data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let the function\n$$\nf(x) = x^3 + ax^2 + bx + c \\quad \\text{for } x \\in \\mathbf{R},\n$$\nwhere \\( a \\), \\( b \\), and \\( c \\) are distinct non-zero integers, and\n$$\nf(a) = a^3 \\quad \\text{and} \\quad f(b) = b^3.\n$$\nFind \\( a + b + c = \\quad \\). Let's think st... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 35049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point P is the intersection of the hyperbola $\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1$ ($a > 0, b > 0$) and the circle $x^2+y^2=a^2+b^2$ in the first quadrant. $F_1$ and $F_2$ are the left and right foci of the hyperbola, respectively, and $|PF_1|=3|PF_2|$. Calculate the eccentrici... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{2}"
} | {
"split": "train",
"index": 35050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jordan ran 2 miles in half the time it took Steve to run 3 miles. If it took Steve 24 minutes to run 3 miles, using the same rates, how many minutes would it take Jordan to run 5 miles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 35051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of $w$ to $x$ is $4:3$, the ratio of $y$ to $z$ is $3:2$, and the ratio of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16:3"
} | {
"split": "train",
"index": 35052,
"solution": "1. **Given Ratios**: We are given the following ratios:\n - $\\frac{w}{x} = \\frac{4}{3}$\n - $\\frac{y}{z} = \\frac{3}{2}$\n - $\\frac{z}{x} = \\frac{1}{6}$\n\n2. **Express in terms of variables**: From these ratios, we can express the relationships as:\n ... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to $\\angle A$, $\\angle B$, and $\\angle C$ are $a$, $b$, and $c$ respectively. Given that $a=1$, $b=1$, and $c= \\sqrt{2}$, then $\\sin A= \\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 35053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The slope angle of the tangent line to the curve $y=\\frac{1}{2}x^2+2$ at the point $\\left(-1, \\frac{5}{2}\\right)$ is what? 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": 35054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system $(xOy)$, there is an ellipse $(C)$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity $e=\\frac{\\sqrt{2}}{2}$. Also, point $P(2,1)$ is on the ellipse $(C)$.\n\n1. Find the equation of the ellipse $(C)$.\n2. If ... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{2}}{2}"
} | {
"split": "train",
"index": 35055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow {a}=(\\sin(2x+ \\frac {\\pi}{6}), 1)$, $\\overrightarrow {b}=( \\sqrt {3}, \\cos(2x+ \\frac {\\pi}{6}))$, and the function $f(x)= \\overrightarrow {a} \\cdot \\overrightarrow {b}$.\n\n(Ⅰ) Find the interval where the function $f(x)$ is monotonical... | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}+2 \\sqrt {2}"
} | {
"split": "train",
"index": 35056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\{2n+1\\}$ ($n\\in\\mathbb{N}^*$) is arranged sequentially in brackets such that the first bracket contains one number, the second bracket contains two numbers, the third bracket contains three numbers, the fourth bracket contains four numbers, the fifth bracket ... | math | {
"style": "rule",
"ground_truth": "2072"
} | {
"split": "train",
"index": 35057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ for which $n^2-19n+99$ is a perfect square.Find the sum of all integers $n$ such that $\\dfrac{12}{n}$ is also an integer.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 35058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An employee arrives at the unit randomly between 7:50 and 8:30. Calculate the probability that he can clock in on time. 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": 35059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two geometric sequences $a_1, a_2, a_3, \\ldots$ and $b_1, b_2, b_3, \\ldots$ have the same common ratio, with $a_1 = 27$, $b_1=99$, and $a_{15}=b_{11}$. Find $a_9$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "363"
} | {
"split": "train",
"index": 35060,
"solution": "Call the common ratio $r.$ Now since the $n$th term of a geometric sequence with first term $x$ and common ratio $y$ is $xy^{n-1},$ we see that $a_1 \\cdot r^{14} = b_1 \\cdot r^{10} \\implies r^4 = \\frac{99}{27} = \\frac{11}{3}.$ But $a_9$ equals $a_1 \\cdot r^8 ... |
deepscaler | [
{
"role": "user",
"content": "The product of all integers whose absolute value is greater than 3 but not greater than 6 is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-14400"
} | {
"split": "train",
"index": 35061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of $| \\sin x + \\cos x + \\tan x + \\cot x + \\sec x + \\csc x |$ for real numbers $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} - 1"
} | {
"split": "train",
"index": 35062,
"solution": "\\textbf{First solution:} Write \\begin{align*} f(x) &= \\sin x + \\cos x + \\tan x + \\cot x + \\sec x + \\csc x \\\\ &= \\sin x + \\cos x + \\frac{1}{\\sin x \\cos x} + \\frac{\\sin x + \\cos x}{\\sin x \\cos x}. \\end{align*} We can write $\\sin x + \\cos x = \\... |
deepscaler | [
{
"role": "user",
"content": "For how many values of $c$ in the interval $[0, 1000]$ does the equation \\[5 \\lfloor x \\rfloor + 3 \\lceil x \\rceil = c\\]have a solution for $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "251"
} | {
"split": "train",
"index": 35063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$, $q$, and $r$ be the roots of $x^3 - 2x^2 - x + 3 = 0$. Find $\\frac{1}{p-2} + \\frac{1}{q-2} + \\frac{1}{r-2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 35064,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $P(x)$ is a polynomial with real coefficients such that $P(t)=P(1) t^{2}+P(P(1)) t+P(P(P(1)))$ for all real numbers $t$. Compute the largest possible value of $P(P(P(P(1))))$. 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": 35065,
"solution": "Let $(a, b, c):=(P(1), P(P(1)), P(P(P(1))))$, so $P(t)=a t^{2}+b t+c$ and we wish to maximize $P(c)$. Then we have that $$\\begin{aligned} a & =P(1)=a+b+c \\\\ b & =P(a)=a^{3}+a b+c \\\\ c & =P(b)=a b^{2}+b^{2}+c \\end{aligned}$$ The first equation implies $c=-b$. ... |
deepscaler | [
{
"role": "user",
"content": "Let \\( g(n) = (n^2 - 2n + 1)^{1/3} + (n^2 - 1)^{1/3} + (n^2 + 2n + 1)^{1/3} \\). Find \\( \\frac{1}{g(1)} + \\frac{1}{g(3)} + \\frac{1}{g(5)} + \\ldots + \\frac{1}{g(999999)} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 35066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two arithmetic sequences \\\\(\\{a_n\\}\\) and \\\\(\\{b_n\\}\\) with the sum of the first \\\\(n\\) terms denoted as \\\\(S_n\\) and \\\\(T_n\\) respectively. If \\\\( \\dfrac {S_n}{T_n}= \\dfrac {2n}{3n+1}\\), then \\\\( \\dfrac {a_2}{b_3+b_7}+ \\dfrac {a_8}{b_4+b_6}=\\... | math | {
"style": "rule",
"ground_truth": "\\dfrac {9}{14}"
} | {
"split": "train",
"index": 35067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression \\[(5^{1001} + 6^{1002})^2 - (5^{1001} - 6^{1002})^2\\] and express it in the form \\(k \\cdot 30^{1001}\\) for some integer \\(k\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 35068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number \\( n \\) such that both \\( n^2 \\) and \\( (n+1)^2 \\) contain the digit 7. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 35069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four students participate in a competition where each chooses one question from two options, A and B. The rules result in the following point system: 21 points for correct A, -21 points for incorrect A, 7 points for correct B, and -7 points for incorrect B. If the total score o... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 35070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Marina solved the quadratic equation $9x^2-18x-720=0$ by completing the square. In the process, she came up with the equivalent equation $$(x+r)^2 = s,$$where $r$ and $s$ are constants.\n\nWhat is $s$? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 35071,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of square units in the area of the triangle.\n\n[asy]size(125);\ndraw( (-10,-2) -- (2,10), Arrows);\ndraw( (0,-2)-- (0,10) ,Arrows);\ndraw( (5,0) -- (-10,0),Arrows);\n\nlabel(\"$l$\",(2,10), NE);\nlabel(\"$x$\", (5,0) , E);\nlabel(\"$y$\", (0,-2) , S);\nfilldraw... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 35072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest integer $n$, greater than $1$, such that $n^{-1}\\pmod{1050}$ is defined? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 35073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC, AB = AC = 10$ and $BC = 12$. Point $D$ lies strictly between $A$ and $B$ on $\\overline{AB}$ and point $E$ lies strictly between $A$ and $C$ on $\\overline{AC}$ so that $AD = DE = EC$. Then $AD$ can be expressed in the form $\\dfrac{p}{q}$, where $p$ and $q$... | math | {
"style": "rule",
"ground_truth": "289"
} | {
"split": "train",
"index": 35074,
"solution": "By the Law of Cosines, we have: \\[\\cos(A) = \\frac{10^2+10^2-12^2}{2*10*10} = \\frac{7}{25}\\] Let $M$ be midpoint of $AE$, then \\[\\frac{7}{25} = \\frac{10-x}{2x} \\iff x =\\frac{250}{39}\\] So, our answer is $250+39=\\boxed{289}$."
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of cubic centimeters in the volume of the cylinder formed by rotating a square with side length 14 centimeters about its vertical line of symmetry. Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "686\\pi"
} | {
"split": "train",
"index": 35075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $r$ be the positive real solution to $x^3 + \\frac{2}{5} x - 1 = 0.$ Find the exact numerical value of\n\\[r^2 + 2r^5 + 3r^8 + 4r^{11} + \\dotsb.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{4}"
} | {
"split": "train",
"index": 35076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are unit vectors, if $|\\overrightarrow{a}-2\\overrightarrow{b}|=\\sqrt{3}$, then the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is ____. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}\\pi"
} | {
"split": "train",
"index": 35077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many even positive integers $n$ less than or equal to 800 is $$(\\sin t - i\\cos t)^n = \\sin nt - i\\cos nt$$ true for all real $t$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 35078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of functions of the form \\( f(x) = ax^3 + bx^2 + cx + d \\) such that\n\\[ f(x)f(-x) = f(x^3). \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 35079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of ordered pairs $(a, b)$ of positive integers such that \\operatorname{gcd}(a, b)=1$. Compute $$\\sum_{(a, b) \\in S}\\left\\lfloor\\frac{300}{2 a+3 b}\\right\\rfloor$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7400"
} | {
"split": "train",
"index": 35080,
"solution": "The key claim is the following. Claim: The sum in the problem is equal to the number of solutions of $2 x+3 y \\leq 300$ where $x, y$ are positive integers. Proof. The sum in the problem is the same as counting the number of triples $(a, b, d)$ of positive integers... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos (\\frac{\\pi}{4}-a) = \\frac{12}{13}$ and $(\\frac{\\pi}{4}-a)$ is a first-quadrant angle, find the value of $\\frac{\\sin (\\frac{\\pi}{2}-2a)}{\\sin (\\frac{\\pi}{4}+a)}$: \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\frac{119}{144}"
} | {
"split": "train",
"index": 35081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The points $A$ , $B$ , $C$ , $D$ , and $E$ lie in one plane and have the following properties: $AB = 12, BC = 50, CD = 38, AD = 100, BE = 30, CE = 40$ . \n\nFind the length of the segment $ED$ . Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "74"
} | {
"split": "train",
"index": 35082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers \\( a, b, c, d \\) belong to the interval \\([-7.5, 7.5]\\). Find the maximum value of the expression \\( a + 2b + c + 2d - ab - bc - cd - da \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 35083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all solutions to the equation $\\sqrt{5+2z} = 11$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 35084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The rules for a race require that all runners start at $A$, touch any part of the 1500-meter wall, and stop at $B$. What is the number of meters in the minimum distance a participant must run? Express your answer to the nearest meter. Assume the distances from A to the nearest ... | math | {
"style": "rule",
"ground_truth": "1803"
} | {
"split": "train",
"index": 35085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The 2-digit integers from 19 to 92 are written consecutively to form the integer \\(N=192021\\cdots9192\\). Suppose that \\(3^k\\) is the highest power of 3 that is a factor of \\(N\\). What is \\(k\\)? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 35086,
"solution": "To find the highest power of 3 that divides $N$, we need to determine the divisibility of $N$ by powers of 3. We can use the fact that a number is divisible by 3 if the sum of its digits is divisible by 3, and it is divisible by 9 if the sum of its digits is divisi... |
deepscaler | [
{
"role": "user",
"content": "Given a hyperbola $F$ with the equation $\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, let $F$ be its left focus. Draw a line perpendicular to one asymptote of the hyperbola passing through $F$, and denote the foot of the perpendicular as $A$ and the intersec... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 35087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $ P(x)$ denotes a polynomial of degree $ n$ such that $ P(k)\\equal{}\\frac{k}{k\\plus{}1}$ for $ k\\equal{}0,1,2,\\ldots,n$, determine $ P(n\\plus{}1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{(-1)^{n+1} + (n+1)}{n+2}"
} | {
"split": "train",
"index": 35088,
"solution": "To solve this problem, we need to determine the value of \\( P(n+1) \\) for the given polynomial \\( P(x) \\) of degree \\( n \\) such that \n\n\\[\nP(k) = \\frac{k}{k+1} \\quad \\text{for } k = 0, 1, 2, \\ldots, n.\n\\]\n\nOur goal is to express \\( P(x) \\) as:\n... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $AB = 10$, $BC = 6$, $CA = 8$, and side $AB$ is extended to a point $P$ such that $\\triangle PCB$ is similar to $\\triangle CAB$. Find the length of $PC$.\n[asy]\ndefaultpen(linewidth(0.7)+fontsize(10));\npair A=origin, P=(1.5,5), B=(10,0), C=P+2.5*dir(P--... | math | {
"style": "rule",
"ground_truth": "4.8"
} | {
"split": "train",
"index": 35089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute \n\\[\n\\sin^6 0^\\circ + \\sin^6 1^\\circ + \\sin^6 2^\\circ + \\dots + \\sin^6 90^\\circ.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{229}{8}"
} | {
"split": "train",
"index": 35090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate: $64^2-36^2$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2800"
} | {
"split": "train",
"index": 35091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two positive numbers $a$, $b$ such that $a<b$. Let $A.M.$ be their arithmetic mean and let $G.M.$ be their positive geometric mean. Then $A.M.$ minus $G.M.$ is always less than: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{(b-a)^2}{8a}"
} | {
"split": "train",
"index": 35092,
"solution": "Given two positive numbers $a$ and $b$ such that $a < b$. We need to find the difference between their arithmetic mean (A.M.) and geometric mean (G.M.) and compare it to the given options.\n\n1. **Calculate A.M. and G.M.:**\n - The arithmetic mean (A.M.) of $a$ a... |
deepscaler | [
{
"role": "user",
"content": "Let $A = 3^7 + \\binom{7}{2}3^5 + \\binom{7}{4}3^3 + \\binom{7}{6}3$, and $B = \\binom{7}{1}3^6 + \\binom{7}{3}3^4 + \\binom{7}{5}3^2 + 1$. Find the value of $A - B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 35093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten points are selected on the positive $x$-axis, $X^+$, and five points are selected on the positive $y$-axis, $Y^+$. The fifty segments connecting the ten points on $X^+$ to the five points on $Y^+$ are drawn. What is the maximum possible number of points of intersection of t... | math | {
"style": "rule",
"ground_truth": "450"
} | {
"split": "train",
"index": 35094,
"solution": "To find the maximum possible number of points of intersection of the fifty segments that could lie in the interior of the first quadrant, we need to consider how these intersections can occur.\n\n1. **Understanding the Setup**:\n - We have 10 points on the positi... |
deepscaler | [
{
"role": "user",
"content": "Compute the limit of the function:\n\n$$\\lim _{x \\rightarrow \\frac{\\pi}{2}} \\frac{e^{\\sin 2 x}-e^{\\tan 2 x}}{\\ln \\left(\\frac{2 x}{\\pi}\\right)}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2\\pi"
} | {
"split": "train",
"index": 35095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kanga labelled the vertices of a square-based pyramid using \\(1, 2, 3, 4,\\) and \\(5\\) once each. For each face, Kanga calculated the sum of the numbers on its vertices. Four of these sums equaled \\(7, 8, 9,\\) and \\(10\\). What is the sum for the fifth face? Let's think s... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 35096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many perfect squares are factors of 180? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 35097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When simplified and expressed with negative exponents, the expression $(x + y)^{ - 1}(x^{ - 1} + y^{ - 1})$ is equal to: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^{ - 1}y^{ - 1}"
} | {
"split": "train",
"index": 35098,
"solution": "1. **Rewrite the expression with negative exponents as fractions:**\n \\[\n (x + y)^{-1}(x^{-1} + y^{-1}) = \\frac{1}{x + y} \\left(\\frac{1}{x} + \\frac{1}{y}\\right)\n \\]\n\n2. **Simplify the expression inside the parentheses:**\n \\[\n \\frac{1}{x} + ... |
deepscaler | [
{
"role": "user",
"content": "During her birthday, her parents have decided to give Laura and her 2 younger brothers new cellphones. However, they are confused between the innumerable service providers. Assuming no child wants a provider that another sibling has, and that there are 20 service providers, in... | math | {
"style": "rule",
"ground_truth": "6840"
} | {
"split": "train",
"index": 35099,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "12 real numbers x and y satisfy \\( 1 + \\cos^2(2x + 3y - 1) = \\frac{x^2 + y^2 + 2(x+1)(1-y)}{x-y+1} \\). Find the minimum value of xy. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{25}"
} | {
"split": "train",
"index": 35100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $2 \\cdot 5^{-1} + 8 \\cdot 11^{-1} \\pmod{56}$.\n\nExpress your answer as an integer from $0$ to $55$, inclusive. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 35101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In December 2022, $550$ cm of snow fell in Winterpark, Canada. What was the average snowfall in cm per minute during that month?\nA) $\\frac{550}{31\\times 24 \\times 60}$\nB) $\\frac{550 \\times 31}{24 \\times 60}$\nC) $\\frac{550 \\times 24 \\times 60}{31}$\nD) $\\frac{31 \\t... | math | {
"style": "rule",
"ground_truth": "\\frac{550}{31\\times 24 \\times 60}"
} | {
"split": "train",
"index": 35102,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"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? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 35103,
"solution": "1. **Identify the original matrix and the changes**: The original $4 \\times 4$ matrix is given by:\n \\[\n \\begin{array}{|c|c|c|c|}\n \\hline\n 1 & 2 & 3 & 4 \\\\\n \\hline\n 8 & 9 & 10 & 11 \\\\\n \\hline\n 15 & 16 & 17 & 18 \\\\\n \\hline\n ... |
deepscaler | [
{
"role": "user",
"content": "Martians measure angles in clerts. There are $500$ clerts in a full circle. How many clerts are there in a right angle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 35104,
"solution": "1. **Understanding the Martian Measurement**: A full circle on Mars is measured as $500$ clerts.\n\n2. **Relating Martian Measurement to Earth Measurement**: On Earth, a full circle is $360^\\circ$. A right angle, which is $90^\\circ$, is $\\frac{1}{4}$ of a full c... |
deepscaler | [
{
"role": "user",
"content": "Let $A = \\{1, 2, \\cdots, 10\\}$. A function $f: A \\rightarrow A$ satisfies $f(x) \\neq x$ for all $x \\in A$, but $f^{21}(x) = x$ for all $x \\in A$. Find the number of such permutations. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "172800"
} | {
"split": "train",
"index": 35105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( A \\) be a set with 225 elements, and \\( A_{1}, A_{2}, \\cdots, A_{11} \\) be 11 subsets of \\( A \\) each containing 45 elements, such that for any \\( 1 \\leq i < j \\leq 11 \\), \\(|A_{i} \\cap A_{j}| = 9\\). Find the minimum value of \\(|A_{1} \\cup A_{2} \\cup \\c... | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 35106,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $9\\times 1$ board is completely covered by $m\\times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $9\\times 1$ boa... | math | {
"style": "rule",
"ground_truth": "838"
} | {
"split": "train",
"index": 35107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eight numbers \\( a_{1}, a_{2}, a_{3}, a_{4} \\) and \\( b_{1}, b_{2}, b_{3}, b_{4} \\) satisfy the equations:\n\n\\[\n\\left\\{\n\\begin{array}{l}\na_{1} b_{1} + a_{2} b_{3} = 1 \\\\\na_{1} b_{2} + a_{2} b_{4} = 0 \\\\\na_{3} b_{1} + a_{4} b_{3} = 0 \\\\\na_{3} b_{2} + a_{4} b... | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 35108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the arithmetic sequence $\\{a_n\\}$, it is known that $a_1=10$, and the sum of the first $n$ terms is $S_n$. If $S_9=S_{12}$, find the maximum value of $S_n$ and the corresponding value of $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 35109,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice is once again very bored in class. On a whim, she chooses three primes $p, q, r$ independently and uniformly at random from the set of primes of at most 30. She then calculates the roots of $p x^{2}+q x+r$. What is the probability that at least one of her roots is an inte... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{200}"
} | {
"split": "train",
"index": 35110,
"solution": "Since all of the coefficients are positive, any root $x$ must be negative. Moreover, by the rational root theorem, in order for $x$ to be an integer we must have either $x=-1$ or $x=-r$. So we must have either $p r^{2}-q r+r=0 \\Longleftrightarrow p r=q-1$ or $p-q+... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n > 1 \\) such that the arithmetic mean of the squares of the integers \\( 1^2, 2^2, 3^2, \\ldots, n^2 \\) is a perfect square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "337"
} | {
"split": "train",
"index": 35111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the geometric sequence $\\{a_{n}\\}$, $3a_{5}-a_{3}a_{7}=0$. If $\\{b_{n}\\}$ is an arithmetic sequence where $b_{5}=a_{5}$, find the sum of the first 9 terms, $S_{9}$, of $\\{b_{n}\\}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 35112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $x$ if a cube's volume is $5x$ cubic units and its surface area is $x$ square units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5400"
} | {
"split": "train",
"index": 35113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when 1,234,567,890 is divided by 99? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 35114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\det \\mathbf{M} = -2,$ then find $ \\det (\\mathbf{M}^4).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 35115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = 2\\sin\\omega x\\cos\\omega x + 2\\sqrt{3}\\sin^2\\omega x - \\sqrt{3}$ ($\\omega > 0$) has the smallest positive period of $\\pi$.\n\n(1) Find the interval of monotonic increase for the function $f(x)$; \n(2) The graph of $f(x)$ is obtained by trans... | math | {
"style": "rule",
"ground_truth": "\\frac{59\\pi}{12}"
} | {
"split": "train",
"index": 35116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that events $A$ and $B$ are independent, and $P(A)=\\frac{1}{2}$, $P(B)=\\frac{2}{3}$, find $P(\\overline{AB})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 35117,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all natural numbers \\( n \\) such that, when writing the numbers \\( n^3 \\) and \\( n^4 \\) side by side in decimal notation, each of the ten digits appears exactly once.\n(Former Yugoslavia Mathematical Olympiad, 1983) Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 35118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be the two real values of $x$ for which\\[\\sqrt[3]{x} + \\sqrt[3]{20 - x} = 2\\]The smaller of the two values can be expressed as $p - \\sqrt{q}$, where $p$ and $q$ are integers. Compute $p + q$.\n Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "118"
} | {
"split": "train",
"index": 35119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $P$ is a point inside rectangle $ABCD$, the distances from $P$ to the vertices of the rectangle are $PA = 5$ inches, $PD = 12$ inches, and $PC = 13$ inches. Find $PB$, which is $x$ inches. 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": 35120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The altitudes of a triangle are $12, 15,$ and $20.$ The largest angle in this triangle is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90^\\circ"
} | {
"split": "train",
"index": 35121,
"solution": "1. **Identify the relationship between the altitudes and the sides of the triangle**: Let $a, b,$ and $c$ be the lengths of the sides of the triangle opposite the altitudes of lengths $12, 15,$ and $20$ respectively. By the formula for the area of a triangle using ... |
deepscaler | [
{
"role": "user",
"content": "Two different numbers are randomly selected from the set $\\{-3, -1, 0, 2, 4\\}$ and then multiplied together. What is the probability that the product of the two numbers chosen is 0? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 35122,
"solution": "There are ten possible pairs of numbers that can be chosen: -3 and $-1$; -3 and $0$; -3 and 2; -3 and $4$; -1 and $0$; -1 and $2$; -1 and $4$; 0 and $2$; 0 and $4$; 2 and 4. Each pair is equally likely to be chosen. Pairs that include 0 (4 pairs) have a product of ... |
deepscaler | [
{
"role": "user",
"content": "Jefferson Middle School has the same number of boys and girls. $\\frac{3}{4}$ of the girls and $\\frac{2}{3}$ of the boys went on a field trip. What fraction of the students on the field trip were girls? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{17}"
} | {
"split": "train",
"index": 35123,
"solution": "1. **Assume the number of boys and girls**: Let the number of boys be $b$ and the number of girls be $g$. Given that Jefferson Middle School has the same number of boys and girls, we have $g = b$.\n\n2. **Calculate the number of students who went on the trip**:\n ... |
deepscaler | [
{
"role": "user",
"content": "In an organization with 200 employees, those over the age of 50 account for 20%, those aged 40-50 make up 30%, and those under 40 account for 50%. If 40 employees are to be sampled, and the systematic sampling method is used—where all employees are randomly numbered 1-200 and e... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 35124,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A student typed out several circles on the computer as follows: ○●○○●○○○●○○○○●○○○○○●… If this pattern continues, forming a series of circles, then the number of ● in the first 120 circles is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 35125,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = \\sin{x} + 2\\cos{x} + 3\\tan{x}$, using radian measure for the variable $x$. Let $r$ be the smallest positive value of $x$ for which $f(x) = 0$. Find $\\lfloor r \\rfloor.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 35126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all positive integers $\\nu$ for which $\\mathop{\\text{lcm}}[\\nu, 18] = 72$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 35127,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a nonnegative integer $n$, let $r_7(n)$ represent the remainder when $n$ is divided by $7$. Determine the $15^{\\text{th}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\\le 3.$$ Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 35128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum $y$-coordinate of a point on the graph of $r = \\sin 2 \\theta.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4 \\sqrt{3}}{9}"
} | {
"split": "train",
"index": 35129,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $E$: $\\frac{x^{2}}{2}+y^{2}=1$ with its right focus $F$, two perpendicular lines passing through $F$ intersect with $E$ at points $A$, $C$ and $B$, $D$.\n\n1. Can the quadrilateral $ABCD$ form a parallelogram? Please explain the reason.\n2. Find the minimum v... | math | {
"style": "rule",
"ground_truth": "\\frac{8 \\sqrt{2}}{3}"
} | {
"split": "train",
"index": 35130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle touches the extensions of two sides $AB$ and $AD$ of square $ABCD$ with a side length of $2-\\sqrt{5-\\sqrt{5}}$ cm. From point $C$, two tangents are drawn to this circle. Find the radius of the circle, given that the angle between the tangents is $72^{\\circ}$ and it ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5 - \\sqrt{5}}"
} | {
"split": "train",
"index": 35131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $x$ and $y$ are complex numbers such that $x+y=1$ and that $x^{20}+y^{20}=20$. Find the sum of all possible values of $x^{2}+y^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-90"
} | {
"split": "train",
"index": 35132,
"solution": "We have $x^{2}+y^{2}+2 x y=1$. Define $a=2 x y$ and $b=x^{2}+y^{2}$ for convenience. Then $a+b=1$ and $b-a=x^{2}+y^{2}-2 x y=(x-y)^{2}=2 b-1$ so that $x, y=\\frac{\\sqrt{2 b-1} \\pm 1}{2}$. Then $x^{20}+y^{20}=\\left(\\frac{\\sqrt{2 b-1}+1}{2}\\right)^{20}+\\left(\... |
deepscaler | [
{
"role": "user",
"content": "Arrange the numbers from 1 to 25 in a random order. Then subtract them sequentially from 1, 2, 3, ..., 25, always subtracting the smaller number from the larger number. What is the maximum number of even differences that can appear among these 25 differences? Let's think step b... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 35133,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The arithmetic mean of two distinct positive integers $x$ and $y$ is a two-digit integer. The geometric mean of $x$ and $y$ is obtained by reversing the digits of the arithmetic mean. What is $|x - y|$? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 35134,
"solution": "1. **Setting up the equations**:\n Let the arithmetic mean of $x$ and $y$ be a two-digit integer, which can be expressed as $10a + b$ where $1 \\leq a \\leq 9$ and $0 \\leq b \\leq 9$. Thus, we have:\n \\[\n \\frac{x + y}{2} = 10a + b\n \\]\n The geometri... |
deepscaler | [
{
"role": "user",
"content": "John is thinking of a number. He gives the following 3 clues. ``My number has 125 as a factor. My number is a multiple of 30. My number is between 800 and 2000.'' What is John's number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1500"
} | {
"split": "train",
"index": 35135,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest common divisor of 9118, 12173, and 33182. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 35136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integer divisors of $1800^{1800}$ are divisible by exactly 180 positive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 35137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When one ounce of water is added to a mixture of acid and water, the new mixture is $20\\%$ acid. When one ounce of acid is added to the new mixture, the result is $33\\frac13\\%$ acid. The percentage of acid in the original mixture is Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "25\\%"
} | {
"split": "train",
"index": 35138,
"solution": "1. **Define Variables:**\n Let $a$ be the original number of ounces of acid and $w$ be the original number of ounces of water in the mixture.\n\n2. **Set Up Equations:**\n - After adding one ounce of water, the mixture becomes $20\\%$ acid. Therefore, the equat... |
deepscaler | [
{
"role": "user",
"content": "What is the product of the numerator and the denominator when $0.\\overline{012}$ is expressed as a fraction in lowest terms? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1332"
} | {
"split": "train",
"index": 35139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a school fundraiser, $3109 was raised. The money was shared equally among 25 charities. The amount received by each charity from the school was: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$124.36"
} | {
"split": "train",
"index": 35140,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $f(x)$ and $g(x)$ are polynomials of degree $4$ and $5$ respectively. What is the degree of $f(x^3) \\cdot g(x^2)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 35141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\arccos (\\sin 1.5).$ All functions are in radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2} - 1.5"
} | {
"split": "train",
"index": 35142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle with dimensions $8 \\times 2 \\sqrt{2}$ and a circle with a radius of 2 have a common center. Find the area of their overlapping region. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\pi + 4"
} | {
"split": "train",
"index": 35143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,\n\\[\n\\sqrt[N]{\\frac{x^{2 N}+1}{2}} \\leqslant a_{n}(x-1)^{2}+x .\n\\]\n[i]Version 2[/i]. For every positive integer $N$, determine the ... | math | {
"style": "rule",
"ground_truth": "{a_n = 2^{n-1}}"
} | {
"split": "train",
"index": 35144,
"solution": "\nWe are tasked with finding the smallest real number \\( a_n \\) for a given positive integer \\( n \\) and \\( N = 2^n \\), such that the inequality\n\n\\[\n\\sqrt[N]{\\frac{x^{2N} + 1}{2}} \\leq a_{n}(x-1)^{2} + x\n\\]\n\nholds for all real \\( x \\).\n\n### Ste... |
deepscaler | [
{
"role": "user",
"content": "In the quadrilateral \\(ABCD\\), diagonals \\(AC\\) and \\(BD\\) intersect at point \\(K\\). Points \\(L\\) and \\(M\\) are the midpoints of sides \\(BC\\) and \\(AD\\), respectively. Segment \\(LM\\) contains point \\(K\\). The quadrilateral \\(ABCD\\) is such that a circle ca... | math | {
"style": "rule",
"ground_truth": "3/2"
} | {
"split": "train",
"index": 35145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Parabola C is defined by the equation y²=2px (p>0). A line l with slope k passes through point P(-4,0) and intersects with parabola C at points A and B. When k=$\\frac{1}{2}$, points A and B coincide.\n\n1. Find the equation of parabola C.\n2. If A is the midpoint of PB, find t... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{11}"
} | {
"split": "train",
"index": 35146,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a circular cone with vertex $V$, and let $A B C$ be a triangle inscribed in the base of the cone, such that $A B$ is a diameter and $A C=B C$. Let $L$ be a point on $B V$ such that the volume of the cone is 4 times the volume of the tetrahedron $A B C L$. Find the valu... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4-\\pi}"
} | {
"split": "train",
"index": 35147,
"solution": "Let $R$ be the radius of the base, $H$ the height of the cone, $h$ the height of the pyramid and let $B L / L V=x / y$. Let [.] denote volume. Then [cone] $=\\frac{1}{3} \\pi R^{2} H$ and $[A B C L]=\\frac{1}{3} \\pi R^{2} h$ and $h=\\frac{x}{x+y} H$. We are given ... |
deepscaler | [
{
"role": "user",
"content": "Find the 6-digit repetend in the decimal representation of $\\frac 3{13}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "230769"
} | {
"split": "train",
"index": 35148,
"solution": ""
} |
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