data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system $xOy$, a polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinate equation of circle $C$ is $\\rho^2 - 2m\\rho\\cos\\theta + 4\\rho\\sin\... | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{3}"
} | {
"split": "train",
"index": 1000,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle \\( \\triangle ABC \\) and a point \\( D \\) on its side \\( BC \\) such that \\( AB = 2BD \\) and \\( AC = 3CD \\), and an ellipse \\( \\Gamma \\) can be constructed with \\( A \\) and \\( D \\) as foci passing through points \\( B \\) and \\( C \\), find the ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{21}}{7}"
} | {
"split": "train",
"index": 1001,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For all non-zero real numbers $x$ and $y$ such that $x-y=xy$, $\\frac{1}{x}-\\frac{1}{y}$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 1002,
"solution": "1. **Start with the given equation and manipulate it:**\n Given that \\(x - y = xy\\), we can rearrange this equation to:\n \\[\n x - y - xy = 0\n \\]\n Adding 1 to both sides, we get:\n \\[\n x - y - xy + 1 = 1\n \\]\n Factoring the left-hand side... |
deepscaler | [
{
"role": "user",
"content": "Ten Cs are written in a row. Some Cs are upper-case and some are lower-case, and each is written in one of two colors, green and yellow. It is given that there is at least one lower-case C, at least one green C, and at least one C that is both upper-case and yellow. Furthermore... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 1003,
"solution": "By the conditions of the problem, we must pick some point in the line where the green Cs transition to yellow, and some point where the upper-case Cs transition to lower-case. We see that the first transition must occur before the second, and that they cannot occur ... |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that $1.2n-4.4<5.2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 1004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\frac{1}{2+\frac{2}{3}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"split": "train",
"index": 1005,
"solution": "Evaluating, $\frac{1}{2+\frac{2}{3}}=\frac{1}{\frac{8}{3}}=\frac{3}{8}$."
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a_{n}\\}$ satisfies $a_{1}=1$, $({{a}\\_{n+1}}-{{a}\\_{n}}={{(-1)}^{n+1}}\\dfrac{1}{n(n+2)})$, find the sum of the first 40 terms of the sequence $\\{(-1)^{n}a_{n}\\}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{20}{41}"
} | {
"split": "train",
"index": 1006,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the function $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ by $$f(x)= \\begin{cases}\\frac{1}{x^{2}+\\sqrt{x^{4}+2 x}} & \\text { if } x \\notin(-\\sqrt[3]{2}, 0] \\\\ 0 & \\text { otherwise }\\end{cases}$$ The sum of all real numbers $x$ for which $f^{10}(x)=1$ can be writt... | math | {
"style": "rule",
"ground_truth": "932"
} | {
"split": "train",
"index": 1007,
"solution": "If $x \\in(-\\sqrt[3]{2}, 0]$, it is evidently not a solution, so let us assume otherwise. Then, we find $$f(x)=\\frac{\\sqrt{x^{4}+2 x}-x^{2}}{2 x}$$ which implies that $x f(x)^{2}+x^{2} f(x)-1 / 2=0$, by reverse engineering the quadratic formula. Therefore, if $x>... |
deepscaler | [
{
"role": "user",
"content": "Consider a $2 \\times n$ grid of points and a path consisting of $2 n-1$ straight line segments connecting all these $2 n$ points, starting from the bottom left corner and ending at the upper right corner. Such a path is called efficient if each point is only passed through onc... | math | {
"style": "rule",
"ground_truth": "\\binom{4030}{2015}"
} | {
"split": "train",
"index": 1008,
"solution": "The general answer is $\\binom{2(n-1)}{n-1}$ : Simply note that the points in each column must be taken in order, and anything satisfying this avoids intersections, so just choose the steps during which to be in the first column."
} |
deepscaler | [
{
"role": "user",
"content": "In the arithmetic sequence $\\{a_n\\}$, it is given that $a_{15}+a_{16}+a_{17}=-45$ and $a_{9}=-36$. Let $S_n$ be the sum of the first $n$ terms.\n$(1)$ Find the minimum value of $S_n$ and the corresponding value of $n$;\n$(2)$ Calculate $T_n=|a_1|+|a_2|+\\cdots+|a_n|$. Let's t... | math | {
"style": "rule",
"ground_truth": "-630"
} | {
"split": "train",
"index": 1009,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Side $\\overline{AB}$ of $\\triangle ABC$ has length $10$. The bisector of angle $A$ meets $\\overline{BC}$ at $D$, and $CD = 3$. The set of all possible values of $AC$ is an open interval $(m,n)$. What is $m+n$? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 1010,
"solution": "1. **Assign Variables and Use the Angle Bisector Theorem**:\n Let $AC = x$. By the Angle Bisector Theorem, we have:\n \\[\n \\frac{AB}{AC} = \\frac{BD}{CD} \\implies \\frac{10}{x} = \\frac{BD}{3}\n \\]\n Solving for $BD$, we get:\n \\[\n BD = \\frac{10... |
deepscaler | [
{
"role": "user",
"content": "If $f(1) = 3$, $f(2)= 12$, and $f(x) = ax^2 + bx + c$, what is the value of $f(3)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 1011,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If three different lines $x+y=1$, $x-y=1$, and $ax+y=1$ cannot form a triangle, then the value of the real number $a$ is. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 1012,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A labor employment service center has 7 volunteers ready to provide labor employment guidance on the streets over the weekend, with 6 people to be arranged for duty on Saturday and Sunday. If 3 people are arranged for each day, there are a total of _______ different arrangement... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 1013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the distance, in units, between the points $(-3, -4)$ and $(4, -5)$? Express your answer in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5\\sqrt{2}"
} | {
"split": "train",
"index": 1014,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose \n\\[\\frac{1}{x^3 - 2x^2 - 13x + 10} = \\frac{A}{x+2} + \\frac{B}{x-1} + \\frac{C}{(x-1)^2}\\] \nwhere $A$, $B$, and $C$ are real constants. What is $A$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 1015,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function, defined on the set of positive integers, is such that \\( f(x y) = f(x) + f(y) \\) for all \\( x \\) and \\( y \\). It is known that \\( f(10) = 14 \\) and \\( f(40) = 20 \\).\n\nWhat is the value of \\( f(500) \\)? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 1016,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right pyramid has a square base that measures 10 cm on each side. Its peak is 12 cm above the center of its base. What is the sum of the lengths of the pyramid's eight edges? Express your answer to the nearest whole number.\n\n[asy]\nsize(150);\ndraw((0,0)--(3,3)--(13,3)--(10... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 1017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Li is ready to complete the question over the weekend: Simplify and evaluate $(3-2x^{2}-5x)-(\\square x^{2}+3x-4)$, where $x=-2$, but the coefficient $\\square$ is unclearly printed.<br/>$(1)$ She guessed $\\square$ as $8$. Please simplify $(3-2x^{2}-5x)-(8x^{2}+3x-4)$ and find... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 1018,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle has dimensions 12 by 15, and a circle centered at one of its vertices has a radius of 15. What is the area of the union of the regions enclosed by the rectangle and the circle? Express your answer in terms of \\( \\pi \\). Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "180 + 168.75\\pi"
} | {
"split": "train",
"index": 1019,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x, y \\) be nonnegative integers such that \\( x + 2y \\) is a multiple of 5, \\( x + y \\) is a multiple of 3, and \\( 2x + y \\geq 99 \\). Find the minimum possible value of \\( 7x + 5y \\). Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "366"
} | {
"split": "train",
"index": 1020,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a die is rolled, event \\( A = \\{1, 2, 3\\} \\) consists of rolling one of the faces 1, 2, or 3. Similarly, event \\( B = \\{1, 2, 4\\} \\) consists of rolling one of the faces 1, 2, or 4.\n\nThe die is rolled 10 times. It is known that event \\( A \\) occurred exactly 6 ti... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{3}"
} | {
"split": "train",
"index": 1021,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid \\(ABCD\\), the bases \\(AD\\) and \\(BC\\) are equal to 8 and 18, respectively. It is known that the circumcircle of triangle \\(ABD\\) is tangent to lines \\(BC\\) and \\(CD\\). Find the perimeter of the trapezoid. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 1022,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A student walks from intersection $A$ to intersection $B$ in a city layout as described: the paths allow him to move only east or south. Every morning, he walks from $A$ to $B$, but now he needs to pass through intersections $C$ and then $D$. The paths from $A$ to $C$ have 3 ea... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{77}"
} | {
"split": "train",
"index": 1023,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(13 points) A rental company owns 100 cars. When the monthly rent for each car is set at 3000 yuan, all cars can be rented out. For every 50 yuan increase in the monthly rent per car, there will be one more car that remains unrented. The maintenance cost for each rented car is ... | math | {
"style": "rule",
"ground_truth": "307050"
} | {
"split": "train",
"index": 1024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Juca is a scout exploring the vicinity of his camp. After collecting fruits and wood, he needs to fetch water from the river and return to his tent. Represent Juca by the letter $J$, the river by the letter $r$, and his tent by the letter $B$. The distance from the feet of the ... | math | {
"style": "rule",
"ground_truth": "180\\sqrt{2}"
} | {
"split": "train",
"index": 1025,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\overline{CH}$ be an altitude of $\\triangle ABC$. Let $R\\,$ and $S\\,$ be the points where the circles inscribed in the triangles $ACH\\,$ and $BCH^{}_{}$ are tangent to $\\overline{CH}$. If $AB = 1995\\,$, $AC = 1994\\,$, and $BC = 1993\\,$, then $RS\\,$ can be express... | math | {
"style": "rule",
"ground_truth": "997"
} | {
"split": "train",
"index": 1026,
"solution": "[asy] unitsize(48); pair A,B,C,H; A=(8,0); B=origin; C=(3,4); H=(3,0); draw(A--B--C--cycle); draw(C--H); label(\"$A$\",A,SE); label(\"$B$\",B,SW); label(\"$C$\",C,N); label(\"$H$\",H,NE); draw(circle((2,1),1)); pair [] x=intersectionpoints(C--H,circle((2,1),1)); dot... |
deepscaler | [
{
"role": "user",
"content": "For any finite set $X$, let $| X |$ denote the number of elements in $X$. Define \\[ S_n = \\sum | A \\cap B | , \\] where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\\left\\{ 1 , 2 , 3, \\cdots , n \\right\\}$ with $|A| = |B|$. For e... | math | {
"style": "rule",
"ground_truth": "245"
} | {
"split": "train",
"index": 1027,
"solution": "We take cases based on the number of values in each of the subsets in the pair. Suppose we have $k$ elements in each of the subsets in a pair (for a total of n elements in the set). The expected number of elements in any random pair will be $n \\cdot \\frac{k}{n} \\... |
deepscaler | [
{
"role": "user",
"content": "In the number \\(2016*****02*\\), each of the 5 stars must be replaced by any of the digits \\(0, 2, 4, 5, 7, 9\\) (digits can repeat) so that the resulting 11-digit number is divisible by 15. How many ways can this be done? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "864"
} | {
"split": "train",
"index": 1028,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer with exactly 14 positive divisors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 1029,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There exist real numbers $t$ and $s$ such that\n\\[\\begin{pmatrix} 2 \\\\ 0 \\end{pmatrix} + t \\begin{pmatrix} 7 \\\\ -5 \\end{pmatrix} = \\begin{pmatrix} 1 \\\\ -1 \\end{pmatrix} + s \\begin{pmatrix} -2 \\\\ 3 \\end{pmatrix}.\\]Enter the ordered pair $(t,s).$ Let's think ste... | math | {
"style": "rule",
"ground_truth": "\\left( -\\frac{5}{11}, \\frac{12}{11} \\right)"
} | {
"split": "train",
"index": 1030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From a large bottle containing 1 liter of alcohol, 1/3 liter of alcohol is poured out, an equal amount of water is added and mixed thoroughly. Then, 1/3 liter of the mixture is poured out, an equal amount of water is added and mixed thoroughly again. Finally, 1/3 liter of the m... | math | {
"style": "rule",
"ground_truth": "8/27"
} | {
"split": "train",
"index": 1031,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the least positive integer that is both $22$ percent less than one integer and $16$ percent greater than another integer. Find the remainder when $N$ is divided by $1000$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "131"
} | {
"split": "train",
"index": 1032,
"solution": "Continuing from Solution 1, we have $N=\\frac{39}{50}k$ and $N=\\frac{29}{25}m$. It follows that $k=\\frac{50}{39}N$ and $m=\\frac{25}{29}N$. Both $m$ and $k$ have to be integers, so, in order for that to be true, $N$ has to cancel the denominators of both $\\frac{5... |
deepscaler | [
{
"role": "user",
"content": "Let \\( m \\in \\mathbf{N}^{*} \\), and let \\( F(m) \\) represent the integer part of \\( \\log_{2} m \\). Determine the value of \\( F(1) + F(2) + \\cdots + F(1024) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8204"
} | {
"split": "train",
"index": 1033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the set of all points $(x,y)$ in the coordinate plane for which one of the coordinates is exactly twice the other. If we were to plot all such points, into how many regions would the resulting graph split up the plane? Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 1034,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence $\\{a_n\\}$ is a geometric sequence, and $a_4 = e$, if $a_2$ and $a_7$ are the two real roots of the equation $$ex^2 + kx + 1 = 0, (k > 2\\sqrt{e})$$ (where $e$ is the base of the natural logarithm),\n1. Find the general formula for $\\{a_n\\}$.\n2. Let ... | math | {
"style": "rule",
"ground_truth": "n = 4"
} | {
"split": "train",
"index": 1035,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let point $O$ be inside $\\triangle ABC$ and satisfy $4\\overrightarrow{OA}+\\overrightarrow{OB}+\\overrightarrow{OC}=\\overrightarrow{0}$. Determine the probability that a randomly thrown bean into $\\triangle ABC$ lands in $\\triangle OBC$. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "\\dfrac{2}{3}"
} | {
"split": "train",
"index": 1036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A jacket was originally priced $\\textdollar 100$ . The price was reduced by $10\\%$ three times and increased by $10\\%$ four times in some order. To the nearest cent, what was the final price? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "106.73"
} | {
"split": "train",
"index": 1037,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The \"Tiao Ri Method\", invented by mathematician He Chengtian during the Southern and Northern Dynasties of China, is an algorithm for finding a more accurate fraction to represent a numerical value. Its theoretical basis is as follows: If the deficient approximate value and t... | math | {
"style": "rule",
"ground_truth": "\\frac{22}{7}"
} | {
"split": "train",
"index": 1038,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the volume in cubic centimeters of a truncated cone formed by cutting a smaller cone from a larger cone. The larger cone has a diameter of 8 cm at the base and a height of 10 cm. The smaller cone, which is cut from the top, has a diameter of 4 cm and a height of 4 cm.... | math | {
"style": "rule",
"ground_truth": "48\\pi"
} | {
"split": "train",
"index": 1039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $| \\overrightarrow{a}|=12$, $| \\overrightarrow{b}|=9$, and $\\overrightarrow{a} \\cdot \\overrightarrow{b}=-54 \\sqrt {2}$, find the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{4}"
} | {
"split": "train",
"index": 1040,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_n = \\frac{10^n-1}{9}$. Define $d_n$ to be the greatest common divisor of $a_n$ and $a_{n+1}$. What is the maximum possible value that $d_n$ can take on? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 1041,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, the smaller circles touch the larger circle and touch each other at the center of the larger circle. The radius of the larger circle is $6.$ What is the area of the shaded region?\n\n[asy]\nsize(100);\nimport graph;\nfilldraw(Circle((0,0),2),mediumgray);\nfilldr... | math | {
"style": "rule",
"ground_truth": "18\\pi"
} | {
"split": "train",
"index": 1042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The parabola $y = x^2+2$ and the hyperbola $y^2 - mx^2 = 1$ are tangent. Find $m.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4+2\\sqrt3"
} | {
"split": "train",
"index": 1043,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sarah's six assignment scores are 87, 90, 86, 93, 89, and 92. What is the arithmetic mean of these six scores? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "89.5"
} | {
"split": "train",
"index": 1044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $\\#N$ by the formula $\\#N = .5(N) + 1$. Calculate $\\#(\\#(\\#50))$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 1045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert $e^{11 \\pi i/2}$ to rectangular form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-i"
} | {
"split": "train",
"index": 1046,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $A, B, C, D, E, F$ are the vertices of a regular hexagon with a side length of 2, and a parabola passes through the points $A, B, C, D$, find the distance from the focus of the parabola to its directrix. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 1047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Is\n\\[f(x) = \\frac{1}{2^x - 1} + \\frac{1}{2}\\]an even function, odd function, or neither?\n\nEnter \"odd\", \"even\", or \"neither\". Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\text{odd}"
} | {
"split": "train",
"index": 1048,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\square + q = 74$ and $\\square + 2q^2 = 180$, what is the value of $\\square$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 1049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An even perfect square in the decimal system is of the form: $\\overline{a b 1 a b}$. What is this perfect square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "76176"
} | {
"split": "train",
"index": 1050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tyler has entered a buffet line in which he chooses one kind of meat, two different vegetables and one dessert. If the order of food items is not important, how many different meals might he choose?\n\n$\\bullet$ Meat: beef, chicken, pork\n\n$\\bullet$ Vegetables: baked bean... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 1051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C D$ be an isosceles trapezoid with $A D=B C=255$ and $A B=128$. Let $M$ be the midpoint of $C D$ and let $N$ be the foot of the perpendicular from $A$ to $C D$. If $\\angle M B C=90^{\\circ}$, compute $\\tan \\angle N B M$. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "\\frac{120}{353}"
} | {
"split": "train",
"index": 1052,
"solution": "Construct $P$, the reflection of $A$ over $C D$. Note that $P, M$, and $B$ are collinear. As $\\angle P N C=\\angle P B C=$ $90^{\\circ}, P N B C$ is cyclic. Thus, $\\angle N B M=\\angle N C P$, so our desired tangent is $\\tan \\angle A C N=\\frac{A N}{C N}$. Note ... |
deepscaler | [
{
"role": "user",
"content": "Let \\( z \\) be a complex number. If \\( \\frac{z-2}{z-\\mathrm{i}} \\) (where \\( \\mathrm{i} \\) is the imaginary unit) is a real number, then the minimum value of \\( |z+3| \\) is \\(\\quad\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 1053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"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 followi... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 1054,
"solution": "By the Chinese remainder theorem, we want the product of the answers modulo 5 and modulo 7 (i.e. when 35 is replaced by 5 and 7, respectively). First we do the modulo 7 case. Since $x^{2}+1$ is irreducible modulo 7 (or more conceptually, in $\\mathbb{F}_{7}[x]$ ), e... |
deepscaler | [
{
"role": "user",
"content": "Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. What is the probability that the four chords form a convex quadrilateral? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{91}"
} | {
"split": "train",
"index": 1055,
"solution": "1. **Count the total number of chords:**\n Given six points on a circle, the number of ways to choose two points to form a chord is given by the combination formula $\\binom{n}{2}$, where $n$ is the number of points. For $n = 6$, we have:\n \\[\n \\binom{6}{2}... |
deepscaler | [
{
"role": "user",
"content": "In the Sweet Tooth store, they are thinking about what promotion to announce before March 8. Manager Vasya suggests reducing the price of a box of candies by $20\\%$ and hopes to sell twice as many goods as usual because of this. Meanwhile, Deputy Director Kolya says it would b... | math | {
"style": "rule",
"ground_truth": "6000"
} | {
"split": "train",
"index": 1056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), \\( \\angle BAC = 90^\\circ \\), \\( AC = AB = 4 \\), and point \\( D \\) is inside \\( \\triangle ABC \\) such that \\( AD = \\sqrt{2} \\). Find the minimum value of \\( BD + CD \\). Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{10}"
} | {
"split": "train",
"index": 1057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle constructed on the side $AD$ of the parallelogram $ABCD$ as its diameter passes through the midpoint of diagonal $AC$ and intersects side $AB$ at point $M$. Find the ratio $AM: AB$ if $AC = 3BD$. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "4/5"
} | {
"split": "train",
"index": 1058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right triangle, instead of having one $90^{\\circ}$ angle and two small angles sum to $90^{\\circ}$, consider now the acute angles are $x^{\\circ}$, $y^{\\circ}$, and a smaller angle $z^{\\circ}$ where $x$, $y$, and $z$ are all prime numbers, and $x^{\\circ} + y^{\\circ} +... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 1059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In pentagon $ABCDE$, $BC=CD=DE=2$ units, $\\angle E$ is a right angle and $m \\angle B = m \\angle C = m \\angle D = 135^\\circ$. The length of segment $AE$ can be expressed in simplest radical form as $a+2\\sqrt{b}$ units. What is the value of $a+b$? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 1060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the polygon shown, each side is perpendicular to its adjacent sides, and all 24 of the sides are congruent. The perimeter of the polygon is 48. Find the area of the polygon. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 1061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The altitudes of an acute isosceles triangle, where \\(AB = BC\\), intersect at point \\(H\\). Find the area of triangle \\(ABC\\), given \\(AH = 5\\) and the altitude \\(AD\\) is 8. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 1062,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two particles move along the edges of equilateral $\\triangle ABC$ in the direction $A\\Rightarrow B\\Rightarrow C\\Rightarrow A,$ starting simultaneously and moving at the same speed. One starts at $A$, and the other starts at the midpoint of $\\overline{BC}$. The midpoint of ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"split": "train",
"index": 1063,
"solution": "1. **Setup and Coordinate Assignment**:\n Without loss of generality (WLOG), let's place point $A$ at the origin of a coordinate system. Assume $\\overline{AB}$ lies on the $x$-axis, and let point $B$ be at $(1, 0)$. Since $\\triangle ABC$ is equilateral, point $C... |
deepscaler | [
{
"role": "user",
"content": "The positive integers $ a$ and $ b$ are such that the numbers $ 15a \\plus{} 16b$ and $ 16a \\minus{} 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... | math | {
"style": "rule",
"ground_truth": "481^2"
} | {
"split": "train",
"index": 1064,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that $\\sin \\alpha+\\sin \\beta=2 \\sin (\\alpha+\\beta)$ and $\\alpha+\\beta \\neq 2 \\pi n (n \\in \\mathbb{Z})$. Find $\\operatorname{tg} \\frac{\\alpha}{2} \\operatorname{tg} \\frac{\\beta}{2}$. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 1065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In square $ABCD$, $AD$ is 4 centimeters, and $M$ is the midpoint of $\\overline{CD}$. Let $O$ be the intersection of $\\overline{AC}$ and $\\overline{BM}$. What is the ratio of $OC$ to $OA$? Express your answer as a common fraction.\n\n[asy]\n\nsize (3cm,3cm);\n\npair A,B,C,D,M... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 1066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a=3$, $b=\\sqrt{3}$, and $A=\\dfrac{\\pi}{3}$, find the measure of angle $B$. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"split": "train",
"index": 1067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A teacher finds that when she offers candy to her class of 30 students, the mean number of pieces taken by each student is 5. If every student takes some candy, what is the greatest number of pieces one student could have taken? Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "121"
} | {
"split": "train",
"index": 1068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $M(4,4)$ is the midpoint of $\\overline{AB}$. If point $A$ has coordinates $(8,4)$, what is the sum of the coordinates of point $B$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 1069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two complementary angles are in a ratio of $3:2$. What is the measure, in degrees, of the smaller angle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 1070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parameterized equation of a line is $$\\begin{cases} x=1+ \\frac {1}{2}t \\\\ y=1+ \\frac { \\sqrt {3}}{2}t\\end{cases}$$ (where $t$ is the parameter), determine the angle of inclination of the line. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 1071,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On side \\(BC\\) of square \\(ABCD\\), point \\(E\\) is chosen such that it divides the segment into \\(BE = 2\\) and \\(EC = 3\\). The circumscribed circle of triangle \\(ABE\\) intersects the diagonal \\(BD\\) a second time at point \\(G\\). Find the area of triangle \\(AGE\\... | math | {
"style": "rule",
"ground_truth": "43.25"
} | {
"split": "train",
"index": 1072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the interval $[0, \\pi]$, a number $\\vartheta$ is randomly selected. The probability that $\\sqrt{2} \\leq \\sqrt{2}\\cos\\vartheta + \\sqrt{2}\\sin\\vartheta \\leq 2$ holds true is $\\_\\_\\_\\_\\_\\_\\_\\_\\cdot$ Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 1073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a given point $P$ on the curve $x^2 - y - \\ln x = 0$, what is 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": 1074,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the midsegment (median) of an isosceles trapezoid, if its diagonal is 25 and its height is 15. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 1075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\\ge a, y\\ge b, z\\ge c.$ Let $\\mathcal{S}$ consist of those trip... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 1076,
"solution": "This problem just requires a good diagram and strong 3D visualization.\n The region in $(x,y,z)$ where $x \\ge \\frac{1}{2}, y \\ge \\frac{1}{3}$ is that of a little triangle on the bottom of the above diagram, of $y \\ge \\frac{1}{3}, z \\ge \\frac{1}{6}$ is the tr... |
deepscaler | [
{
"role": "user",
"content": "Given acute angles $α$ and $β$ that satisfy: $\\cos α= \\frac {1}{3}$ and $\\cos (α+β)=- \\frac {1}{3}$, find $\\cos (α-β)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{23}{27}"
} | {
"split": "train",
"index": 1077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular parking lot has a diagonal of $25$ meters and an area of $168$ square meters. In meters, what is the perimeter of the parking lot? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 1078,
"solution": "1. **Identify the given information and set up equations:**\n Let the sides of the rectangular parking lot be $a$ and $b$. The problem states that the diagonal is $25$ meters and the area is $168$ square meters. Using the Pythagorean theorem for the diagonal, we h... |
deepscaler | [
{
"role": "user",
"content": "If $x + x^2 + x^3 + \\ldots + x^9 + x^{10} = a_0 + a_1(1 + x) + a_2(1 + x)^2 + \\ldots + a_9(1 + x)^9 + a_{10}(1 + x)^{10}$, then $a_9 = \\_\\_\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"split": "train",
"index": 1079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\alpha$ is an angle in the second quadrant, simplify $$\\frac { \\sqrt {1+2\\sin(5\\pi-\\alpha)\\cos(\\alpha-\\pi)}}{\\sin\\left(\\alpha - \\frac {3}{2}\\pi \\right)- \\sqrt {1-\\sin^{2}\\left( \\frac {3}{2}\\pi+\\alpha\\right)}}.$$ Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 1080,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sphere is centered at a point with integer coordinates and passes through the three points $(2,0,0)$, $(0,4,0),(0,0,6)$, but not the origin $(0,0,0)$. If $r$ is the smallest possible radius of the sphere, compute $r^{2}$. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 1081,
"solution": "Let $(x, y, z)$ be the center of the sphere. By the given condition, we have $$(x-2)^{2}+y^{2}+z^{2}=x^{2}+(y-4)^{2}+z^{2}=x^{2}+y^{2}+(z-6)^{2}$$ Subtracting $x^{2}+y^{2}+z^{2}$ yields $$x^{2}-(x-2)^{2}=y^{2}-(y-4)^{2}=z^{2}-(z-6)^{2}$$ or $$4(x-1)=8(y-2)=12(z-3)$$... |
deepscaler | [
{
"role": "user",
"content": "Lily is taking a 30-question, multiple-choice Biology quiz. Each question offers four possible answers. Lily guesses on the last six questions. What is the probability that she will get at least two of these last six questions wrong? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "\\frac{4077}{4096}"
} | {
"split": "train",
"index": 1082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mrs. Kučerová was on a seven-day vacation, and Káta walked her dog and fed her rabbits during this time. Káta received a large cake and 700 CZK as compensation. After another vacation, this time lasting four days, Káta received the same cake and 340 CZK for the same tasks.\n\nW... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 1083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "According to census statistics, the probability of a woman of childbearing age giving birth to a boy or a girl is equal. If a second child is allowed, calculate the probability that a woman of childbearing age will have two girls. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{4}"
} | {
"split": "train",
"index": 1084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integer solutions does the equation \\(\\sqrt{x} + \\sqrt{y} = \\sqrt{1960}\\) have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 1085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ and $BCFG$ be two faces of a cube with $AB=12$. A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P$, which is 7 units from $\\overline{BG}$ and 5 units from $\\overline{BC}$. The beam continues to be reflected off the faces of the cube.... | math | {
"style": "rule",
"ground_truth": "230"
} | {
"split": "train",
"index": 1086,
"solution": "When a light beam reflects off a surface, the path is like that of a ball bouncing. Picture that, and also imagine X, Y, and Z coordinates for the cube vertices. The coordinates will all involve 0's and 12's only, so that means that the X, Y, and Z distance traveled... |
deepscaler | [
{
"role": "user",
"content": "Below is the graph of $y = a \\sin (bx + c)$ for some constants $a$, $b$, and $c$. Assume $a>0$ and $b>0$. Find the smallest possible value of $c$ if it is given that the graph reaches its minimum at $x = 0$. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{2}"
} | {
"split": "train",
"index": 1087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a point P $(x, y)$ on the ellipse $\\frac{x^2}{4} + \\frac{y^2}{9} = 1$, find the minimum distance from point P to the line $2x + y - 10 = 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 1088,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Roger the ant is traveling on a coordinate plane, starting at $(0,0)$. Every second, he moves from one lattice point to a different lattice point at distance 1, chosen with equal probability. He will continue to move until he reaches some point $P$ for which he could have reach... | math | {
"style": "rule",
"ground_truth": "1103"
} | {
"split": "train",
"index": 1089,
"solution": "Roger is guaranteed to be able to take at least one step. Suppose he takes that step in a direction $u$. Let $e_{1}$ be the expectation of the number of additional steps Roger will be able to take after that first move. Notice that Roger is again guaranteed to be ab... |
deepscaler | [
{
"role": "user",
"content": "How many elements are in the set obtained by transforming $\\{(0,0),(2,0)\\} 14$ times? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "477"
} | {
"split": "train",
"index": 1090,
"solution": "Transforming it $k \\geq 1$ times yields the diamond $\\{(n, m):|n-1|+|m| \\leq k+1\\}$ with the points $(1, k),(1, k+1),(1,-k),(1,-k-1)$ removed (this can be seen inductively). So we get $(k+1)^{2}+k^{2}-4$ lattice points, making the answer 477."
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive multiple of $225$ that can be written using\ndigits $0$ and $1$ only? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11111111100"
} | {
"split": "train",
"index": 1091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ , $b$ , $c$ , $d$ , $e$ be positive reals satisfying \\begin{align*} a + b &= c a + b + c &= d a + b + c + d &= e.\\end{align*} If $c=5$ , compute $a+b+c+d+e$ .\n\n*Proposed by Evan Chen* Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 1092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer which when divided by 5 gives a remainder of 4, when divided by 6 gives a remainder of 5, when divided by 7 gives a remainder of 6, when divided by 8 gives a remainder of 7, when divided by 9 gives a remainder of 8, and when divided by 10 give... | math | {
"style": "rule",
"ground_truth": "2519"
} | {
"split": "train",
"index": 1093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two cars, Car A and Car B, start simultaneously from points A and B, respectively, and move towards each other. After 6 hours, the distance they have traveled is 3/5 of the distance between points A and B. Car A travels at a speed of 42 km per hour, which is 1/7 less than Car B... | math | {
"style": "rule",
"ground_truth": "780"
} | {
"split": "train",
"index": 1094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the domain of the expression $\\frac{\\sqrt{x-2}}{\\sqrt{5-x}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "[2,5)"
} | {
"split": "train",
"index": 1095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A 12-slice pizza was made with only pepperoni and mushroom toppings, and every slice has at least one topping. Only six slices have pepperoni, and exactly ten slices have mushrooms. How many slices have both pepperoni and mushrooms? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 1096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the angles $\\angle B = 30^\\circ$ and $\\angle A = 90^\\circ$ are known. Point $K$ is marked on side $AC$, and points $L$ and $M$ are marked on side $BC$ such that $KL = KM$ (point $L$ is on segment $BM$).\n\nFind the length of segment $LM$, given that $AK =... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 1097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to arrange the $6$ permutations of the tuple $(1, 2, 3)$ in a sequence, such that each pair of adjacent permutations contains at least one entry in common? \nFor example, a valid such sequence is given by $(3, 2, 1) - (2, 3, 1) - (2, 1, 3) - (1, 2, ... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 1098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the number of sequences of positive integers $\\left(a_{1}, a_{2}, a_{3}, \\ldots, a_{15}\\right)$ for which the polynomials $$x^{2}-a_{i} x+a_{i+1}$$ each have an integer root for every $1 \\leq i \\leq 15$, setting $a_{16}=a_{1}$. Estimate $N$. An estimate of $E$ w... | math | {
"style": "rule",
"ground_truth": "1409"
} | {
"split": "train",
"index": 1099,
"solution": "We note that $a_{i+1}=x\\left(a_{i}-x\\right)$ for some positive integer $x$, so $a_{i+1} \\geq a_{i}-1$. So, the only way $a_{i}$ can decrease is decreasing by 1. As it cannot decrease that quickly, we will make the assumption that if $a_{i} \\geq 10, a_{i+1}=a_{i}... |
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