data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Consider the equation $F O R T Y+T E N+T E N=S I X T Y$, where each of the ten letters represents a distinct digit from 0 to 9. Find all possible values of $S I X T Y$."
}
] | math | {
"style": "rule",
"ground_truth": "31486"
} | {
"index": 7000,
"solution": "Since $Y+N+N$ ends in $Y$, $N$ must be 0 or 5. But if $N=5$ then $T+E+E+1$ ends in T, which is impossible, so $N=0$ and $E=5$. Since $F \\neq S$ we must have $O=9, R+T+T+1>10$, and $S=F+1$. Now $I \\neq 0$, so it must be that $I=1$ and $R+T+T+1>20$. Thus $R$ and $T$ are 6 and 7, 6 and ... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the equation $p = 15q^2 - 5$. Determine the value of $q$ when $p = 40$.\nA) $q = 1$\nB) $q = 2$\nC) $q = \\sqrt{3}$\nD) $q = \\sqrt{6}$"
}
] | math | {
"style": "rule",
"ground_truth": "q = \\sqrt{3}"
} | {
"index": 7001,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the equation $x^2 + 14x = 32$. Find the values of $a$ and $b$ such that the positive solution of the equation has the form $\\sqrt{a}-b$, where $a$ and $b$ are positive natural numbers. Calculate $a+b$."
}
] | math | {
"style": "rule",
"ground_truth": "88"
} | {
"index": 7002,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the equation $x^2 + 16x = 100$. The positive solution can also be written in the form $\\sqrt{a} - b$ for positive natural numbers $a$ and $b$. Find $a + b$."
}
] | math | {
"style": "rule",
"ground_truth": "172"
} | {
"index": 7003,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the expanded hexagonal lattice shown below, where each point is one unit from its nearest neighbor. Determine the number of equilateral triangles whose vertices lie on this lattice.\n```asy\nsize(100);\ndot(origin);\ndot(dir(30) + dir(90));\ndot(dir(90));\ndot(dir(90) ... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 7004,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the expression \\(1 \\ast 2 \\ast 3 \\ast 4 \\ast 5 \\ast 6\\).\nEach star in the expression is to be replaced with either ' + ' or ' \\times '.\n\\(N\\) is the largest possible value of the expression. What is the largest prime factor of \\(N\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "103"
} | {
"index": 7005,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 7006,
"solution": "To solve this problem, we need to count all the triangles formed by the lines in the given figure. The figure consists of a square, its diagonals, and segments joining the midpoints of opposite sides. Let's analyze the figure and count the triangles of different sizes.\n\n1. **Identify... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the following data from a new season graph, showing the number of home runs hit in April by the top hitters in the baseball league:\n- 5 players hit 6 home runs each.\n- 6 players hit 8 home runs each.\n- 4 players hit 10 home runs each.\n\nCalculate the mean number of... | math | {
"style": "rule",
"ground_truth": "\\frac{118}{15}"
} | {
"index": 7007,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the following diagram showing a rectangular grid of dots consisting of 3 rows and 4 columns. How many rectangles can be formed in this grid?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 7008,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the following flowchart: INPUT $\\rightarrow$ Subtract $8 \\rightarrow \\square \\rightarrow$ Divide by $2 \\rightarrow \\square$ Add $16 \\rightarrow$ OUTPUT. If the OUTPUT is 32, what was the INPUT?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 7009,
"solution": "We start from the OUTPUT and work back to the INPUT. Since the OUTPUT 32 is obtained from adding 16 to the previous number, then the previous number is $32 - 16 = 16$. Since 16 is obtained by dividing the previous number by 2, then the previous number is $2 \\times 16$ or 32. Since 32 ... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the following function $g(x)$ defined as\\[(x^{2^{2008}-1}-1)g(x) = (x+1)(x^2+1)(x^4+1)\\cdots (x^{2^{2007}}+1) - 1\\]Find $g(2)$."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 7010,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the following infinite geometric series: $$\\frac{7}{8}-\\frac{14}{27}+\\frac{28}{81}-\\dots$$ Find the common ratio of this series."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{3}"
} | {
"index": 7011,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the following multiplicative magic square, where each row, column, and diagonal product equals the same value:\n\n$\\begin{tabular}{|c|c|c|} \\hline 75 & \\textit{b} & \\textit{c} \\\\ \\hline \\textit{d} & \\textit{e} & \\textit{f} \\\\ \\hline \\textit{g} & \\textit{... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 7012,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the following pair of equations:\n\\[120x^4 + ax^3 + bx^2 + cx + 18 = 0\\] and \n\\[18x^5 + dx^4 + ex^3 + fx^2 + gx + 120 = 0\\]\nThese equations have a common rational root $k$ which is not an integer and is positive. Determine $k$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 7013,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the following scenario where three stores offer different discounts on a television priced at $149.99$:\n\\begin{tabular}{|l|l|}\n\\hline\n\\textbf{Store} & \\textbf{Sale Price for Television $Y$} \\\\\n\\hline\nValue Market & $\\$10$~off the list price~$\\$149.99$ \\\... | math | {
"style": "rule",
"ground_truth": "3500"
} | {
"index": 7014,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the following sequence $$\\left(a_{n}\\right)_{n=1}^{\\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1, \\ldots)$$ Find all pairs $(\\alpha, \\beta)$ of positive real numbers such that $\\lim _{n \\rightarrow \\infty} \\frac{\\sum_{k=1}^{n} a_{k}}{n^{\\alpha}}=\\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "(\\alpha, \\beta)=\\left(\\frac{3}{2}, \\frac{\\sqrt{2}}{3}\\right)"
} | {
"index": 7015,
"solution": "Let $N_{n}=\\binom{n+1}{2}$ (then $a_{N_{n}}$ is the first appearance of number $n$ in the sequence) and consider limit of the subsequence $$b_{N_{n}}:=\\frac{\\sum_{k=1}^{N_{n}} a_{k}}{N_{n}^{\\alpha}}=\\frac{\\sum_{k=1}^{n} 1+\\cdots+k}{\\binom{n+1}{2}^{\\alpha}}=\\frac{\\sum_{k=1}^{... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the following sequence of sets of natural numbers. The first set \\( I_{0} \\) consists of two ones, 1,1. Then, between these numbers, we insert their sum \\( 1+1=2 \\); we obtain the set \\( I_{1}: 1,2,1 \\). Next, between each pair of numbers in \\( I_{1} \\) we inse... | math | {
"style": "rule",
"ground_truth": "1972"
} | {
"index": 7016,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the following transformation of the Cartesian plane: choose a lattice point and rotate the plane $90^\\circ$ counterclockwise about that lattice point. Is it possible, through a sequence of such transformations, to take the triangle with vertices $(0,0)$, $(1,0)$ and $... | math | {
"style": "rule",
"ground_truth": "$\\text { No }$"
} | {
"index": 7017,
"solution": "\nTo determine if it is possible to transform the triangle with vertices \\((0,0)\\), \\((1,0)\\), and \\((0,1)\\) into the triangle with vertices \\((0,0)\\), \\((1,0)\\), and \\((1,1)\\) through a sequence of 90° counterclockwise rotations about lattice points, we analyze the effects... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function\n\\[ f(x) = \\max \\{-8x - 29, 3x + 2, 7x - 4\\} \\] defined for all real $x$. Let $q(x)$ be a quadratic polynomial tangent to the graph of $f$ at three distinct points with $x$-coordinates $a_1$, $a_2$, $a_3$. Find $a_1 + a_2 + a_3$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{163}{22}"
} | {
"index": 7018,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function\n\\[f(x) = \\max \\{-11x - 37, x - 1, 9x + 3\\}\\]defined for all real $x.$ Let $p(x)$ be a quadratic polynomial tangent to the graph of $f$ at three distinct points with $x$-coordinates $x_1,$ $x_2,$ $x_3.$ Find $x_1 + x_2 + x_3.$"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{11}{2}"
} | {
"index": 7019,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function $ f: \\mathbb{N}_0\\to\\mathbb{N}_0$, where $ \\mathbb{N}_0$ is the set of all non-negative\nintegers, defined by the following conditions :\n\n $ (i)$ $ f(0) \\equal{} 0$; $ (ii)$ $ f(2n) \\equal{} 2f(n)$ and $ (iii)$ $ f(2n \\plus{} 1) \\equal{} n \\p... | math | {
"style": "rule",
"ground_truth": "2^k - 1"
} | {
"index": 7020,
"solution": "\n### Part (a)\n\nWe have the function \\( f: \\mathbb{N}_0 \\rightarrow \\mathbb{N}_0 \\) defined by:\n- \\( f(0) = 0 \\)\n- \\( f(2n) = 2f(n) \\)\n- \\( f(2n + 1) = n + 2f(n) \\)\n\nWe need to determine the sets:\n- \\( L = \\{ n \\mid f(n) < f(n + 1) \\} \\)\n- \\( E = \\{ n \\mid f... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function $f(x) = 2x^2 - 4x + 9$. Evaluate $2f(3) + 3f(-3)$."
}
] | math | {
"style": "rule",
"ground_truth": "147"
} | {
"index": 7021,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function $f(x) = \\cos(2x + \\frac{\\pi}{3}) + \\sqrt{3}\\sin 2x + 2m$, where $x \\in \\mathbb{R}$ and $m \\in \\mathbb{R}$.\n(I) Determine the smallest positive period of $f(x)$ and its intervals of monotonic increase.\n(II) If $f(x)$ has a minimum value of 0 when... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4}"
} | {
"index": 7022,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function $f(x) = x^2 +2\\sqrt{x}$. Evaluate $2f(2) - f(8)$."
}
] | math | {
"style": "rule",
"ground_truth": "-56"
} | {
"index": 7023,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function $f(x) = x^3 + 3\\sqrt{x}$. Evaluate $3f(3) - 2f(9)$."
}
] | math | {
"style": "rule",
"ground_truth": "-1395 + 9\\sqrt{3}"
} | {
"index": 7024,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function $f(x)=\\cos^2x+a\\sin x- \\frac{a}{4}- \\frac{1}{2}$, where $0 \\leq x \\leq \\frac{\\pi}{2}$ and $a > 0$.\n(1) Express the maximum value $M(a)$ of $f(x)$ in terms of $a$.\n(2) Find the value of $a$ when $M(a)=2$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"index": 7025,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function $f: \\mathbb{N}_{0} \\rightarrow \\mathbb{N}_{0}$, where $\\mathbb{N}_{0}$ is the set of all non-negative integers, defined by the following conditions: (i) $f(0)=0$, (ii) $f(2n)=2f(n)$ and (iii) $f(2n+1)=n+2f(n)$ for all $n \\geq 0$. (a) Determine the thr... | math | {
"style": "rule",
"ground_truth": "a_{k}=k2^{k-1}-2^{k}+1"
} | {
"index": 7026,
"solution": "(a) Let $L_{1}:=\\{2k: k>0\\}, \\quad E_{1}:=\\{0\\} \\cup\\{4k+1: k \\geq 0\\}, \\quad \\text { and } \\quad G_{1}:=\\{4k+3: k \\geq 0\\}$. We will show that $L_{1}=L, E_{1}=E$, and $G_{1}=G$. It suffices to verify that $L_{1} \\subseteq E, E_{1} \\subseteq E$, and $G_{1} \\subseteq G... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function $g(x) = \\frac{ax+b}{cx+d}$, where $a$, $b$, $c$, and $d$ are nonzero real numbers. Assume $g(5) = 5$ and $g(25) = 25$, and it also satisfies $g(g(x)) = x$ for all values except $\\frac{-d}{c}$. Find the unique number that is not in the range of $g$."
}
... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 7027,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function $g(x) = \\frac{x^2}{2} + 2x - 1$. Determine the sum of all distinct numbers $x$ such that $g(g(g(x))) = 1$."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 7028,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function $g(x)$ represented by the line segments in the graph below. The graph consists of four line segments as follows: connecting points (-3, -4) to (-1, 0), (-1, 0) to (0, -1), (0, -1) to (2, 3), and (2, 3) to (3, 2). Find the sum of the $x$-coordinates where $... | math | {
"style": "rule",
"ground_truth": "-1.5"
} | {
"index": 7029,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function $g(x)$ satisfying \n\\[ g(xy) = 2g(x)g(y) \\]\nfor all real numbers $x$ and $y$ and $g(0) = 2.$ Find $g(10)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 7030,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function $g(x)=3x-4$. For what value of $a$ is $g(a)=0$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 7031,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function $y=a\\sqrt{1-x^2} + \\sqrt{1+x} + \\sqrt{1-x}$ ($a\\in\\mathbb{R}$), and let $t= \\sqrt{1+x} + \\sqrt{1-x}$ ($\\sqrt{2} \\leq t \\leq 2$).\n\n(1) Express $y$ as a function of $t$, denoted as $m(t)$.\n\n(2) Let the maximum value of the function $m(t)$ be $g... | math | {
"style": "rule",
"ground_truth": "a = 1"
} | {
"index": 7032,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function $z(x, y)$ describing the paraboloid $z=(2 x-y)^{2}-2 y^{2}-3 y$. Archimedes and Brahmagupta are playing a game. Archimedes first chooses $x$. Afterwards, Brahmagupta chooses $y$. Archimedes wishes to minimize $z$ while Brahmagupta wishes to maximize $z$. A... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{8}"
} | {
"index": 7033,
"solution": "Viewing $x$ as a constant and completing the square, we find that $z =-\\left(y+\\frac{4 x+3}{2}\\right)^{2}+\\left(\\frac{4 x+3}{2}\\right)^{2}+4 x^{2}$. Brahmagupta wishes to maximize $z$, so regardless of the value of $x$, he will pick $y=-\\frac{4 x+3}{2}$. The expression for $z$ t... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function \\( g(x) = \\left\\{ \\begin{aligned} x-3 & \\quad \\text{ if } x < 5 \\\\ \\sqrt{x-1} & \\quad \\text{ if } x \\ge 5 \\end{aligned} \\right. \\). Find the value of \\( g^{-1}(-6) + g^{-1}(-5) + \\dots + g^{-1}(4) + g^{-1}(5) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "58"
} | {
"index": 7034,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function \\( g(x) = \\sum_{k=3}^{12} (\\lfloor kx \\rfloor - k \\lfloor x \\rfloor) \\) where \\( \\lfloor r \\rfloor \\) denotes the greatest integer less than or equal to \\( r \\). Determine how many distinct values \\( g(x) \\) can take for \\( x \\ge 0 \\).\nA... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 7035,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function \\( y = g(x) = \\frac{x^2}{Ax^2 + Bx + C} \\), where \\( A, B, \\) and \\( C \\) are integers. The function has vertical asymptotes at \\( x = -1 \\) and \\( x = 2 \\), and for all \\( x > 4 \\), it is true that \\( g(x) > 0.5 \\). Determine the value of \... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 7036,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function defined piecewise by\n\\[ f(x) = \\left\\{ \\begin{aligned} \n2x + 1 & \\quad \\text{if } x < 1 \\\\\nx^2 & \\quad \\text{if } x \\ge 1 \n\\end{aligned} \\right. \\]\nDetermine the value of \\( f^{-1}(-3) + f^{-1}(-1) + f^{-1}(1) + f^{-1}(3) + f^{-1}(9) \\... | math | {
"style": "rule",
"ground_truth": "1 + \\sqrt{3}"
} | {
"index": 7037,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the function:\n\\[ f(x) = \\left\\{ \n\\begin{aligned} \n3x + 1 & \\quad \\text{ if } x \\leq 2 \\\\\nx^2 & \\quad \\text{ if } x > 2 \n\\end{aligned} \n\\right.\\]\nThe function has an inverse $f^{-1}.$ Find the value of $f^{-1}(-5) + f^{-1}(0) + \\dots + f^{-1}(8) + ... | math | {
"style": "rule",
"ground_truth": "22 + 2\\sqrt{2}"
} | {
"index": 7038,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the geometric sequence $3$, $\\dfrac{9}{2}$, $\\dfrac{27}{4}$, $\\dfrac{81}{8}$, $\\ldots$. Find the eighth term of the sequence. Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6561}{128}"
} | {
"index": 7039,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the geometric sequence $5$, $\\dfrac{15}{4}$, $\\dfrac{45}{16}$, $\\dfrac{135}{64}$, $\\ldots$. Find the tenth term of the sequence. Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{98415}{262144}"
} | {
"index": 7040,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the given functions: $$\\begin{array}{ccc}\nf(x) & = & 5x^2 - \\frac{1}{x}+ 3\\\\\ng(x) & = & x^2-k\n\\end{array}$$If $f(2) - g(2) = 2$, what is the value of $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "k = \\frac{-33}{2}"
} | {
"index": 7041,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the graph in 3-space of $0=xyz(x+y)(y+z)(z+x)(x-y)(y-z)(z-x)$. This graph divides 3-space into $N$ connected regions. What is $N$?"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 7042,
"solution": "Note that reflecting for each choice of sign for $x, y, z$, we get new regions. Therefore, we can restrict to the case where $x, y, z>0$. In this case, the sign of the expression only depends on $(x-y)(y-z)(z-x)$. It is easy to see that for this expression, every one of the $3!=6$ orde... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the graph of $y=f(x)$, which consists of five line segments as described below:\n- From $(-5, -4)$ to $(-3, 0)$\n- From $(-3, 0)$ to $(-1, -1)$\n- From $(-1, -1)$ to $(1, 3)$\n- From $(1, 3)$ to $(3, 2)$\n- From $(3, 2)$ to $(5, 6)$\n\nWhat is the sum of the $x$-coordi... | math | {
"style": "rule",
"ground_truth": "4.35"
} | {
"index": 7043,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the graph of \\( y = g(x) \\), with \\( 1 \\) unit between grid lines, where \\( g(x) = \\frac{(x-4)(x-2)(x)(x+2)(x+4)(x+6)}{720} - 2.5 \\), defined only on the shown domain. \n\nDetermine the sum of all integers \\( c \\) for which the equation \\( g(x) = c \\) has ex... | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 7044,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the graph on $1000$ vertices $v_1, v_2, ...v_{1000}$ such that for all $1 \\le i < j \\le 1000$ , $v_i$ is connected to $v_j$ if and only if $i$ divides $j$ . Determine the minimum number of colors that must be used to color the vertices of this graph such... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 7045,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the graphs $y=Ax^2$ and $y^2+3=x^2+4y$, where $A$ is a positive constant and $x$ and $y$ are real variables. In how many points do the two graphs intersect?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 7046,
"solution": "#### Step-by-step Algebraic Approach:\n\n1. **Substitute $y = Ax^2$ into the second equation:**\n \\[\n y^2 + 3 = x^2 + 4y \\implies (Ax^2)^2 + 3 = x^2 + 4Ax^2\n \\]\n Simplifying, we get:\n \\[\n A^2x^4 + 3 = x^2 + 4Ax^2\n \\]\n Rearranging terms:\n \\[\n A^2x^4 - ... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the hyperbola $C$: $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > 0, b > 0)$. Its right focus and one endpoint of its conjugate axis are $F$ and $A$, respectively. Let $P$ be a point on the left branch of the hyperbola $C$. If the minimum value of the perimeter of $\\tr... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{85}}{7}"
} | {
"index": 7047,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the hyperbola $x^{2}-y^{2}=8$ with left and right foci denoted as $F_{1}$ and $F_{2}$, respectively. Let $P_{n}(x_{n},y_{n})$ be a sequence of points on its right branch such that $|P_{n+1}F_{2}|=|P_{n}F_{1}|$ and $P_{1}F_{2} \\perp F_{1}F_{2}$. Determine the value of ... | math | {
"style": "rule",
"ground_truth": "8064"
} | {
"index": 7048,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the infinite series $1 - \\frac{1}{3} - \\frac{1}{9} + \\frac{1}{27} - \\frac{1}{81} - \\frac{1}{243} + \\frac{1}{729} - \\cdots$. Calculate the sum $S$ of this series."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{26}"
} | {
"index": 7049,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the infinite series $1 - \\frac{1}{3} - \\frac{1}{9} + \\frac{1}{27} - \\frac{1}{81} - \\frac{1}{243} + \\frac{1}{729} - \\cdots$. Let \\( T \\) be the sum of this series. Find \\( T \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{26}"
} | {
"index": 7050,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the infinite series $1 - \\frac{1}{3} - \\frac{1}{9} + \\frac{1}{27} - \\frac{1}{81} - \\frac{1}{243} + \\frac{1}{729} - \\frac{1}{2187} - \\cdots$. Evaluate the sum of this series."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{26}"
} | {
"index": 7051,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the infinite series \\(S\\) represented by \\(2 - 1 - \\frac{1}{3} + \\frac{1}{9} - \\frac{1}{27} - \\frac{1}{81} + \\frac{1}{243} - \\cdots\\). Find the sum \\(S\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 7052,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the infinite series defined by the following progression:\n\\[2 + \\frac{1}{3} + \\frac{1}{9} + \\frac{1}{3^2} + \\frac{1}{9^2} + \\frac{1}{3^3} + \\frac{1}{9^3} + \\cdots\\]\nDetermine the limit of this series as it extends to infinity.\nA) $\\frac{1}{3}$\nB) $3$\nC) ... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{8}"
} | {
"index": 7053,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the infinite series: $1 - \\frac{1}{3} - \\frac{1}{9} + \\frac{1}{27} - \\frac{1}{81} - \\frac{1}{243} + \\frac{1}{729} - \\cdots$. Let $T$ be the limiting sum of this series. Find $T$.\n**A)** $\\frac{3}{26}$ \n**B)** $\\frac{15}{26}$ \n**C)** $\\frac{27}{26}$ \n**... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{26}"
} | {
"index": 7054,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the integer \\[M = 8 + 88 + 888 + 8888 + \\cdots + \\underbrace{88\\ldots 88}_\\text{150 digits}.\\] Find the sum of the digits of $M$."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 7055,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the integer \\[N = 9 + 99 + 999 + 9999 + \\cdots + \\underbrace{99\\ldots 99}_\\text{321 digits}.\\]Find the sum of the digits of $N$."
}
] | math | {
"style": "rule",
"ground_truth": "342"
} | {
"index": 7056,
"solution": "Observe how adding results in the last term but with a $1$ concatenated in front and also a $1$ subtracted ($09$, $108$, $1107$, $11106$). Then for any index of terms, $n$, the sum is $11...10-n$, where the first term is of length $n+1$. Here, that is $\\boxed{342}$.\n ~BJHHar"
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the integer\\[N = 9 + 99 + 999 + 9999 + \\cdots + \\underbrace{99\\ldots 99}_\\text{321 digits}.\\]Find the sum of the digits of $N$."
}
] | math | {
"style": "rule",
"ground_truth": "342"
} | {
"index": 7057,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the largest solution to the equation \\[\\log_{10x^2} 10 + \\log_{100x^3} 10 = -2.\\]Find the value of $\\frac{1}{x^{12}},$ writing your answer in decimal representation."
}
] | math | {
"style": "rule",
"ground_truth": "10000000"
} | {
"index": 7058,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the largest solution to the equation \\[\\log_{5x^3} 5 + \\log_{25x^4} 5 = -1.\\] Find the value of \\( x^{10} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "0.0000001024"
} | {
"index": 7059,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the line $15x + 6y = 90$ which forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?\nA) $21$\nB) $35$\nC) $41$\nD) $21 + 10\\sqrt{\\frac{1}{29}}$"
}
] | math | {
"style": "rule",
"ground_truth": "21 + 10\\sqrt{\\frac{1}{29}}"
} | {
"index": 7060,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the line $18x + 9y = 162$ forming a triangle with the coordinate axes. Calculate the sum of the lengths of the altitudes of this triangle.\nA) 21.21 \nB) 42.43 \nC) 63.64 \nD) 84.85 \nE) 105.06"
}
] | math | {
"style": "rule",
"ground_truth": "42.43"
} | {
"index": 7061,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the line parameterized by\n\\begin{align*} \nx&= 4t + 2,\\\\ \ny& = t+2.\n\\end{align*}Find a vector $\\begin{pmatrix}a \\\\ b \\end{pmatrix}$ pointing from the origin to this line that is parallel to $\\begin{pmatrix}2 \\\\1 \\end{pmatrix}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix}6\\\\3\\end{pmatrix}"
} | {
"index": 7062,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the lines \\begin{align*}\ny&=3x+5 \\\\ 2y&=4x+5 \\\\ 3y&=9x-2 \\\\ 2y&=x-3 \\\\ 4y&=x-5.\n\\end{align*}Let's say that a pair of lines is $\\emph{good}$ if the two lines are either parallel or perpendicular to each other. Among all the pairs of lines shown, how many p... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 7063,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the matrix \n\\[\\mathbf{N} = \\begin{pmatrix} 2x & -y & z \\\\ y & x & -2z \\\\ y & -x & z \\end{pmatrix}\\]\nand it is known that $\\mathbf{N}^T \\mathbf{N} = \\mathbf{I}$. Find $x^2 + y^2 + z^2$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 7064,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the multiplication of the two numbers $1,002,000,000,000,000,000$ and $999,999,999,999,999,999$. Calculate the number of digits in the product of these two numbers."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 7065,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the non-decreasing sequence of positive integers\n\\[1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,\\cdots\\]\nin which the $n^{th}$ positive integer appears $n$ times. The remainder when the $1993^{rd}$ term is divided by $5$ is"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 7066,
"solution": "1. **Identify the sequence pattern**: The given sequence is such that each integer $n$ appears $n$ times. This means the sequence looks like:\n \\[1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \\ldots\\]\n\n2. **Determine the position of each integer**: The position where each integer $n$ ends can b... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the number $99,\\!999,\\!999,\\!999$ squared. Following a pattern observed in previous problems, determine how many zeros are in the decimal expansion of this number squared."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 7067,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the numbers $\\{24,27,55,64,x\\}$ . Given that the mean of these five numbers is prime and the median is a multiple of $3$ , compute the sum of all possible positive integral values of $x$ ."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 7068,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the operation \"minus the reciprocal of,\" defined by $a \\diamond b = a - \\frac{1}{b}$. What is $((1 \\diamond 2) \\diamond 3) - (1 \\diamond (2 \\diamond 3))$?"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{30}"
} | {
"index": 7069,
"solution": "1. **Calculate \\(1 \\diamond 2\\):**\n \\[\n 1 \\diamond 2 = 1 - \\frac{1}{2} = \\frac{2}{2} - \\frac{1}{2} = \\frac{1}{2}\n \\]\n\n2. **Calculate \\((1 \\diamond 2) \\diamond 3\\):**\n \\[\n \\left(\\frac{1}{2}\\right) \\diamond 3 = \\frac{1}{2} - \\frac{1}{3} = \\frac{3}{6... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the operation $*$ defined by the following table:\n\\begin{tabular}{c|cccc} * & 1 & 2 & 3 & 4 \\\\ \\hline 1 & 1 & 2 & 3 & 4 \\\\ 2 & 2 & 4 & 1 & 3 \\\\ 3 & 3 & 1 & 4 & 2 \\\\ 4 & 4 & 3 & 2 & 1 \\end{tabular}\nFor example, $3*2=1$. Then $(2*4)*(1*3)=$"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 7070,
"solution": "To solve the problem, we need to evaluate $(2*4)*(1*3)$ using the operation $*$ defined by the given table.\n\n1. **Find $2*4$ and $1*3$ using the table:**\n - From the table, $2*4 = 3$.\n - From the table, $1*3 = 3$.\n\n2. **Now find $(2*4)*(1*3)$ using the results from step 1:**\... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyram... | math | {
"style": "rule",
"ground_truth": "408"
} | {
"index": 7071,
"solution": "The formed tetrahedron has pairwise parallel planar and oppositely equal length ($4\\sqrt{13},15,17$) edges and can be inscribed in a parallelepiped (rectangular box) with the six tetrahedral edges as non-intersecting diagonals of the box faces. Let the edge lengths of the parallelepip... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the parabola consisting of the points $(x, y)$ in the real plane satisfying $(y+x)=(y-x)^{2}+3(y-x)+3$. Find the minimum possible value of $y$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 7072,
"solution": "Let $w=y-x$. Adding $w$ to both sides and dividing by two gives $$y=\\frac{w^{2}+4w+3}{2}=\\frac{(w+2)^{2}-1}{2}$$ which is minimized when $w=-2$. This yields $y=-\\frac{1}{2}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the parallelogram with vertices $(10,45)$, $(10,114)$, $(28,153)$, and $(28,84)$. A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
}
] | math | {
"style": "rule",
"ground_truth": "118"
} | {
"index": 7073,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the parallelogram with vertices $(10,45)$, $(10,114)$, $(28,153)$, and $(28,84)$. A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n$."
}
] | math | {
"style": "rule",
"ground_truth": "118"
} | {
"index": 7074,
"solution": "You can clearly see that a line that cuts a parallelogram into two congruent pieces must go through the center of the parallelogram. Taking the midpoint of $(10,45)$, and $(28,153)$ gives $(19,99)$, which is the center of the parallelogram. Thus the slope of the line must be $\\frac{99... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the parametric equations for a curve given by\n\\begin{align*}\nx &= \\cos t + \\frac{t}{3}, \\\\\ny &= \\sin t.\n\\end{align*}\nDetermine how many times the graph intersects itself between $x = 0$ and $x = 60$."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 7075,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the paths from \\((0,0)\\) to \\((6,3)\\) that only take steps of unit length up and right. Compute the sum of the areas bounded by the path, the \\(x\\)-axis, and the line \\(x=6\\) over all such paths."
}
] | math | {
"style": "rule",
"ground_truth": "756"
} | {
"index": 7076,
"solution": "We see that the sum of the areas under the path is equal the sum of the areas above the path. Thus, the sum of the areas under the path is half the area of the rectangle times the number of paths, which is \\(\\frac{18\\binom{9}{3}}{2}=756\\)."
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the permutation of $1, 2, \\cdots, 20$ as $\\left(a_{1} a_{2} \\cdots a_{20}\\right)$. Perform the following operation on this permutation: swap the positions of any two numbers. The goal is to transform this permutation into $(1, 2, \\cdots, 20)$. Let $k_{a}$ denote t... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 7077,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the points $A(0,12), B(10,9), C(8,0),$ and $D(-4,7).$ There is a unique square $S$ such that each of the four points is on a different side of $S.$ Let $K$ be the area of $S.$ Find the remainder when $10K$ is divided by $1000$."
}
] | math | {
"style": "rule",
"ground_truth": "936"
} | {
"index": 7078,
"solution": "Let $(a,b)$ denote a normal vector of the side containing $A$. Note that $\\overline{AC}, \\overline{BD}$ intersect and hence must be opposite vertices of the square. The lines containing the sides of the square have the form $ax+by=12b$, $ax+by=8a$, $bx-ay=10b-9a$, and $bx-ay=-4b-7a$.... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the polynomial\n\\[P(x)=x^3+3x^2+6x+10.\\]\nLet its three roots be $a$ , $b$ , $c$ . Define $Q(x)$ to be the monic cubic polynomial with roots $ab$ , $bc$ , $ca$ . Compute $|Q(1)|$ .\n\n*Proposed by Nathan Xiong*"
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 7079,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the polynomial $49x^3 - 105x^2 + 63x - 10 = 0$ whose roots are in arithmetic progression. Determine the difference between the largest and smallest roots.\nA) $\\frac{2}{7}$\nB) $\\frac{1}{7}$\nC) $\\frac{3\\sqrt{11}}{7}$\nD) $\\frac{2\\sqrt{11}}{7}$\nE) $\\frac{4\\sqr... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{11}}{7}"
} | {
"index": 7080,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the polynomial $Q(x) = 3x^3 + dx^2 + ex + f$. The polynomial has the property that the reciprocal of the sum of its zeros, the product of its zeros, and the sum of the coefficients are all equal. The $y$-intercept of the graph of $y = Q(x)$ is 9. Find the value of $e$.... | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"index": 7081,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the polynomial \\( P(x)=x^{3}+x^{2}-x+2 \\). Determine all real numbers \\( r \\) for which there exists a complex number \\( z \\) not in the reals such that \\( P(z)=r \\)."
}
] | math | {
"style": "rule",
"ground_truth": "r>3, r<49/27"
} | {
"index": 7082,
"solution": "Because such roots to polynomial equations come in conjugate pairs, we seek the values \\( r \\) such that \\( P(x)=r \\) has just one real root \\( x \\). Considering the shape of a cubic, we are interested in the boundary values \\( r \\) such that \\( P(x)-r \\) has a repeated zero.... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the polynomial \\( p(x) = x^n + n x^{n-1} + a_2 x^{n-2} + \\cdots + a_n \\) having all real roots. If \\( r_1^{16} + r_2^{16} + \\cdots + r_n^{16} = n \\), where the \\( r_j \\) are the roots of \\( p(x) \\), find all such roots."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 7083,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the polynomials $P\\left(x\\right)=16x^4+40x^3+41x^2+20x+16$ and $Q\\left(x\\right)=4x^2+5x+2$ . If $a$ is a real number, what is the smallest possible value of $\\frac{P\\left(a\\right)}{Q\\left(a\\right)}$ ?\n\n*2016 CCA Math Bonanza Team #6*"
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"index": 7084,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the polynomials \\[f(x)=1-12x+3x^2-4x^3+5x^4\\] and \\[g(x)=3-2x-6x^3+9x^4.\\] Find $c$ such that the polynomial $f(x)+cg(x)$ has degree 3."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{9}"
} | {
"index": 7085,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the polynomials $P(x) = x^{6} - x^{5} - x^{3} - x^{2} - x$ and $Q(x) = x^{4} - x^{3} - x^{2} - 1.$ Given that $z_{1},z_{2},z_{3},$ and $z_{4}$ are the roots of $Q(x) = 0,$ find $P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4}).$"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 7086,
"solution": "Let $S_k=z_1^k+z_2^k+z_3^k+z_4^k$ then by Vieta's Formula we have \\[S_{-1}=\\frac{z_1z_2z_3+z_1z_3z_4+z_1z_2z_4+z_1z_2z_3}{z_1z_2z_3z_4}=0\\] \\[S_0=4\\] \\[S_1=1\\] \\[S_2=3\\] By Newton's Sums we have \\[a_4S_k+a_3S_{k-1}+a_2S_{k-2}+a_1S_{k-1}+a_0S_{k-4}=0\\]\n Applying the formula ... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the quadratic equation $2x^2 - 5x + m = 0$. Find the value of $m$ such that the sum of the roots of the equation is maximized while ensuring that the roots are real and rational."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{8}"
} | {
"index": 7087,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the quadratic equation $x^{2}-(r+7) x+r+87=0$ where $r$ is a real number. This equation has two distinct real solutions $x$ which are both negative exactly when $p<r<q$, for some real numbers $p$ and $q$. What is the value of $p^{2}+q^{2}$?"
}
] | math | {
"style": "rule",
"ground_truth": "8098"
} | {
"index": 7088,
"solution": "A quadratic equation has two distinct real solutions exactly when its discriminant is positive. For the quadratic equation $x^{2}-(r+7) x+r+87=0$, the discriminant is $\\Delta=(r+7)^{2}-4(1)(r+87)=r^{2}+14 r+49-4 r-348=r^{2}+10 r-299$. Since $\\Delta=r^{2}+10 r-299=(r+23)(r-13)$ which ... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the region $A$ in the complex plane that consists of all points $z$ such that both $\\frac{z}{40}$ and $\\frac{40}{\\overline{z}}$ have real and imaginary parts between $0$ and $1$, inclusive. What is the integer that is nearest the area of $A$?"
}
] | math | {
"style": "rule",
"ground_truth": "572"
} | {
"index": 7089,
"solution": "Let $z=a+bi \\implies \\frac{z}{40}=\\frac{a}{40}+\\frac{b}{40}i$. Since $0\\leq \\frac{a}{40},\\frac{b}{40}\\leq 1$ we have the inequality \\[0\\leq a,b \\leq 40\\]which is a square of side length $40$.\n Also, $\\frac{40}{\\overline{z}}=\\frac{40}{a-bi}=\\frac{40a}{a^2+b^2}+\\frac{40... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the region $A^{}_{}$ in the complex plane that consists of all points $z^{}_{}$ such that both $\\frac{z^{}_{}}{40}$ and $\\frac{40^{}_{}}{\\overline{z}}$ have real and imaginary parts between $0^{}_{}$ and $1^{}_{}$, inclusive. Find the area of $A.$"
}
] | math | {
"style": "rule",
"ground_truth": "1200 - 200 \\pi"
} | {
"index": 7090,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the region \\(B\\) in the complex plane consisting of all points \\(z\\) such that both \\(\\frac{z}{50}\\) and \\(\\frac{50}{\\overline{z}}\\) have real and imaginary parts between 0 and 1, inclusive. Find the area of \\(B\\)."
}
] | math | {
"style": "rule",
"ground_truth": "2500 - 312.5 \\pi"
} | {
"index": 7091,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the sequence\n\n$$\na_{n}=\\cos (\\underbrace{100 \\ldots 0^{\\circ}}_{n-1})\n$$\n\nFor example, $a_{1}=\\cos 1^{\\circ}, a_{6}=\\cos 100000^{\\circ}$.\n\nHow many of the numbers $a_{1}, a_{2}, \\ldots, a_{100}$ are positive?"
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"index": 7092,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the sequence \n$1,-2,3,-4,5,-6,\\ldots,$\nwhose $n$th term is $(-1)^{n+1}\\cdot n$. What is the average of the first $200$ terms of the sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "-0.5"
} | {
"index": 7093,
"solution": "1. **Identify the sequence**: The given sequence is $1, -2, 3, -4, 5, -6, \\ldots$, where the $n$th term is given by $a_n = (-1)^{n+1} \\cdot n$.\n\n2. **Pairing terms for summation**: We observe that the sequence alternates in sign. Pairing the terms, we have:\n \\[\n (1 + (-2)) +... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the sequence $(a_k)_{k\\ge 1}$ of positive rational numbers defined by $a_1 = \\frac{2020}{2021}$ and for $k\\ge 1$, if $a_k = \\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then\n\\[a_{k+1} = \\frac{m + 18}{n+19}.\\]Determine the sum of all positive... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 7094,
"solution": "We know that $a_{1}=\\tfrac{t}{t+1}$ when $t=2020$ so $1$ is a possible value of $j$. Note also that $a_{2}=\\tfrac{2038}{2040}=\\tfrac{1019}{1020}=\\tfrac{t}{t+1}$ for $t=1019$. Then $a_{2+q}=\\tfrac{1019+18q}{1020+19q}$ unless $1019+18q$ and $1020+19q$ are not relatively prime which ... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the sequence $\\sqrt{2}, \\sqrt{5}, 2\\sqrt{2}, \\sqrt{11}, \\ldots$. Determine the position of $\\sqrt{41}$ in this sequence."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 7095,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the sequence $\\{a\\_n\\}(n\\geqslant 1,n\\in\\mathbb{N})$ that satisfies $a\\_1=2$, $a\\_2=6$, and $a\\_{n+2}-2a\\_{n+1}+a\\_n=2$. If $[x]$ represents the largest integer not exceeding $x$, then the value of $\\left[\\frac{2017}{a\\_1}+\\frac{2017}{a\\_2}+...+\\frac{2... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 7096,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the sequence \\( 5, 55, 555, 5555, 55555, \\ldots \\). Are any of the numbers in this sequence divisible by 495; if so, what is the smallest such number?"
}
] | math | {
"style": "rule",
"ground_truth": "555555555555555555"
} | {
"index": 7097,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the sequence \\(\\left\\{a_{n}\\right\\}\\) defined by \\(a_{0}=\\frac{1}{2}\\) and \\(a_{n+1}=a_{n}+\\frac{a_{n}^{2}}{2023}\\) for \\(n = 0, 1, \\ldots\\). Find the integer \\(k\\) such that \\(a_{k} < 1 < a_{k+1}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "2023"
} | {
"index": 7098,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the sequence created by intermixing the following sets of numbers: the first $1000$ odd numbers, and the squares of the first $100$ integers. What is the median of the new list of $1100$ numbers?\n- $1, 3, 5, \\ldots, 1999$\n- $1^2, 2^2, \\ldots, 100^2$\nA) $1089$\nB) ... | math | {
"style": "rule",
"ground_truth": "1100"
} | {
"index": 7099,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.