} |
math | [
{
"content": "Given vectors $$\\overrightarrow {m}=( \\sqrt {3}\\sin x+\\cos x,1), \\overrightarrow {n}=(\\cos x,-f(x)), \\overrightarrow {m}\\perp \\overrightarrow {n}$$.\n(1) Find the monotonic intervals of $f(x)$;\n(2) Given that $A$ is an internal angle of $\\triangle ABC$, and $$f\\left( \\frac {A}{2}\\rig... | {
"ground_truth": "\\frac { \\sqrt {3}-1}{4}",
"style": "rule"
} | {
"index": "5916",
"split": "train"
} |
math | [
{
"content": "What is the remainder when $(x + 1)^{2010}$ is divided by $x^2 + x + 1$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "5917",
"split": "train"
} |
math | [
{
"content": "A room has a floor with dimensions \\(7 \\times 8\\) square meters, and the ceiling height is 4 meters. A fly named Masha is sitting in one corner of the ceiling, while a spider named Petya is in the opposite corner of the ceiling. Masha decides to travel to visit Petya by the shortest route that ... | {
"ground_truth": "\\sqrt{265}",
"style": "rule"
} | {
"index": "5918",
"split": "train"
} |
math | [
{
"content": "Define an odd function f(x) on ℝ that satisfies f(x+1) is an even function, and when x ∈ [0,1], f(x) = x(3-2x). Evaluate f(31/2). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "5919",
"split": "train"
} |
math | [
{
"content": "Six distinct positive integers are randomly chosen between $1$ and $2006$, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of $5$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "5920",
"split": "train"
} |
math | [
{
"content": "Antônio needs to find a code with 3 different digits \\( A, B, C \\). He knows that \\( B \\) is greater than \\( A \\), \\( A \\) is less than \\( C \\), and also:\n\\[\n\\begin{array}{cccc}\n & B & B \\\\\n+ & A & A \\\\\n\\hline\n& C & C \\\\\n\\end{array} = 242\n\\]\n\nWhat is the code that An... | {
"ground_truth": "232",
"style": "rule"
} | {
"index": "5921",
"split": "train"
} |
math | [
{
"content": "Consider a regular octagon. How many triangles can be formed whose vertices are the vertices of the octagon? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "56",
"style": "rule"
} | {
"index": "5922",
"split": "train"
} |
math | [
{
"content": "Let \\( a \\) and \\( b \\) be real numbers, and consider the function \\( f(x) = x^{3} + a x^{2} + b x \\). If there exist three real numbers \\( x_{1}, x_{2}, x_{3} \\) such that \\( x_{1} + 1 \\leqslant x_{2} \\leqslant x_{3} - 1 \\), and \\( f(x_{1}) = f(x_{2}) = f(x_{3}) \\), find the minimum... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "5923",
"split": "train"
} |
math | [
{
"content": "For any positive integer $a$ , define $M(a)$ to be the number of positive integers $b$ for which $a+b$ divides $ab$ . Find all integer(s) $a$ with $1\\le a\\le 2013$ such that $M(a)$ attains the largest possible value in the range of $a$ . Please output the final answer within \\box... | {
"ground_truth": "1680",
"style": "rule"
} | {
"index": "5924",
"split": "train"
} |
math | [
{
"content": "A sequence of positive integers \\(a_{1}, a_{2}, \\ldots\\) is such that for each \\(m\\) and \\(n\\) the following holds: if \\(m\\) is a divisor of \\(n\\) and \\(m < n\\), then \\(a_{m}\\) is a divisor of \\(a_{n}\\) and \\(a_{m} < a_{n}\\). Find the least possible value of \\(a_{2000}\\). Plea... | {
"ground_truth": "128",
"style": "rule"
} | {
"index": "5925",
"split": "train"
} |
math | [
{
"content": "**Q13.** Determine the greatest value of the sum $M=11xy+3x+2012yz$ , where $x,y,z$ are non negative integers satisfying condition $x+y+z=1000.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "503000000",
"style": "rule"
} | {
"index": "5926",
"split": "train"
} |
math | [
{
"content": "In a right tetrahedron \\(ABCD\\), \\(DA\\), \\(DB\\), and \\(DC\\) are mutually perpendicular. Let \\(S\\) and \\(R\\) represent its surface area and the radius of the circumscribed sphere, respectively. What is the maximum value of \\(\\frac{S}{R^2}\\)? Please output the final answer within \\bo... | {
"ground_truth": "\\frac{2}{3}(3+\\sqrt{3})",
"style": "rule"
} | {
"index": "5927",
"split": "train"
} |
math | [
{
"content": "Points $P$ and $Q$ are on a circle with radius $7$ and $PQ = 8$. Point $R$ is the midpoint of the minor arc $PQ$. Calculate the length of line segment $PR$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{32}",
"style": "rule"
} | {
"index": "5928",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{M}$ be a matrix, and let $\\mathbf{v}$ and $\\mathbf{w}$ be vectors, such that\n\\[\\mathbf{M} \\mathbf{v} = \\begin{pmatrix} 2 \\\\ 3 \\end{pmatrix} \\quad \\text{and} \\quad \\mathbf{M} \\mathbf{w} = \\begin{pmatrix} -2 \\\\ -5 \\end{pmatrix}.\\]Compute $\\mathbf{M} (\\mathbf{v} + ... | {
"ground_truth": "\\begin{pmatrix} -4 \\\\ -12 \\end{pmatrix}",
"style": "rule"
} | {
"index": "5929",
"split": "train"
} |
math | [
{
"content": "If $x$ satisfies $x^2 + 3x + \\frac{3}x + \\frac{1}{x^2} = 26$ and $x$ can be written as $a + \\sqrt{b}$ where $a$ and $b$ are positive integers, then find $a + b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "5930",
"split": "train"
} |
math | [
{
"content": "A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and \"flips\" every 3rd locker (\"fipping\") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it. ... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "5931",
"split": "train"
} |
math | [
{
"content": "How, without any measuring tools, can you measure 50 cm from a string that is $2/3$ meters long? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "5932",
"split": "train"
} |
math | [
{
"content": "How many nondecreasing sequences $a_{1}, a_{2}, \\ldots, a_{10}$ are composed entirely of at most three distinct numbers from the set $\\{1,2, \\ldots, 9\\}$ (so $1,1,1,2,2,2,3,3,3,3$ and $2,2,2,2,5,5,5,5,5,5$ are both allowed)? Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "3357",
"style": "rule"
} | {
"index": "5933",
"split": "train"
} |
math | [
{
"content": "Given the numbers 1, 2, 3, 4, 5, and 6, calculate the probability that three randomly selected numbers can be arranged in an arithmetic sequence in a certain order. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{10}",
"style": "rule"
} | {
"index": "5934",
"split": "train"
} |
math | [
{
"content": "A positive integer is said to be bi-digital if it uses two different digits, with each digit used exactly twice. For example, 1331 is bi-digital, whereas 1113, 1111, 1333, and 303 are not. Determine the exact value of the integer \\( b \\), the number of bi-digital positive integers. Please output... | {
"ground_truth": "243",
"style": "rule"
} | {
"index": "5935",
"split": "train"
} |
math | [
{
"content": "In Cologne, there were three brothers who had 9 containers of wine. The first container had a capacity of 1 quart, the second contained 2 quarts, with each subsequent container holding one more quart than the previous one, so the last container held 9 quarts. The task is to divide the wine equally... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "5936",
"split": "train"
} |
math | [
{
"content": "Patrick and Anderson are having a snowball fight. Patrick throws a snowball at Anderson which is shaped like a sphere with a radius of 10 centimeters. Anderson catches the snowball and uses the snow from the snowball to construct snowballs with radii of 4 centimeters. Given that the total volume o... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "5937",
"split": "train"
} |
math | [
{
"content": "At the end of the day, a bakery had $\\frac{6}{7}$ of a pie left over. Louie, Duey and Huey split the pie, with each taking home the same amount of leftover pie. How much pie did Louie take home? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{7}",
"style": "rule"
} | {
"index": "5938",
"split": "train"
} |
math | [
{
"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 point falls into $D$ ... | {
"ground_truth": "\\frac{4}{\\pi}",
"style": "rule"
} | {
"index": "5939",
"split": "train"
} |
math | [
{
"content": "$$\n\\text{Consider the system of inequalities:}\n\\begin{cases}\nx + 2y \\leq 6 \\\\\n3x + y \\geq 3 \\\\\nx \\leq 4 \\\\\ny \\geq 0\n\\end{cases}\n$$\nDetermine the number of units in the length of the longest side of the polygonal region formed by this system. Express your answer in simplest ra... | {
"ground_truth": "2\\sqrt{5}",
"style": "rule"
} | {
"index": "5940",
"split": "train"
} |
math | [
{
"content": "In the equation $w^3+x^3+y^3=z^3$, $w^3$, $x^3$, $y^3$, and $z^3$ are distinct, consecutive positive perfect cubes listed in ascending order. What is the smallest possible value of $z$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "5941",
"split": "train"
} |
math | [
{
"content": "Country music songs are added to a playlist so that now $40\\%$ of the songs are Country. If the ratio of Hip Hop songs to Pop songs remains the same, what percentage of the total number of songs are now Hip Hop? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "39\\%",
"style": "rule"
} | {
"index": "5942",
"split": "train"
} |
math | [
{
"content": "Given real numbers \\(a_{1}, a_{2}, \\cdots, a_{n}\\) which are all greater than 0 (where \\(n\\) is a natural number no less than 4) and satisfy the equation \\(a_{1} + a_{2} + \\cdots + a_{n} = 1\\), find the maximum value of the sum \\(S = \\sum_{k=1}^{n} \\frac{a_{k}^{2}}{a_{k} + a_{k+1} + a_{... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "5943",
"split": "train"
} |
math | [
{
"content": "Let $S$ be the set of nonzero real numbers. Let $f : S \\to \\mathbb{R}$ be a function such that\n\n(i) $f(1) = 1,$\n(ii) $f \\left( \\frac{1}{x + y} \\right) = f \\left( \\frac{1}{x} \\right) + f \\left( \\frac{1}{y} \\right)$ for all $x,$ $y \\in S$ such that $x + y \\in S,$ and\n(iii) $(x + y)... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "5944",
"split": "train"
} |
math | [
{
"content": "A fair coin is flipped $8$ times. What is the probability that at least $6$ consecutive flips come up heads? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{128}",
"style": "rule"
} | {
"index": "5945",
"split": "train"
} |
math | [
{
"content": "If $a$ is a nonzero integer and $b$ is a positive number such that $ab^2=\\log_{10} b$, what is the median of the set $\\{0, 1, a, b,\n1/b\\}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "b",
"style": "rule"
} | {
"index": "5946",
"split": "train"
} |
math | [
{
"content": "What is 25% of 60? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "5947",
"split": "train"
} |
math | [
{
"content": "If $m$ and $n$ are odd integers, how many terms in the expansion of $(m+n)^6$ are odd? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "5948",
"split": "train"
} |
math | [
{
"content": "Fill the five numbers $2015, 2016, 2017, 2018, 2019$ into the five boxes labeled \" $D, O, G, C, W$ \" such that $D+O+G=C+O+W$. How many different ways can this be done? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "5949",
"split": "train"
} |
math | [
{
"content": "Niall's four children have different integer ages under 18. The product of their ages is 882. What is the sum of their ages? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "5950",
"split": "train"
} |
math | [
{
"content": "Two workers started paving paths in a park simultaneously from point $A$. The first worker paves the section $A-B-C$ and the second worker paves the section $A-D-E-F-C$. They worked at constant speeds and finished at the same time, having spent 9 hours on the job. It is known that the second worke... | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "5951",
"split": "train"
} |
math | [
{
"content": "Find all values of \\( x \\) for which the greater of the numbers \\( \\sqrt{\\frac{x}{2}} \\) and \\( \\operatorname{tg} x \\) is not greater than 1. Provide the total length of the intervals on the number line that satisfy this condition, rounding the result to the nearest hundredth if necessary... | {
"ground_truth": "1.21",
"style": "rule"
} | {
"index": "5952",
"split": "train"
} |
math | [
{
"content": "The number of distinct points common to the curves $x^2+4y^2=1$ and $4x^2+y^2=4$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "5953",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive real number $r$ with the following property: For every choice of $2023$ unit vectors $v_1,v_2, \\dots ,v_{2023} \\in \\mathbb{R}^2$ , a point $p$ can be found in the plane such that for each subset $S$ of $\\{1,2, \\dots , 2023\\}$ , the sum $$ \\sum_{i \\in ... | {
"ground_truth": "\\frac{2023}{2}",
"style": "rule"
} | {
"index": "5954",
"split": "train"
} |
math | [
{
"content": "Compute $({11011_{(2)}} - {101_{(2)}} = )$\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_$(.$ (represented in binary) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10110_{(2)}",
"style": "rule"
} | {
"index": "5955",
"split": "train"
} |
math | [
{
"content": "The positive integer divisors of 294, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 21? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "5956",
"split": "train"
} |
math | [
{
"content": "The ratio of the areas of two squares is $\\frac{50}{98}$. After rationalizing the denominator, express the simplified form of the ratio of their side lengths in the form $\\frac{a \\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. Find the sum $a+b+c$. Please output the final answer within \\bo... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "5957",
"split": "train"
} |
math | [
{
"content": "Let $A B C$ be a triangle with $A B=13, B C=14$, and $C A=15$. We construct isosceles right triangle $A C D$ with $\\angle A D C=90^{\\circ}$, where $D, B$ are on the same side of line $A C$, and let lines $A D$ and $C B$ meet at $F$. Similarly, we construct isosceles right triangle $B C E$ with $... | {
"ground_truth": "-\\frac{5}{13}",
"style": "rule"
} | {
"index": "5958",
"split": "train"
} |
math | [
{
"content": "For what values of the parameter \\( a \\in \\mathbb{R} \\) does the maximum distance between the roots of the equation\n\n\\[ a \\tan^{3} x + \\left(2-a-a^{2}\\right) \\tan^{2} x + \\left(a^{2}-2a-2\\right) \\tan x + 2a = 0 \\]\n\nbelonging to the interval \\(\\left(-\\frac{\\pi}{2} , \\frac{\\pi... | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "5959",
"split": "train"
} |
math | [
{
"content": "The average of the numbers \\(1, 2, 3, \\dots, 100, x\\) is \\(50x\\). What is \\(x\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5050}{5049}",
"style": "rule"
} | {
"index": "5960",
"split": "train"
} |
math | [
{
"content": "What is the coefficient of $x^5$ when $$2x^5 - 4x^4 + 3x^3 - x^2 + 2x - 1$$ is multiplied by $$x^3 + 3x^2 - 2x + 4$$ and the like terms are combined? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "5961",
"split": "train"
} |
math | [
{
"content": "Simplify first, then evaluate: $\\left(\\frac{x}{x-1}-1\\right) \\div \\frac{{x}^{2}-1}{{x}^{2}-2x+1}$, where $x=\\sqrt{5}-1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "5962",
"split": "train"
} |
math | [
{
"content": "Find the solutions to $z^3 = -8.$ Enter the solutions, separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2, 1 + i \\sqrt{3}, 1 - i \\sqrt{3}",
"style": "rule"
} | {
"index": "5963",
"split": "train"
} |
math | [
{
"content": "The region between the graph of $y = f (x)$ and the $x$-axis, shaded in this figure, has an area of 10 square units. What will be the area between the graph of $y = 3f (x -2)$ and the $x$-axis?\n\n[asy]\ndefaultpen(linewidth(0.75));\nfill((10,0)..(30,20)..(40,15)--(50,40)..(58,39)--(70,0)--cycle,g... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "5964",
"split": "train"
} |
math | [
{
"content": "The fifth, sixth and seventh terms of an arithmetic sequence are 5, 9 and 13, respectively. What is the sum of the first four terms of the sequence? [asy]\nunitsize(8mm);\ndefaultpen(linewidth(.7pt)+fontsize(12pt));\n\nlabel(\"\\underline{\\quad}, \\underline{\\quad}, \\underline{\\quad}, \\underl... | {
"ground_truth": "-20",
"style": "rule"
} | {
"index": "5965",
"split": "train"
} |
math | [
{
"content": "How many solutions does the equation $\\tan x = \\tan (\\tan x)$ have in the interval $0 \\le x \\le \\tan^{-1} 1000$? Assume $\\tan \\theta > \\theta$ for $0 < \\theta < \\frac{\\pi}{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "318",
"style": "rule"
} | {
"index": "5966",
"split": "train"
} |
math | [
{
"content": "Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "5967",
"split": "train"
} |
math | [
{
"content": "The five solutions to the equation\\[(z-1)(z^2+2z+4)(z^2+4z+6)=0\\] may be written in the form $x_k+y_ki$ for $1\\le k\\le 5,$ where $x_k$ and $y_k$ are real. Let $\\mathcal E$ be the unique ellipse that passes through the points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4),$ and $(x_5,y_5)$. The ecce... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "5968",
"split": "train"
} |
math | [
{
"content": "A positive two-digit number is odd and is a multiple of 9. The product of its digits is a perfect square. What is this two-digit number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "99",
"style": "rule"
} | {
"index": "5969",
"split": "train"
} |
math | [
{
"content": "The arithmetic mean of seven numbers is 42. If three new numbers \\( x, y, \\) and \\( z \\) are added to the list, the mean of the ten-member list becomes 50. What is the mean of \\( x, y, \\) and \\( z \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{206}{3}",
"style": "rule"
} | {
"index": "5970",
"split": "train"
} |
math | [
{
"content": "Find\n\\[\\sum_{n = 1}^\\infty \\frac{n^2 + n - 1}{(n + 2)!}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "5971",
"split": "train"
} |
math | [
{
"content": "Write in ascending order the multiples of 3 which, when 1 is added, are perfect squares, i.e., $3, 15, 24, 48, \\ldots$ What is the multiple of 3 in the $2006^{\\mathrm{th}}$ position? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9060099",
"style": "rule"
} | {
"index": "5972",
"split": "train"
} |
math | [
{
"content": "Calculate the value of $\\displaystyle\\frac{245^2-225^2}{20}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "470",
"style": "rule"
} | {
"index": "5973",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_1=1$, $a_n=\\log_n(n+1)$ $(n\\geqslant 2,n\\in \\mathbb{N}^*)$, define: $k$ $(k\\in \\mathbb{N}^*)$ that makes the product $a_1\\cdot a_2\\cdot \\cdots \\cdot a_k$ an integer as \"simple number\". Then, the sum of all \"simple numbers\" within the interva... | {
"ground_truth": "2035",
"style": "rule"
} | {
"index": "5974",
"split": "train"
} |
math | [
{
"content": "2001 coins, each valued at 1, 2, or 3, are arranged in a row. The coins are placed such that:\n- Between any two coins of value 1, there is at least one other coin.\n- Between any two coins of value 2, there are at least two other coins.\n- Between any two coins of value 3, there are at least thre... | {
"ground_truth": "501",
"style": "rule"
} | {
"index": "5975",
"split": "train"
} |
math | [
{
"content": "A $4\\times 4$ block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "5976",
"split": "train"
} |
math | [
{
"content": "Determine the length of $BC$ in an acute triangle $ABC$ with $\\angle ABC = 45^{\\circ}$ , $OG = 1$ and $OG \\parallel BC$ . (As usual $O$ is the circumcenter and $G$ is the centroid.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "5977",
"split": "train"
} |
math | [
{
"content": "Gerard cuts a large rectangle into four smaller rectangles. The perimeters of three of these smaller rectangles are 16, 18, and 24. What is the perimeter of the fourth small rectangle?\nA) 8\nB) 10\nC) 12\nD) 14\nE) 16 Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "5978",
"split": "train"
} |
math | [
{
"content": "Simplify first, then find the value of the algebraic expression $\\frac{a}{{{a^2}-2a+1}}÷({1+\\frac{1}{{a-1}}})$, where $a=\\sqrt{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}+1",
"style": "rule"
} | {
"index": "5979",
"split": "train"
} |
math | [
{
"content": "Eight teams participated in a football tournament, and each team played exactly once against each other team. If a match was drawn then both teams received 1 point; if not then the winner of the match was awarded 3 points and the loser received no points. At the end of the tournament the total num... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "5980",
"split": "train"
} |
math | [
{
"content": "The distance between A and C is the absolute value of (k-7) plus the distance between B and C is the square root of ((k-4)^2 + (-1)^2). Find the value of k that minimizes the sum of these two distances. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{11}{2}",
"style": "rule"
} | {
"index": "5981",
"split": "train"
} |
math | [
{
"content": "In a bag, there are three balls of different colors: red, yellow, and blue, each color having one ball. Each time a ball is drawn from the bag, its color is recorded and then the ball is put back. The drawing stops when all three colors of balls have been drawn, what is the probability of stopping... | {
"ground_truth": "\\frac{14}{81}",
"style": "rule"
} | {
"index": "5982",
"split": "train"
} |
math | [
{
"content": "Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $P$ and the point $\\left(\\frac58, \\frac38 \\right)$ is greater than or equal to $\\frac12$ can be writte... | {
"ground_truth": "171",
"style": "rule"
} | {
"index": "5983",
"split": "train"
} |
math | [
{
"content": "In trapezoid \\(ABCD\\), the side \\(AD\\) is perpendicular to the bases and is equal to 9. \\(CD\\) is 12, and the segment \\(AO\\), where \\(O\\) is the point of intersection of the diagonals of the trapezoid, is equal to 6. Find the area of triangle \\(BOC\\). Please output the final answer wit... | {
"ground_truth": "\\frac{108}{5}",
"style": "rule"
} | {
"index": "5984",
"split": "train"
} |
math | [
{
"content": "In the coordinate plane, a point is called a $\\text{lattice point}$ if both of its coordinates are integers. Let $A$ be the point $(12,84)$ . Find the number of right angled triangles $ABC$ in the coordinate plane $B$ and $C$ are lattice points, having a right angle at vertex $A$ ... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "5985",
"split": "train"
} |
math | [
{
"content": "For certain pairs $(m,n)$ of positive integers with $m\\geq n$ there are exactly $50$ distinct positive integers $k$ such that $|\\log m - \\log k| < \\log n$. Find the sum of all possible values of the product $mn$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "5986",
"split": "train"
} |
math | [
{
"content": "Giselle will combine blue paint, green paint, and white paint in the ratio $3:2:4$, respectively. If she uses $12$ quarts of white paint, how many quarts of green paint should she use? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "5987",
"split": "train"
} |
math | [
{
"content": "Let side $AD$ of convex quadrilateral $ABCD$ be extended through $D$, and let side $BC$ be extended through $C$, to meet in point $E.$ Let $S$ be the degree-sum of angles $CDE$ and $DCE$, and let $S'$ represent the degree-sum of angles $BAD$ and $ABC.$ If $r=S/S'$, then: Please output the final an... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "5988",
"split": "train"
} |
math | [
{
"content": "Find the value of $x$ that satisfies $\\frac{\\sqrt{5x}}{\\sqrt{3(x-1)}}=2$. Express your answer in simplest fractional form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{12}{7}",
"style": "rule"
} | {
"index": "5989",
"split": "train"
} |
math | [
{
"content": "A square $EFGH$ is inscribed in the region bounded by the parabola $y = x^2 - 6x + 5$ and the $x$-axis. Find the area of square $EFGH$.\n\\[\n\\text{[asy]}\nunitsize(0.8 cm);\n\nreal parab (real x) {\n return(x^2 - 6*x + 5);\n}\n\npair E, F, G, H;\nreal x = -1 + sqrt(3);\n\nE = (3 - x,0);\nF = (3... | {
"ground_truth": "24 - 8\\sqrt{5}",
"style": "rule"
} | {
"index": "5990",
"split": "train"
} |
math | [
{
"content": "Find the range of the function \\[f(x) = \\frac{x}{x^2-x+1},\\]where $x$ can be any real number. (Give your answer in interval notation.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "[-\\tfrac13, 1]",
"style": "rule"
} | {
"index": "5991",
"split": "train"
} |
math | [
{
"content": "Calculate: \\( 1 \\times 2 - 3 \\times 4 + 5 \\times 6 - 7 \\times 8 + \\cdots + 2015 \\times 2016 \\) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2033136",
"style": "rule"
} | {
"index": "5992",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, $BC = 23$, $CA = 27$, and $AB = 30$. Points $V$ and $W$ are on $\\overline{AC}$ with $V$ on $\\overline{AW}$, points $X$ and $Y$ are on $\\overline{BC}$ with $X$ on $\\overline{CY}$, and points $Z$ and $U$ are on $\\overline{AB}$ with $Z$ on $\\overline{BU}$. In addition, the po... | {
"ground_truth": "318",
"style": "rule"
} | {
"index": "5993",
"split": "train"
} |
math | [
{
"content": "A school is hosting a Mathematics Culture Festival, and it was recorded that on that day, there were more than 980 (at least 980 and less than 990) students visiting. Each student visits the school for a period of time and then leaves, and once they leave, they do not return. Regardless of how the... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "5994",
"split": "train"
} |
math | [
{
"content": "A rectangle with dimensions 100 cm by 150 cm is tilted so that one corner is 20 cm above a horizontal line, as shown. To the nearest centimetre, the height of vertex $Z$ above the horizontal line is $(100+x) \\mathrm{cm}$. What is the value of $x$? Please output the final answer within \\boxed{}."... | {
"ground_truth": "67",
"style": "rule"
} | {
"index": "5995",
"split": "train"
} |
math | [
{
"content": "Let $f : \\mathbb{R} \\to \\mathbb{R}$ be a function such that\n\\[f(f(x) + y) = f(x + y) + xf(y) - xy - x + 1\\]for all real numbers $x$ and $y.$\n\nLet $n$ be the number of possible values of $f(1),$ and let $s$ be the sum of all possible values of $f(1).$ Find $n \\times s.$ Please output the ... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "5996",
"split": "train"
} |
math | [
{
"content": "Simplify $\\displaystyle\\frac{2+2i}{-3+4i}$. Express your answer as a complex number in the form $a+bi$, where $a$ and $b$ are real numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{25} - \\frac{14}{25}i",
"style": "rule"
} | {
"index": "5997",
"split": "train"
} |
math | [
{
"content": "One line is parameterized by\n\\[\\begin{pmatrix} -1 + s \\\\ 3 - ks \\\\ 1 + ks \\end{pmatrix}.\\]Another line is parameterized by\n\\[\\begin{pmatrix} t/2 \\\\ 1 + t \\\\ 2 - t \\end{pmatrix}.\\]If the lines are coplanar (i.e. there is a plane that contains both lines), then find $k.$ Please out... | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "5998",
"split": "train"
} |
math | [
{
"content": "How many points on the parabola \\( y = x^2 \\) (other than the origin) have the property that the tangent at these points intersects both coordinate axes at points with integer coordinates whose absolute values do not exceed 2020? Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "44",
"style": "rule"
} | {
"index": "5999",
"split": "train"
} |
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