data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Define the sequence $ (a_p)_{p\\ge0}$ as follows: $ a_p\\equal{}\\displaystyle\\frac{\\binom p0}{2\\cdot 4}\\minus{}\\frac{\\binom p1}{3\\cdot5}\\plus{}\\frac{\\binom p2}{4\\cdot6}\\minus{}\\ldots\\plus{}(\\minus{}1)^p\\cdot\\frac{\\binom pp}{(p\\plus{}2)(p\\plus{}4)}$ .\r\n... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"index": 7500,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the sequence $(b_i)$ by $b_{n+2} = \\frac{b_n + 2011}{1 + b_{n+1}}$ for $n \\geq 1$ with all terms being positive integers. Determine the minimum possible value of $b_1 + b_2$."
}
] | math | {
"style": "rule",
"ground_truth": "2012"
} | {
"index": 7501,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the sequence $\\left\\{a_{n}\\right\\}$ such that $a_{n}$ is the last digit of the sum $1 + 2 + \\cdots + n$. Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\\left\\{a_{n}\\right\\}$. Find $S_{2016}$."
}
] | math | {
"style": "rule",
"ground_truth": "7066"
} | {
"index": 7502,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the sequence $\\{x_{i}\\}_{i \\geq 0}$ by $x_{0}=2009$ and $x_{n}=-\\frac{2009}{n} \\sum_{k=0}^{n-1} x_{k}$ for all $n \\geq 1$. Compute the value of $\\sum_{n=0}^{2009} 2^{n} x_{n}$"
}
] | math | {
"style": "rule",
"ground_truth": "2009"
} | {
"index": 7503,
"solution": "We have $-\\frac{n x_{n}}{2009}=x_{n-1}+x_{n-2}+\\ldots+x_{0}=x_{n-1}+\\frac{(n-1) x_{n-1}}{2009}$, which yields the recursion $x_{n}=\\frac{n-2010}{n} x_{n-1}$. Unwinding this recursion, we find $x_{n}=(-1)^{n} \\cdot 2009$. $\\binom{2008}{n}$. Thus $\\sum_{k=0}^{2009} 2^{n} x_{n} =\\... |
DeepScaleR | [
{
"role": "user",
"content": "Define the sequence $a_1, a_2, a_3, \\ldots$ by $a_n = \\sum\\limits_{k=1}^n \\sin{k}$, where $k$ represents radian measure. Find the index of the 100th term for which $a_n < 0$."
}
] | math | {
"style": "rule",
"ground_truth": "628"
} | {
"index": 7504,
"solution": "If $n = 1$, $a_n = \\sin(1) > 0$. Then if $n$ satisfies $a_n < 0$, $n \\ge 2$, and \\[a_n = \\sum_{k=1}^n \\sin(k) = \\cfrac{1}{\\sin{1}} \\sum_{k=1}^n\\sin(1)\\sin(k) = \\cfrac{1}{2\\sin{1}} \\sum_{k=1}^n\\cos(k - 1) - \\cos(k + 1) = \\cfrac{1}{2\\sin(1)} [\\cos(0) + \\cos(1) - \\cos(... |
DeepScaleR | [
{
"role": "user",
"content": "Define the sequence $a_{1}, a_{2} \\ldots$ as follows: $a_{1}=1$ and for every $n \\geq 2$, $a_{n}= \\begin{cases}n-2 & \\text { if } a_{n-1}=0 \\\\ a_{n-1}-1 & \\text { if } a_{n-1} \\neq 0\\end{cases}$. A non-negative integer $d$ is said to be jet-lagged if there are non-nega... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 7505,
"solution": "Let $N=n+r$, and $M=n$. Then $r=N-M$, and $s=a_{N}-a_{M}$, and $d=r+s=\\left(a_{N}+N\\right)-\\left(a_{M}+M\\right)$. So we are trying to find the number of possible values of $\\left(a_{N}+N\\right)-\\left(a_{M}+M\\right)$, subject to $N \\geq M$ and $a_{N} \\geq a_{M}$. Divide the $a... |
DeepScaleR | [
{
"role": "user",
"content": "Define the sequence $b_1, b_2, b_3, \\ldots$ by $b_n = \\sum\\limits_{k=1}^n \\cos{k}$, where $k$ represents radian measure. Find the index of the 100th term for which $b_n < 0$."
}
] | math | {
"style": "rule",
"ground_truth": "632"
} | {
"index": 7506,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the sequence $b_{0}, b_{1}, \\ldots, b_{59}$ by $$ b_{i}= \\begin{cases}1 & \\text { if } \\mathrm{i} \\text { is a multiple of } 3 \\\\ 0 & \\text { otherwise }\\end{cases} $$ Let \\left\\{a_{i}\\right\\} be a sequence of elements of \\{0,1\\} such that $$ b_{n} \\equiv... | math | {
"style": "rule",
"ground_truth": "0, 3, 5, 6"
} | {
"index": 7507,
"solution": "Try the four possible combinations of values for $a_{0}$ and $a_{1}$. Since we can write $a_{n} \\equiv$ $b_{n-1}-a_{n-2}-a_{n-1}$, these two numbers completely determine the solution $\\left\\{a_{i}\\right\\}$ beginning with them (if there is one). For $a_{0}=a_{1}=0$, we can check th... |
DeepScaleR | [
{
"role": "user",
"content": "Define the sequence $f_{1}, f_{2}, \\ldots:[0,1) \\rightarrow \\mathbb{R}$ of continuously differentiable functions by the following recurrence: $$ f_{1}=1 ; \\quad f_{n+1}^{\\prime}=f_{n} f_{n+1} \\quad \\text { on }(0,1), \\quad \\text { and } \\quad f_{n+1}(0)=1 $$ Show that... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{1-x}"
} | {
"index": 7508,
"solution": "First of all, the sequence $f_{n}$ is well defined and it holds that $$ f_{n+1}(x)=e^{\\int_{0}^{x} f_{n}(t) \\mathrm{d} t} \\tag{2} $$ The mapping $\\Phi: C([0,1)) \\rightarrow C([0,1))$ given by $$ \\Phi(g)(x)=e^{\\int_{0}^{x} g(t) \\mathrm{d} t} $$ is monotone, i.e. if $f<g$ on $(0,... |
DeepScaleR | [
{
"role": "user",
"content": "Define the sequence \\( b_1, b_2, b_3, \\ldots \\) by \\( b_n = \\sum\\limits_{k=1}^n \\cos{k} \\), where \\( k \\) represents radian measure. Find the index of the 50th term for which \\( b_n < 0 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "314"
} | {
"index": 7509,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the sequence \\(\\{a_n\\}\\) where \\(a_n = n^3 + 4\\) for \\(n \\in \\mathbf{N}_+\\). Let \\(d_n = \\gcd(a_n, a_{n+1})\\), which is the greatest common divisor of \\(a_n\\) and \\(a_{n+1}\\). Find the maximum value of \\(d_n\\)."
}
] | math | {
"style": "rule",
"ground_truth": "433"
} | {
"index": 7510,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the sequence \\left\\{x_{i}\\right\\}_{i \\geq 0} by $x_{0}=x_{1}=x_{2}=1$ and $x_{k}=\\frac{x_{k-1}+x_{k-2}+1}{x_{k-3}}$ for $k>2$. Find $x_{2013}$."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 7511,
"solution": "We have $x_{3}=\\frac{1+1+1}{1}=3, x_{4}=\\frac{3+1+1}{1}=5, x_{5}=\\frac{5+3+1}{1}=9, x_{6}=\\frac{9+5+1}{3}=5$. By the symmetry of our recurrence (or just further computation-it doesn't matter much), $x_{7}=3$ and $x_{8}=x_{9}=x_{10}=1$, so our sequence has period 8. Thus $x_{2013}=x... |
DeepScaleR | [
{
"role": "user",
"content": "Define the sequence of positive integers $\\left\\{a_{n}\\right\\}$ as follows. Let $a_{1}=1, a_{2}=3$, and for each $n>2$, let $a_{n}$ be the result of expressing $a_{n-1}$ in base $n-1$, then reading the resulting numeral in base $n$, then adding 2 (in base $n$). For example,... | math | {
"style": "rule",
"ground_truth": "23097"
} | {
"index": 7512,
"solution": "We claim that for nonnegative integers $m$ and for $0 \\leq n<3 \\cdot 2^{m}, a_{3 \\cdot 2^{m}+n}=\\left(3 \\cdot 2^{m}+n\\right)(m+2)+2 n$. We will prove this by induction; the base case for $a_{3}=6$ (when $m=0$, $n=0$) is given in the problem statement. Now, suppose that this is tr... |
DeepScaleR | [
{
"role": "user",
"content": "Define the sequences $(a_n),(b_n)$ by\n\\begin{align*}\n& a_n, b_n > 0, \\forall n\\in\\mathbb{N_+} \\\\ \n& a_{n+1} = a_n - \\frac{1}{1+\\sum_{i=1}^n\\frac{1}{a_i}} \\\\ \n& b_{n+1} = b_n + \\frac{1}{1+\\sum_{i=1}^n\\frac{1}{b_i}}\n\\end{align*}\n1) If $a_{100}b_{100} = a_{101... | math | {
"style": "rule",
"ground_truth": "199"
} | {
"index": 7513,
"solution": "\n\nDefine the sequences \\( (a_n) \\) and \\( (b_n) \\) by\n\\[\n\\begin{align*}\n& a_n, b_n > 0, \\forall n \\in \\mathbb{N_+}, \\\\\n& a_{n+1} = a_n - \\frac{1}{1 + \\sum_{i=1}^n \\frac{1}{a_i}}, \\\\\n& b_{n+1} = b_n + \\frac{1}{1 + \\sum_{i=1}^n \\frac{1}{b_i}}.\n\\end{align*}\n\\... |
DeepScaleR | [
{
"role": "user",
"content": "Define the ☆ operation: observe the following expressions: \n$1$☆$2=1\\times 3+2=5$; \n$4$☆$(-1)=4\\times 3-1=11$; \n$(-5)$☆$3=(-5)\\times 3+3=-12$; \n$(-6)$☆$(-3)=(-6)\\times 3-3=-21\\ldots$ \n$(1)-1$☆$2=$______, $a$☆$b=$______; \n$(2)$ If $a \\lt b$, then $a$☆$b-b$☆$a$ ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 7514,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define two binary operations on real numbers where $a \\otimes b = \\frac{a+b}{a-b}$ and $b \\oplus a = \\frac{b-a}{b+a}$. Compute the value of $(8\\otimes 6) \\oplus 2$.\nA) $\\frac{5}{9}$ \nB) $\\frac{7}{9}$ \nC) $\\frac{12}{9}$ \nD) $\\frac{1}{9}$ \nE) $\\frac{14}{9}$"
... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"index": 7515,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define: For any three-digit natural number $m$, if $m$ satisfies that the tens digit is $1$ greater than the hundreds digit, and the units digit is $1$ greater than the tens digit, then this three-digit number is called an \"upward number\"; for any three-digit natural number $... | math | {
"style": "rule",
"ground_truth": "531"
} | {
"index": 7516,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define: In the sequence $\\{a_{n}\\}$, if $\\frac{{a}_{n+2}}{{a}_{n+1}}-\\frac{{a}_{n+1}}{{a}_{n}}=d(n∈{N}^{*})$, where $d$ is a constant, then the sequence $\\{a_{n}\\}$ is called a \"geometric difference\" sequence. Given a \"geometric difference\" sequence $\\{a_{n}\\}$ wher... | math | {
"style": "rule",
"ground_truth": "3363"
} | {
"index": 7517,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define: \\( a \\oplus b = a \\times b \\), \\( c \\bigcirc d = d \\times d \\times d \\times \\cdots \\times d \\) (d multiplied c times). Find \\( (5 \\oplus 8) \\oplus (3 \\bigcirc 7) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "13720"
} | {
"index": 7518,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define: \\(\\triangle a = a + (a + 1) + (a + 2) + \\cdots + (2a - 2) + (2a - 1)\\). For example: \\(\\triangle 5 = 5 + 6 + 7 + 8 + 9\\). What is the result of \\(\\triangle 1 + \\triangle 2 + \\triangle 3 + \\cdots + \\triangle 19 + \\triangle 20\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "4200"
} | {
"index": 7519,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Defined on $\\mathbf{R}$, the function $f$ satisfies\n$$\nf(1+x)=f(9-x)=f(9+x).\n$$\n\nGiven $f(0)=0$, and $f(x)=0$ has $n$ roots in the interval $[-4020, 4020]$, find the minimum value of $n$."
}
] | math | {
"style": "rule",
"ground_truth": "2010"
} | {
"index": 7520,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Definition: If the line $l$ is tangent to the graphs of the functions $y=f(x)$ and $y=g(x)$, then the line $l$ is called the common tangent line of the functions $y=f(x)$ and $y=g(x)$. If the functions $f(x)=a\\ln x (a>0)$ and $g(x)=x^{2}$ have exactly one common tangent line, ... | math | {
"style": "rule",
"ground_truth": "2e"
} | {
"index": 7521,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Delete all perfect squares from the sequence of positive integers $1, 2, 3, \\cdots$. Find the 2003rd term of the new sequence."
}
] | math | {
"style": "rule",
"ground_truth": "2048"
} | {
"index": 7522,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Denis has cards with numbers from 1 to 50. How many ways are there to choose two cards such that the difference of the numbers on the cards is 11, and their product is divisible by 5?\n\nThe order of the selected cards does not matter: for example, selecting cards with numbers ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 7523,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Denote $S$ as the subset of $\\{1,2,3,\\dots,1000\\}$ with the property that none of the sums of two different elements in $S$ is in $S$. Find the maximum number of elements in $S$."
}
] | math | {
"style": "rule",
"ground_truth": "501"
} | {
"index": 7524,
"solution": "\nDenote \\( S \\) as a subset of \\( \\{ 1, 2, 3, \\ldots, 1000 \\} \\) with the property that no sum of two different elements in \\( S \\) is itself an element of \\( S \\). We wish to find the maximum number of elements in \\( S \\).\n\nTo address this problem, consider the possibi... |
DeepScaleR | [
{
"role": "user",
"content": "Denote $\\mathbb{Z}_{>0}=\\{1,2,3,...\\}$ the set of all positive integers. Determine all functions $f:\\mathbb{Z}_{>0}\\rightarrow \\mathbb{Z}_{>0}$ such that, for each positive integer $n$,\n$\\hspace{1cm}i) \\sum_{k=1}^{n}f(k)$ is a perfect square, and\n$\\vspace{0.1cm}$\n$\... | math | {
"style": "rule",
"ground_truth": "f(n) = n^3"
} | {
"index": 7525,
"solution": "\nWe are tasked with finding all functions \\( f: \\mathbb{Z}_{>0} \\rightarrow \\mathbb{Z}_{>0} \\) that satisfy the following conditions for each positive integer \\( n \\):\n\n1. \\( \\sum_{k=1}^{n} f(k) \\) is a perfect square.\n2. \\( f(n) \\) divides \\( n^3 \\).\n\nGiven the ref... |
DeepScaleR | [
{
"role": "user",
"content": "Denote $\\phi=\\frac{1+\\sqrt{5}}{2}$ and consider the set of all finite binary strings without leading zeroes. Each string $S$ has a \"base-$\\phi$ \" value $p(S)$. For example, $p(1101)=\\phi^{3}+\\phi^{2}+1$. For any positive integer $n$, let $f(n)$ be the number of such str... | math | {
"style": "rule",
"ground_truth": "\\frac{25+3 \\sqrt{69}}{2}"
} | {
"index": 7526,
"solution": "We write everything in base $\\phi$. Notice that $$\\frac{\\phi^{48 n}-1}{\\phi^{48}-1}=10 \\ldots 010 \\ldots 01 \\ldots 10 \\ldots 01$$ where there are $n-1$ blocks of 47 zeros each. We can prove that every valid base-$\\phi$ representation comes from replacing a consecutive string 1... |
DeepScaleR | [
{
"role": "user",
"content": "Denote by $P(n)$ the greatest prime divisor of $n$. Find all integers $n\\geq 2$ for which \\[P(n)+\\lfloor\\sqrt{n}\\rfloor=P(n+1)+\\lfloor\\sqrt{n+1}\\rfloor\\]"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 7527,
"solution": "\nWe seek all integers \\( n \\geq 2 \\) such that the greatest prime divisor of \\( n \\), denoted \\( P(n) \\), together with the integer part of the square root of \\( n \\), satisfies the equation:\n\n\\[\nP(n) + \\lfloor \\sqrt{n} \\rfloor = P(n+1) + \\lfloor \\sqrt{n+1} \\rfloor.... |
DeepScaleR | [
{
"role": "user",
"content": "Denote by $\\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \\mathbb{Q}^+ \\mapsto \\mathbb{Q}^+$ which satisfy the following equation for all $x, y \\in \\mathbb{Q}^+:$ \\[f\\left( f(x)^2y \\right) = x^3 f(xy).\\]\n\n[i]"
}
] | math | {
"style": "rule",
"ground_truth": "{f(x) = \\frac{1}{x}}"
} | {
"index": 7528,
"solution": "\nTo solve the functional equation for all functions \\( f : \\mathbb{Q}^+ \\to \\mathbb{Q}^+ \\) such that for all \\( x, y \\in \\mathbb{Q}^+ \\),\n\n\\[\nf(f(x)^2 y) = x^3 f(xy),\n\\]\n\nwe proceed with the following steps:\n\n**Step 1: Simplify the equation using a special substitu... |
DeepScaleR | [
{
"role": "user",
"content": "Denote by $\\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n \\geq 0$, let $P_n$ be the subset of $\\mathbb{Z}^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2 + y^2 = 2^k$ for some integer ... | math | {
"style": "rule",
"ground_truth": "5n+1"
} | {
"index": 7529,
"solution": "The answer is $5n+1$.\n\nWe first determine the set $P_n$. Let $Q_n$ be the set of points in $\\mathbb{Z}^2$ of the form $(0, \\pm 2^k)$ or $(\\pm 2^k, 0)$ for some $k \\leq n$. Let $R_n$ be the set of points in $\\mathbb{Z}^2$ of the form $(\\pm 2^k, \\pm 2^k)$ for some $k \\leq n$ (t... |
DeepScaleR | [
{
"role": "user",
"content": "Denote by \\( f(n) \\) the integer obtained by reversing the digits of a positive integer \\( n \\). Find the greatest integer that is certain to divide \\( n^{4} - f(n)^{4} \\) regardless of the choice of \\( n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"index": 7530,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Density is defined as the ratio of mass to volume. There are two cubes. The second cube is made from a material with twice the density of the first, and the side length of the second cube is 100% greater than the side length of the first. By what percentage is the mass of the s... | math | {
"style": "rule",
"ground_truth": "1500"
} | {
"index": 7531,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Derek and Julia are two of 64 players at a casual basketball tournament. The players split up into 8 teams of 8 players at random. Each team then randomly selects 2 captains among their players. What is the probability that both Derek and Julia are captains?"
}
] | math | {
"style": "rule",
"ground_truth": "5/84"
} | {
"index": 7532,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Derek fills a square $10$ by $10$ grid with $50$ $1$ s and $50$ $2$ s. He takes the product of the numbers in each of the $10$ rows. He takes the product of the numbers in each of the $10$ columns. He then sums these $20$ products up to get an integer $N.$ Fin... | math | {
"style": "rule",
"ground_truth": "640"
} | {
"index": 7533,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Derek is deciding between two different-sized pizzas at his favorite restaurant. The menu lists a 14-inch pizza and an 18-inch pizza. Calculate the percent increase in area if Derek chooses the 18-inch pizza over the 14-inch pizza."
}
] | math | {
"style": "rule",
"ground_truth": "65.31\\%"
} | {
"index": 7534,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Descartes's Blackjack: How many integer lattice points (points of the form $(m, n)$ for integers $m$ and $n$) lie inside or on the boundary of the disk of radius 2009 centered at the origin?"
}
] | math | {
"style": "rule",
"ground_truth": "12679605"
} | {
"index": 7535,
"solution": "The number of lattice points inside or on the boundary of a circle with radius $r$ centered at the origin can be approximated using the formula $\\pi r^2 + \\text{error term}$. For a circle with radius 2009, this results in approximately 12679605 lattice points."
} |
DeepScaleR | [
{
"role": "user",
"content": "Describe how to place the vertices of a triangle in the faces of a cube in such a way that the shortest side of the triangle is the biggest possible."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 7536,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Design a computer operation program:\n1. Initial values \\( x = 3 \\), \\( y = 0 \\).\n2. \\( x = x + 2 \\).\n3. \\( y = y + x \\).\n4. If \\( y \\geqslant 10000 \\), proceed to (5); otherwise, go back to (2).\n5. Print \\( x \\).\n6. Stop running. What will be the printed resu... | math | {
"style": "rule",
"ground_truth": "201"
} | {
"index": 7537,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Design a set of stamps with the following requirements: The set consists of four stamps of different denominations, with denominations being positive integers. Moreover, for any denomination value among the consecutive integers 1, 2, ..., R, it should be possible to achieve it ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 7538,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Detached calculation.\n\n327 + 46 - 135\n\n1000 - 582 - 128\n\n(124 - 62) × 6\n\n500 - 400 ÷ 5"
}
] | math | {
"style": "rule",
"ground_truth": "420"
} | {
"index": 7539,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine $3x_4+2x_5$ if $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$ satisfy the system of equations below.\n$2x_1+x_2+x_3+x_4+x_5=6$\n$x_1+2x_2+x_3+x_4+x_5=12$\n$x_1+x_2+2x_3+x_4+x_5=24$\n$x_1+x_2+x_3+2x_4+x_5=48$\n$x_1+x_2+x_3+x_4+2x_5=96$"
}
] | math | {
"style": "rule",
"ground_truth": "181"
} | {
"index": 7540,
"solution": "Adding all five equations gives us $6(x_1 + x_2 + x_3 + x_4 + x_5) = 6(1 + 2 + 4 + 8 + 16)$ so $x_1 + x_2 + x_3 + x_4 + x_5 = 31$. Subtracting this from the fourth given equation gives $x_4 = 17$ and subtracting it from the fifth given equation gives $x_5 = 65$, so our answer is $3\\cd... |
DeepScaleR | [
{
"role": "user",
"content": "Determine $S$, the sum of all the real coefficients of the expansion of $(1+ix)^{2020}$, and find $\\log_2(S)$."
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"index": 7541,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine $\\sqrt[4]{105413504}$ without a calculator."
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 7542,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine $\\sqrt[5]{102030201}$ without a calculator."
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 7543,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine $\\sqrt[6]{1061520150601}$ without a calculator."
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 7544,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine $\\sqrt[7]{218618940381251}$ without a calculator."
}
] | math | {
"style": "rule",
"ground_truth": "102"
} | {
"index": 7545,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine $x^2+y^2+z^2+w^2$ if\n$\\frac{x^2}{2^2-1}+\\frac{y^2}{2^2-3^2}+\\frac{z^2}{2^2-5^2}+\\frac{w^2}{2^2-7^2}=1$\n$\\frac{x^2}{4^2-1}+\\frac{y^2}{4^2-3^2}+\\frac{z^2}{4^2-5^2}+\\frac{w^2}{4^2-7^2}=1$\n$\\frac{x^2}{6^2-1}+\\frac{y^2}{6^2-3^2}+\\frac{z^2}{6^2-5^2}+\\frac{w^2... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 7546,
"solution": "As in Solution 1, we have\n$(t-1)(t-9)(t-25)(t-49)-x^2(t-9)(t-25)(t-49)-y^2(t-1)(t-25)(t-49)$ $-z^2(t-1)(t-9)(t-49)-w^2(t-1)(t-9)(t-25)$\n$=(t-4)(t-16)(t-36)(t-64)$\n Now the coefficient of $t^3$ on both sides must be equal. So instead of expanding it fully, we will find what the coeff... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all \\(\\alpha \\in \\mathbb{R}\\) such that for every continuous function \\(f:[0,1] \\rightarrow \\mathbb{R}\\), differentiable on \\((0,1)\\), with \\(f(0)=0\\) and \\(f(1)=1\\), there exists some \\(\\xi \\in(0,1)\\) such that \\(f(\\xi)+\\alpha=f^{\\prime}(\\xi)\... | math | {
"style": "rule",
"ground_truth": "\\(\\alpha = \\frac{1}{e-1}\\)"
} | {
"index": 7547,
"solution": "First consider the function \\(h(x)=\\frac{e^{x}-1}{e-1}\\), which has the property that \\(h^{\\prime}(x)=\\frac{e^{x}}{e-1}\\). Note that \\(h \\in V\\) and that \\(h^{\\prime}(x)-h(x)=1 /(e-1)\\) is constant. As such, \\(\\alpha=1 /(e-1)\\) is the only possible value that could poss... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all four-digit numbers $\\overline{abcd}$ which are perfect squares and for which the equality holds: $\\overline{ab}=3 \\cdot \\overline{cd} + 1$ ."
}
] | math | {
"style": "rule",
"ground_truth": "2809"
} | {
"index": 7548,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine all functions $f : \\mathbb{R}^2 \\to\\mathbb {R}$ for which \\[f(A)+f(B)+f(C)+f(D)=0,\\]whenever $A,B,C,D$ are the vertices of a square with side-length one."
}
] | math | {
"style": "rule",
"ground_truth": "$f(x)=0$"
} | {
"index": 7549,
"solution": "To determine all functions \\( f : \\mathbb{R}^2 \\to \\mathbb{R} \\) for which \n\n\\[\nf(A) + f(B) + f(C) + f(D) = 0,\n\\]\n\nwhenever \\( A, B, C, D \\) are the vertices of a square with side-length one, consider the following steps:\n\n1. **Translation Invariance**: The property ho... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all functions $f: \\mathbb{Q} \\to \\mathbb{Q}$ such that\n$$f(2xy + \\frac{1}{2}) + f(x-y) = 4f(x)f(y) + \\frac{1}{2}$$\nfor all $x,y \\in \\mathbb{Q}$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x^2 + \\frac{1}{2}"
} | {
"index": 7550,
"solution": "\nLet \\( f: \\mathbb{Q} \\to \\mathbb{Q} \\) be a function such that\n\\[\nf(2xy + \\frac{1}{2}) + f(x-y) = 4f(x)f(y) + \\frac{1}{2}\n\\]\nfor all \\( x, y \\in \\mathbb{Q} \\).\n\nFirst, we denote the given functional equation as \\( P(x, y) \\):\n\\[\nP(x, y): f(2xy + \\frac{1}{2}) ... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all functions $f: \\mathbb{R} \\to \\mathbb{R}$ such that \n$$ f(x^3) + f(y)^3 + f(z)^3 = 3xyz $$\nfor all real numbers $x$, $y$ and $z$ with $x+y+z=0$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x"
} | {
"index": 7551,
"solution": "\nWe are tasked with finding all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that satisfy the equation \n\n\\[\nf(x^3) + f(y)^3 + f(z)^3 = 3xyz\n\\]\n\nfor all real numbers \\( x \\), \\( y \\), and \\( z \\) such that \\( x + y + z = 0 \\).\n\nFirst, we consider substituting spe... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all integers $ k\\ge 2$ such that for all pairs $ (m$, $ n)$ of different positive integers not greater than $ k$, the number $ n^{n\\minus{}1}\\minus{}m^{m\\minus{}1}$ is not divisible by $ k$."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\text{ and } 3"
} | {
"index": 7552,
"solution": "Let us analyze the problem, which requires us to determine all integers \\( k \\ge 2 \\) such that for all pairs \\( (m, n) \\) of different positive integers not greater than \\( k \\), the expression \\( n^{n-1} - m^{m-1} \\) is not divisible by \\( k \\).\n\n### Step 1: Understand t... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all integers $ n > 1$ such that\n\\[ \\frac {2^n \\plus{} 1}{n^2}\n\\]\nis an integer."
}
] | math | {
"style": "rule",
"ground_truth": "n=\\boxed {1,3}"
} | {
"index": 7553,
"solution": "\nLet us consider the problem of finding all integers \\( n > 1 \\) such that the expression\n\n\\[\n\\frac{2^n + 1}{n^2}\n\\]\n\nis an integer. We need to identify those values of \\( n \\) for which \\( n^2 \\mid (2^n + 1) \\).\n\nFirst, let us examine small values of \\( n \\):\n\n1... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all integers $k$ such that there exists infinitely many positive integers $n$ [b]not[/b] satisfying \n\\[n+k |\\binom{2n}{n}\\]"
}
] | math | {
"style": "rule",
"ground_truth": "k \\neq 1"
} | {
"index": 7554,
"solution": "\nDetermine all integers \\( k \\) such that there exist infinitely many positive integers \\( n \\) not satisfying \n\\[\nn + k \\mid \\binom{2n}{n}.\n\\]\n\nWe claim that all integers \\( k \\neq 1 \\) satisfy the desired property.\n\nFirst, recall that \\(\\frac{1}{n + 1} \\binom{2n... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all integers $k\\geqslant 1$ with the following property: given $k$ different colours, if each integer is coloured in one of these $k$ colours, then there must exist integers $a_1<a_2<\\cdots<a_{2023}$ of the same colour such that the differences $a_2-a_1,a_3-a_2,\\do... | math | {
"style": "rule",
"ground_truth": "1 \\text{ and } 2"
} | {
"index": 7555,
"solution": "\nTo solve this problem, we need to determine integers \\( k \\geqslant 1 \\) such that when the integers are coloured with \\( k \\) different colours, there exist integers \\( a_1 < a_2 < \\cdots < a_{2023} \\) all of the same colour, where the differences \\( a_2-a_1, a_3-a_2, \\ldo... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all integral solutions of \\[ a^2\\plus{}b^2\\plus{}c^2\\equal{}a^2b^2.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "(0, 0, 0)"
} | {
"index": 7556,
"solution": "\nWe are tasked with finding all integral solutions to the equation:\n\n\\[\na^2 + b^2 + c^2 = a^2b^2.\n\\]\n\nFirst, let's rewrite the equation and rearrange the terms:\n\n\\[\na^2b^2 - a^2 - b^2 = c^2.\n\\]\n\nThis suggests that \\( c^2 \\) must be non-negative, which means \\( a^2b^... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all pairs $(a, b)$ of integers with the property that the numbers $a^{2}+4 b$ and $b^{2}+4 a$ are both perfect squares."
}
] | math | {
"style": "rule",
"ground_truth": "(-4,-4),(-5,-6),(-6,-5),(0, k^{2}),(k^{2}, 0),(k, 1-k)"
} | {
"index": 7557,
"solution": "Without loss of generality, assume that $|b| \\leq|a|$. If $b=0$, then $a$ must be a perfect square. So $(a=k^{2}, b=0)$ for each $k \\in \\mathbb{Z}$ is a solution. Now we consider the case $b \\neq 0$. Because $a^{2}+4 b$ is a perfect square, the quadratic equation $x^{2}+a x-b=0$ ha... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all pairs $(h, s)$ of positive integers with the following property: If one draws $h$ horizontal lines and another $s$ lines which satisfy (i) they are not horizontal, (ii) no two of them are parallel, (iii) no three of the $h+s$ lines are concurrent, then the number ... | math | {
"style": "rule",
"ground_truth": "(995,1),(176,10),(80,21)"
} | {
"index": 7558,
"solution": "Let $a_{h, s}$ the number of regions formed by $h$ horizontal lines and $s$ another lines as described in the problem. Let $\\mathcal{F}_{h, s}$ be the union of the $h+s$ lines and pick any line $\\ell$. If it intersects the other lines in $n$ (distinct!) points then $\\ell$ is partiti... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all pairs of positive integers $(m,n)$ such that $(1+x^n+x^{2n}+\\cdots+x^{mn})$ is divisible by $(1+x+x^2+\\cdots+x^{m})$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\(\\gcd(m+1, n) = 1\\)"
} | {
"index": 7559,
"solution": "Denote the first and larger polynomial to be $f(x)$ and the second one to be $g(x)$ . In order for $f(x)$ to be divisible by $g(x)$ they must have the same roots. The roots of $g(x)$ are the (m+1)th roots of unity, except for 1. When plugging into $f(x)$ , the root of unity is a root o... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all polynomials $P(x)$ with real coefficients such that\n$P(x)^2 + P\\left(\\frac{1}{x}\\right)^2= P(x^2)P\\left(\\frac{1}{x^2}\\right)$ for all $x$."
}
] | math | {
"style": "rule",
"ground_truth": "P(x) = 0"
} | {
"index": 7560,
"solution": "\nTo solve the problem, we need to determine all polynomials \\( P(x) \\) with real coefficients satisfying the equation:\n\n\\[\nP(x)^2 + P\\left(\\frac{1}{x}\\right)^2 = P(x^2)P\\left(\\frac{1}{x^2}\\right)\n\\]\n\nfor all \\( x \\).\n\n### Step 1: Analyze the Equation\n\nLet's start... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors."
}
] | math | {
"style": "rule",
"ground_truth": "130"
} | {
"index": 7561,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing its last three decimal digits."
}
] | math | {
"style": "rule",
"ground_truth": "32768"
} | {
"index": 7562,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine all positive integers $ n\\geq 2$ that satisfy the following condition: for all $ a$ and $ b$ relatively prime to $ n$ we have \\[a \\equiv b \\pmod n\\qquad\\text{if and only if}\\qquad ab\\equiv 1 \\pmod n.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "2, 3, 4, 6, 8, 12, 24"
} | {
"index": 7563,
"solution": "To determine all positive integers \\( n \\geq 2 \\) that satisfy the given condition, we need to analyze when \\( a \\equiv b \\pmod{n} \\) if and only if \\( ab \\equiv 1 \\pmod{n} \\) for all \\( a \\) and \\( b \\) that are relatively prime to \\( n \\).\n\n### Step 1: Analyze the ... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \\cdots$ defined by \\[ a_0 = M + \\frac{1}{2} \\qquad \\textrm{and} \\qquad a_{k+1} = a_k\\lfloor a_k \\rfloor \\quad \\textrm{for} \\, k = 0, 1, 2, \\cdots \\] contains at least one integer ter... | math | {
"style": "rule",
"ground_truth": "M > 1"
} | {
"index": 7564,
"solution": "\nConsider the sequence \\( a_0, a_1, a_2, \\ldots \\) defined by:\n\n\\[\na_0 = M + \\frac{1}{2}\n\\]\n\nand\n\n\\[\na_{k+1} = a_k \\lfloor a_k \\rfloor \\quad \\text{for} \\quad k = 0, 1, 2, \\ldots\n\\]\n\nWe are tasked with finding all positive integers \\( M \\) such that at least... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all positive integers $n$ for which the equation $$x^{n}+(2+x)^{n}+(2-x)^{n}=0$$ has an integer as a solution."
}
] | math | {
"style": "rule",
"ground_truth": "n=1"
} | {
"index": 7565,
"solution": "If $n$ is even, $x^{n}+(2+x)^{n}+(2-x)^{n}>0$, so $n$ is odd. For $n=1$, the equation reduces to $x+(2+x)+(2-x)=0$, which has the unique solution $x=-4$. For $n>1$, notice that $x$ is even, because $x, 2-x$, and $2+x$ have all the same parity. Let $x=2 y$, so the equation reduces to $$... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$."
}
] | math | {
"style": "rule",
"ground_truth": "n = 2^k"
} | {
"index": 7566,
"solution": "\nWe want to determine all positive integers \\( n \\) for which there exists an integer \\( m \\) such that \\( 2^n - 1 \\mid m^2 + 9 \\).\n\nTo solve this problem, we start by expressing the divisibility condition explicitly:\n\n\\[\n2^n - 1 \\mid m^2 + 9 \\quad \\Rightarrow \\quad m... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all positive integers $n$, $n\\ge2$, such that the following statement is true:\nIf $(a_1,a_2,...,a_n)$ is a sequence of positive integers with $a_1+a_2+\\cdots+a_n=2n-1$, then there is block of (at least two) consecutive terms in the sequence with their (arithmetic) ... | math | {
"style": "rule",
"ground_truth": "2, 3"
} | {
"index": 7567,
"solution": "\nTo determine all positive integers \\( n \\), \\( n \\ge 2 \\), such that the following statement is true:\nIf \\((a_1, a_2, \\ldots, a_n)\\) is a sequence of positive integers with \\( a_1 + a_2 + \\cdots + a_n = 2n - 1 \\), then there is a block of (at least two) consecutive terms ... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all positive integers relatively prime to all the terms of the infinite sequence \\[ a_n=2^n+3^n+6^n -1,\\ n\\geq 1. \\]"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 7568,
"solution": "\nTo solve the problem, we need to determine all positive integers that are relatively prime to every term of the sequence defined by:\n\n\\[\na_n = 2^n + 3^n + 6^n - 1, \\quad n \\geq 1.\n\\]\n\n**Step 1: Understanding the sequence properties**\n\nTo determine an integer relatively pr... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all quadruples $(x,y,z,t)$ of positive integers such that\n\\[ 20^x + 14^{2y} = (x + 2y + z)^{zt}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "(1, 1, 3, 1)"
} | {
"index": 7569,
"solution": "\nTo solve the problem of determining all quadruples \\((x, y, z, t)\\) of positive integers such that:\n\n\\[\n20^x + 14^{2y} = (x + 2y + z)^{zt},\n\\]\n\nwe begin by analyzing the constraints in the given equation with small values for \\(x\\), \\(y\\), \\(z\\), and \\(t\\). Our task... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all quadruplets ($x, y, z, t$) of positive integers, such that $12^x + 13^y - 14^z = 2013^t$."
}
] | math | {
"style": "rule",
"ground_truth": "(1, 3, 2, 1)"
} | {
"index": 7570,
"solution": "\nTo solve the problem of determining all quadruplets \\((x, y, z, t)\\) of positive integers such that:\n\n\\[\n12^x + 13^y - 14^z = 2013^t\n\\]\n\nwe will start by analyzing the problem using the reference answer \\((1, 3, 2, 1)\\). This gives us:\n\n\\[\n12^1 + 13^3 - 14^2 = 2013^1\... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all rational numbers \\(a\\) for which the matrix \\(\\left(\\begin{array}{cccc} a & -a & -1 & 0 \\\\ a & -a & 0 & -1 \\\\ 1 & 0 & a & -a \\\\ 0 & 1 & a & -a \\end{array}\\right)\\) is the square of a matrix with all rational entries."
}
] | math | {
"style": "rule",
"ground_truth": "a=0"
} | {
"index": 7571,
"solution": "We will show that the only such number is \\(a=0\\). Let \\(A=\\left(\\begin{array}{cccc} a & -a & -1 & 0 \\\\ a & -a & 0 & -1 \\\\ 1 & 0 & a & -a \\\\ 0 & 1 & a & -a \\end{array}\\right)\\) and suppose that \\(A=B^{2}\\). It is easy to compute the characteristic polynomial of \\(A\\),... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all real numbers $q$ for which the equation $x^4 -40x^2 +q = 0$ has four real solutions which form an arithmetic progression"
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 7572,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine all real numbers $ a$ such that the inequality $ |x^2 + 2ax + 3a|\\le2$ has exactly one solution in $ x$."
}
] | math | {
"style": "rule",
"ground_truth": "1, 2"
} | {
"index": 7573,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine all real numbers $a$ such that the inequality $|x^{2}+2 a x+3 a| \\leq 2$ has exactly one solution in $x$."
}
] | math | {
"style": "rule",
"ground_truth": "1,2"
} | {
"index": 7574,
"solution": "Let $f(x)=x^{2}+2 a x+3 a$. Note that $f(-3 / 2)=9 / 4$, so the graph of $f$ is a parabola that goes through $(-3 / 2,9 / 4)$. Then, the condition that $|x^{2}+2 a x+3 a| \\leq 2$ has exactly one solution means that the parabola has exactly one point in the strip $-1 \\leq y \\leq 1$, ... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all real numbers $x$ for which the function $$g(x) = \\frac{1}{2+\\frac{1}{1+\\frac{1}{x-1}}}$$ is undefined, and find their sum."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 7575,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine all real values of $A$ for which there exist distinct complex numbers $x_{1}, x_{2}$ such that the following three equations hold: $$ x_{1}(x_{1}+1) =A $$ x_{2}(x_{2}+1) =A $$ x_{1}^{4}+3 x_{1}^{3}+5 x_{1} =x_{2}^{4}+3 x_{2}^{3}+5 x_{2} $$"
}
] | math | {
"style": "rule",
"ground_truth": "\\[ A = -7 \\]"
} | {
"index": 7576,
"solution": "Applying polynomial division, $$ x_{1}^{4}+3 x_{1}^{3}+5 x_{1} =\\left(x_{1}^{2}+x_{1}-A\\right)\\left(x_{1}^{2}+2 x_{1}+(A-2)\\right)+(A+7) x_{1}+A(A-2) =(A+7) x_{1}+A(A-2) .$$ Thus, in order for the last equation to hold, we need $(A+7) x_{1}=(A+7) x_{2}$, from which it follows that ... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all real values of the parameter $a$ for which the equation\n\\[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\\]\nhas exactly four distinct real roots that form a geometric progression."
}
] | math | {
"style": "rule",
"ground_truth": "$\\boxed{a=170}$"
} | {
"index": 7577,
"solution": "\nTo determine for which values of the parameter \\( a \\) the equation \n\n\\[\n16x^4 - ax^3 + (2a + 17)x^2 - ax + 16 = 0\n\\]\n\nhas exactly four distinct real roots that form a geometric progression, we follow these steps:\n\n1. **Geometric Progression of Roots**: Let the roots be \... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all six-digit numbers \\( p \\) that satisfy the following properties:\n(1) \\( p, 2p, 3p, 4p, 5p, 6p \\) are all six-digit numbers;\n(2) Each of the six-digit numbers' digits is a permutation of \\( p \\)'s six digits."
}
] | math | {
"style": "rule",
"ground_truth": "142857"
} | {
"index": 7578,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine all such pairs pf positive integers $(a, b)$ such that $a + b + (gcd (a, b))^ 2 = lcm (a, b) = 2 \\cdot lcm(a -1, b)$, where $lcm (a, b)$ denotes the smallest common multiple, and $gcd (a, b)$ denotes the greatest common divisor of numbers $a, b$."
}
] | math | {
"style": "rule",
"ground_truth": "(2, 3) \\text{ and } (6, 15)"
} | {
"index": 7579,
"solution": "\nLet us determine all pairs of positive integers \\( (a, b) \\) such that:\n\n\\[\na + b + (\\gcd(a, b))^2 = \\mathrm{lcm}(a, b) = 2 \\cdot \\mathrm{lcm}(a-1, b)\n\\]\n\nwhere \\(\\mathrm{lcm}(a, b)\\) is the least common multiple and \\(\\gcd(a, b)\\) is the greatest common divisor o... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all the functions $f : \\mathbb{R} \\to \\mathbb{R}$ such that\n\\[ f(x^2 + f(y)) = f(f(x)) + f(y^2) + 2f(xy) \\]\nfor all real numbers $x$ and $y$."
}
] | math | {
"style": "rule",
"ground_truth": "$f(x) = x^2 \\text{ and } f(x) = 0$"
} | {
"index": 7580,
"solution": "\nTo determine all functions \\( f : \\mathbb{R} \\to \\mathbb{R} \\) satisfying the functional equation:\n\n\\[\nf(x^2 + f(y)) = f(f(x)) + f(y^2) + 2f(xy)\n\\]\n\nfor all real numbers \\( x \\) and \\( y \\), we will go through the following steps:\n\n### Step 1: Substitution and Init... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \\frac{ n^p + 1 }{p^n + 1} $ is an integer."
}
] | math | {
"style": "rule",
"ground_truth": "$(p,n)=(p,p),(2,4)$"
} | {
"index": 7581,
"solution": "\nTo solve the problem of finding all pairs \\((p, n)\\) of a prime number \\(p\\) and a positive integer \\(n\\) for which \\(\\frac{n^p + 1}{p^n + 1}\\) is an integer, we start by analyzing the expression:\n\n\\[\n\\frac{n^p + 1}{p^n + 1}.\n\\]\n\n**Step 1: Initial observation**\n\nW... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all three-digit numbers $N$ having the property that $N$ is divisible by $11,$ and $\\frac{N}{11}$ is equal to the sum of the squares of the digits of $N.$"
}
] | math | {
"style": "rule",
"ground_truth": "550"
} | {
"index": 7582,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine all triples $(p, q, r)$ of positive integers, where $p, q$ are also primes, such that $\\frac{r^2-5q^2}{p^2-1}=2$."
}
] | math | {
"style": "rule",
"ground_truth": "(3, 2, 6)"
} | {
"index": 7583,
"solution": "\nTo find all triples \\((p, q, r)\\) of positive integers, where \\(p, q\\) are also primes, such that:\n\n\\[\n\\frac{r^2 - 5q^2}{p^2 - 1} = 2,\n\\]\n\nwe start by rearranging the equation:\n\n\\[\nr^2 - 5q^2 = 2(p^2 - 1).\n\\]\n\nThis can be further rewritten as:\n\n\\[\nr^2 = 5q^2 ... |
DeepScaleR | [
{
"role": "user",
"content": "Determine all triplets of real numbers $(x, y, z)$ satisfying the system of equations $x^{2} y+y^{2} z =1040$, $x^{2} z+z^{2} y =260$, $(x-y)(y-z)(z-x) =-540$."
}
] | math | {
"style": "rule",
"ground_truth": "(16,4,1),(1,16,4)"
} | {
"index": 7584,
"solution": "Call the three equations $(1),(2),(3) \\cdot(1) /(2)$ gives $y=4 z .(3)+(1)-(2)$ gives $\\left(y^{2}-z^{2}\\right) x=15 z^{2} x=240$ so $z^{2} x=16$. Therefore $z(x+2 z)^{2}=x^{2} z+z^{2} y+4 z^{2} x=\\frac{81}{5}$, $z(x-2 z)^{2}=x^{2} z+z^{2} y-4 z^{2} x=\\frac{49}{5}$ so $\\left|\\fr... |
DeepScaleR | [
{
"role": "user",
"content": "Determine an expression for the area of $\\triangle QCA$ in terms of $p$. Your answer should be simplified as much as possible. [asy]\nsize(5cm);defaultpen(fontsize(9));\npair o = (0, 0); pair q = (0, 12); pair b = (12, 0);\npair a = (2, 12); pair t = (2, 0); pair c = (0, 9);\n... | math | {
"style": "rule",
"ground_truth": "12-p"
} | {
"index": 7585,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine both the ratio of the volume of the cone to the volume of the cylinder and the ratio of their lateral surface areas. A cone and a cylinder have the same height of 10 cm. However, the cone's base radius is half that of the cylinder's. The radius of the cylinder is 8 cm... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{116}}{40}"
} | {
"index": 7586,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine by how many times the number \\((2014)^{2^{2014}} - 1\\) is greater than the number written in the following form:\n\\[\n\\left(\\left((2014)^{2^0} + 1\\right) \\cdot \\left((2014)^{2^1} + 1\\right) \\cdot \\left((2014)^{2^2} + 1\\right) \\ldots \\cdot \\left((2014)^{... | math | {
"style": "rule",
"ground_truth": "2013"
} | {
"index": 7587,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine constants $\\alpha$ and $\\beta$ such that $\\frac{x-\\alpha}{x+\\beta} = \\frac{x^2 - 96x + 2210}{x^2 + 65x - 3510}$. What is $\\alpha + \\beta$?"
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"index": 7588,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine for how many different values of $p<2000$, there exists a quadrilateral $ABCD$ with integer side lengths, perimeter $p$, right angles at $D$ and $C$, $AB=3$, and $CD=AD$."
}
] | math | {
"style": "rule",
"ground_truth": "996"
} | {
"index": 7589,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many \"super prime dates\" occurred in 2007, where a \"super prime date\" is defined as a date where both the month and day are prime numbers, and additionally, the day is less than or equal to the typical maximum number of days in the respective prime month."
}... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 7590,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many 4-digit numbers are mountain numbers, where mountain numbers are defined as having their middle two digits larger than any other digits in the number. For example, 3942 and 5732 are mountain numbers."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 7591,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many different ways there are to assign the elements of the set \\( M = \\{1, 2, 3, 4, 5\\} \\) into three ordered sets \\( A, B,\\) and \\( C \\) such that the following conditions are satisfied: each element must belong to at least one of the sets, the intersect... | math | {
"style": "rule",
"ground_truth": "1230"
} | {
"index": 7592,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many distinct four-digit positive integers exist such that the product of their digits equals 18."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 7593,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many divisors of \\(9!\\) are multiples of 3."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"index": 7594,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many five-letter words can be formed such that they begin and end with the same letter and the third letter is always a vowel."
}
] | math | {
"style": "rule",
"ground_truth": "87880"
} | {
"index": 7595,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many hours it will take Carl to mow the lawn, given that the lawn measures 120 feet by 100 feet, the mower's swath is 30 inches wide with an overlap of 6 inches, and Carl walks at a rate of 4000 feet per hour."
}
] | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"index": 7596,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many integer palindromes are between 200 and 700."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 7597,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many integer values of $n$ between 1 and 180 inclusive ensure that the decimal representation of $\\frac{n}{180}$ terminates."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 7598,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine how many integer values of $x$ satisfy $\\lceil{\\sqrt{x}}\\rceil = 20$."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 7599,
"solution": ""
} |
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