data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Find a four-digit number that is a perfect square if its first two digits are the same as its last two digits."
}
] | math | {
"style": "rule",
"ground_truth": "7744"
} | {
"index": 9100,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a four-digit number that is a perfect square, in which its digits can be grouped into two pairs of equal digits."
}
] | math | {
"style": "rule",
"ground_truth": "7744"
} | {
"index": 9101,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a four-digit number that is a perfect square, knowing that the first two digits, as well as the last two digits, are each equal to each other."
}
] | math | {
"style": "rule",
"ground_truth": "7744"
} | {
"index": 9102,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a monic cubic polynomial $P(x)$ with integer coefficients such that\n\\[P(\\sqrt[3]{2} + 1) = 0.\\](A polynomial is monic if its leading coefficient is 1.)"
}
] | math | {
"style": "rule",
"ground_truth": "x^3 - 3x^2 + 3x - 3"
} | {
"index": 9103,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a monic polynomial of degree $4,$ in $x,$ with rational coefficients such that $\\sqrt{2} +\\sqrt{3}$ is a root of the polynomial."
}
] | math | {
"style": "rule",
"ground_truth": "x^4-10x^2+1"
} | {
"index": 9104,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a monic quartic polynomial, in $x,$ with rational coefficients such that $2+\\sqrt{2}$ and $1-\\sqrt{3}$ are roots of the polynomial."
}
] | math | {
"style": "rule",
"ground_truth": "x^4-6x^3+8x^2+4x-4"
} | {
"index": 9105,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a natural number \\( N \\) that is divisible by 5 and 49, and has exactly 10 divisors, including 1 and \\( N \\)."
}
] | math | {
"style": "rule",
"ground_truth": "12005"
} | {
"index": 9106,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a natural number of the form \\( n = 2^{x} 3^{y} 5^{z} \\), knowing that half of this number has 30 fewer divisors, a third has 35 fewer divisors, and a fifth has 42 fewer divisors than the number itself."
}
] | math | {
"style": "rule",
"ground_truth": "2^6 * 3^5 * 5^4"
} | {
"index": 9107,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a necessary and sufficient condition on the natural number $ n$ for the equation\n\\[ x^n \\plus{} (2 \\plus{} x)^n \\plus{} (2 \\minus{} x)^n \\equal{} 0\n\\]\nto have a integral root."
}
] | math | {
"style": "rule",
"ground_truth": "n=1"
} | {
"index": 9108,
"solution": "\nTo solve the problem and find the necessary and sufficient condition for the natural number \\( n \\) such that the equation\n\n\\[\nx^n + (2 + x)^n + (2 - x)^n = 0\n\\]\n\nhas an integral root, we proceed as follows:\n\n### Step 1: Analyze the Case \\( n = 1 \\)\n\nSubstitute \\( n ... |
DeepScaleR | [
{
"role": "user",
"content": "Find a nonzero $p$ such that $px^2-12x+4=0$ has only one solution."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 9109,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a nonzero monic polynomial $P(x)$ with integer coefficients and minimal degree such that $P(1-\\sqrt[3]{2}+\\sqrt[3]{4})=0$. (A polynomial is called monic if its leading coefficient is 1.)"
}
] | math | {
"style": "rule",
"ground_truth": "x^{3}-3x^{2}+9x-9"
} | {
"index": 9110,
"solution": "Note that $(1-\\sqrt[3]{2}+\\sqrt[3]{4})(1+\\sqrt[3]{2})=3$, so $1-\\sqrt[3]{2}+\\sqrt[3]{4}=\\frac{3}{1+\\sqrt[3]{2}}$. Now, if $f(x)=x^{3}-2$, we have $f(\\sqrt[3]{2})=0$, so if we let $g(x)=f(x-1)=(x-1)^{3}-2=x^{3}-3x^{2}+3x-3$, then $g(1+\\sqrt[3]{2})=f(\\sqrt[3]{2})=0$. Finally, w... |
DeepScaleR | [
{
"role": "user",
"content": "Find a nonzero polynomial $P(x,y)$ such that $P(\\lfloor a \\rfloor, \\lfloor 2a \\rfloor) = 0$ for all real numbers $a$. (Note: $\\lfloor \\nu \\rfloor$ is the greatest integer less than or equal to $\\nu$.)"
}
] | math | {
"style": "rule",
"ground_truth": "(y-2x)(y-2x-1)"
} | {
"index": 9111,
"solution": "Take $P(x,y) = (y-2x)(y-2x-1)$. To see that this works, first note that if $m = \\lfloor a \\rfloor$, then $2m$ is an integer less than or equal to $2a$, so $2m \\leq \\lfloor 2a \\rfloor$. On the other hand, $m+1$ is an integer strictly greater than $a$, so $2m+2$ is an integer strict... |
DeepScaleR | [
{
"role": "user",
"content": "Find a number \\( N \\) with five digits, all different and none zero, which equals the sum of all distinct three-digit numbers whose digits are all different and are all digits of \\( N \\)."
}
] | math | {
"style": "rule",
"ground_truth": "35964"
} | {
"index": 9112,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a number such that when it is multiplied by its reverse, the product is 78445."
}
] | math | {
"style": "rule",
"ground_truth": "145"
} | {
"index": 9113,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a polynomial $ p\\left(x\\right)$ with real coefficients such that\n\n$ \\left(x\\plus{}10\\right)p\\left(2x\\right)\\equal{}\\left(8x\\minus{}32\\right)p\\left(x\\plus{}6\\right)$\n\nfor all real $ x$ and $ p\\left(1\\right)\\equal{}210$."
}
] | math | {
"style": "rule",
"ground_truth": "2(x + 4)(x - 4)(x - 8)"
} | {
"index": 9114,
"solution": "\nTo find the polynomial \\( p(x) \\) with real coefficients satisfying the functional equation:\n\n\\[\n(x + 10)p(2x) = (8x - 32)p(x + 6)\n\\]\n\nfor all real \\( x \\), and given the condition \\( p(1) = 210 \\), we start by analyzing the equation.\n\n### Step 1: Analyze and Simplify... |
DeepScaleR | [
{
"role": "user",
"content": "Find a positive integer $n$ with five non-zero different digits, which satisfies to be equal to the sum of all the three-digit numbers that can be formed using the digits of $n$ ."
}
] | math | {
"style": "rule",
"ground_truth": "35964"
} | {
"index": 9115,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a positive integer \\( n \\) less than 2006 such that \\( 2006n \\) is a multiple of \\( 2006 + n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "1475"
} | {
"index": 9116,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a positive integer \\( n \\) with 1000 digits, none of which are 0, such that we can group the digits into 500 pairs so that the sum of the products of the numbers in each pair divides \\( n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "111...111211221122112211221122112211221122112211221122112211221122112211221122112"
} | {
"index": 9117,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a positive integer that is divisible by 14 and has a square root between 25 and 25.3."
}
] | math | {
"style": "rule",
"ground_truth": "630"
} | {
"index": 9118,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a positive integer that is divisible by 18 and has a square root between 26 and 26.2."
}
] | math | {
"style": "rule",
"ground_truth": "684"
} | {
"index": 9119,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a positive integer that is divisible by 18 and whose cube root is a number between 8 and 8.1."
}
] | math | {
"style": "rule",
"ground_truth": "522"
} | {
"index": 9120,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a positive integer that is divisible by 20 and whose cube root is a number between 8.2 and 8.3."
}
] | math | {
"style": "rule",
"ground_truth": "560"
} | {
"index": 9121,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a positive integer that is divisible by 21 and has a square root between 30 and 30.5."
}
] | math | {
"style": "rule",
"ground_truth": "903"
} | {
"index": 9122,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a positive integer whose last digit is 2, and if this digit is moved from the end to the beginning, the new number is twice the original number."
}
] | math | {
"style": "rule",
"ground_truth": "105263157894736842"
} | {
"index": 9123,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a positive integral solution to the equation $\\frac{1+3+5+\\dots+(2n-1)}{2+4+6+\\dots+2n}=\\frac{115}{116}$."
}
] | math | {
"style": "rule",
"ground_truth": "115"
} | {
"index": 9124,
"solution": "1. **Identify the series in the numerator and denominator:**\n The numerator is the sum of the first $n$ odd numbers, which can be expressed as:\n \\[\n 1 + 3 + 5 + \\dots + (2n-1)\n \\]\n The denominator is the sum of the first $n$ even numbers, which can be expressed as:\n ... |
DeepScaleR | [
{
"role": "user",
"content": "Find a quadratic with rational coefficients and quadratic term $x^2$ that has $\\sqrt{3}-2$ as a root."
}
] | math | {
"style": "rule",
"ground_truth": "x^2+4x+1"
} | {
"index": 9125,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a real number $t$ such that for any set of 120 points $P_1, \\ldots P_{120}$ on the boundary of a unit square, there exists a point $Q$ on this boundary with $|P_1Q| + |P_2Q| + \\cdots + |P_{120}Q| = t$."
}
] | math | {
"style": "rule",
"ground_truth": "30(1 + \\sqrt{5})"
} | {
"index": 9126,
"solution": "\nWe need to find a real number \\( t \\) such that for any set of 120 points \\( P_1, \\ldots, P_{120} \\) on the boundary of a unit square, there exists a point \\( Q \\) on this boundary with \\( |P_1Q| + |P_2Q| + \\cdots + |P_{120}Q| = t \\).\n\nDefine \\(\\mathcal{U}\\) to be a se... |
DeepScaleR | [
{
"role": "user",
"content": "Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative."
}
] | math | {
"style": "rule",
"ground_truth": "(-7,-7,18,-7,-7,-7,18,-7,-7,18,-7,-7,-7,18,-7,-7)"
} | {
"index": 9127,
"solution": "Suppose it is possible to have more than 16 terms in the sequence. Let $a_{1}, a_{2}, \\ldots, a_{17}$ be the first 17 terms of the sequence. Consider the following array of terms in the sequence: \\begin{tabular}{lllllllllll} $a_{1}$ & $a_{2}$ & $a_{3}$ & $a_{4}$ & $a_{5}$ & $a_{6}$ &... |
DeepScaleR | [
{
"role": "user",
"content": "Find a six-digit number $\\overline{xy243z}$ that is divisible by 396."
}
] | math | {
"style": "rule",
"ground_truth": "432432"
} | {
"index": 9128,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a six-digit number where the first digit is 6 times less than the sum of all the digits to its right, and the second digit is 6 times less than the sum of all the digits to its right."
}
] | math | {
"style": "rule",
"ground_truth": "769999"
} | {
"index": 9129,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a ten-digit number where the first digit indicates how many times the digit 0 appears in the number, the second digit indicates how many times the digit 1 appears, and so forth, with the tenth digit indicating how many times the digit 9 appears in the number.\n\nGeneralize... | math | {
"style": "rule",
"ground_truth": "6210001000"
} | {
"index": 9130,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a three-digit number equal to the sum of the tens digit, the square of the hundreds digit, and the cube of the units digit.\n\nFind the number \\(\\overline{abcd}\\) that is a perfect square, if \\(\\overline{ab}\\) and \\(\\overline{cd}\\) are consecutive numbers, with \\... | math | {
"style": "rule",
"ground_truth": "357"
} | {
"index": 9131,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a three-digit number whose square is a six-digit number, such that each subsequent digit from left to right is greater than the previous one."
}
] | math | {
"style": "rule",
"ground_truth": "367"
} | {
"index": 9132,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a unit vector that is orthogonal to both $\\begin{pmatrix} 1 \\\\ 1 \\\\ 0 \\end{pmatrix}$ and $\\begin{pmatrix} 1 \\\\ 0 \\\\ 2 \\end{pmatrix}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -2/3 \\\\ 2/3 \\\\ 1/3 \\end{pmatrix}"
} | {
"index": 9133,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find a whole number, $M$, such that $\\frac{M}{5}$ is strictly between 9.5 and 10.5."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 9134,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all $a,$ $0^\\circ < a < 360^\\circ,$ such that $\\cos a,$ $\\cos 2a,$ and $\\cos 3a$ form an arithmetic sequence, in that order. Enter the solutions, separated by commas, in degrees."
}
] | math | {
"style": "rule",
"ground_truth": "45^\\circ, 135^\\circ, 225^\\circ, 315^\\circ"
} | {
"index": 9135,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all $c$ which satisfy $$\\frac{c}{3} \\le 2+c < -2(1+c).$$Express your answer in interval notation, simplifying any fractions which occur in your answer."
}
] | math | {
"style": "rule",
"ground_truth": "\\left[-3,-\\frac{4}{3}\\right)"
} | {
"index": 9136,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all $f : \\mathbb{N} \\to \\mathbb{N} $ such that $f(a) + f(b)$ divides $2(a + b - 1)$ for all $a, b \\in \\mathbb{N}$.\n\nRemark: $\\mathbb{N} = \\{ 1, 2, 3, \\ldots \\} $ denotes the set of the positive integers."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = 1 \\text{ or } f(x) = 2x - 1."
} | {
"index": 9137,
"solution": "\nWe need to find all functions \\( f : \\mathbb{N} \\to \\mathbb{N} \\) such that \\( f(a) + f(b) \\) divides \\( 2(a + b - 1) \\) for all \\( a, b \\in \\mathbb{N} \\).\n\nBegin by analyzing the given divisibility condition:\n\\[\nf(a) + f(b) \\mid 2(a + b - 1).\n\\]\n\n### Step 1: S... |
DeepScaleR | [
{
"role": "user",
"content": "Find all $t$ such that $x-t$ is a factor of $4x^2 + 11x - 3$."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 9138,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all $t$ such that $x-t$ is a factor of $6x^2+13x-5.$\n\nEnter your answer as a list separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{2}"
} | {
"index": 9139,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all $x$ such that $\\lfloor \\lfloor 2x \\rfloor - 1/2 \\rfloor = \\lfloor x + 2 \\rfloor.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\left[ \\frac{5}{2}, \\frac{7}{2} \\right)"
} | {
"index": 9140,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all $x$ such that $x^2+5x<6$. Express your answer in interval notation."
}
] | math | {
"style": "rule",
"ground_truth": "(-6, 1)"
} | {
"index": 9141,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all $y \\in [0,\\pi]$ for which \\[\\sin(x+y)\\leq \\sin(x)+\\sin(y)\\]for every $x$ between $0$ and $\\pi$, inclusive."
}
] | math | {
"style": "rule",
"ground_truth": "[0,\\pi]"
} | {
"index": 9142,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all 10-digit numbers \\( a_0 a_1 \\ldots a_9 \\) such that for each \\( k \\), \\( a_k \\) is the number of times that the digit \\( k \\) appears in the number."
}
] | math | {
"style": "rule",
"ground_truth": "6210001000"
} | {
"index": 9143,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all 4-digit numbers $\\overline{abcd}$ that are multiples of $11$ , such that the 2-digit number $\\overline{ac}$ is a multiple of $7$ and $a + b + c + d = d^2$ ."
}
] | math | {
"style": "rule",
"ground_truth": "3454"
} | {
"index": 9144,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all 4-digit numbers $n$ , such that $n=pqr$ , where $p<q<r$ are distinct primes, such that $p+q=r-q$ and $p+q+r=s^2$ , where $s$ is a prime number."
}
] | math | {
"style": "rule",
"ground_truth": "2015"
} | {
"index": 9145,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all 6-digit multiples of 22 of the form $5d5,\\!22e$ where $d$ and $e$ are digits. What is the maximum value of $d$?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 9146,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all \\( a_{0} \\in \\mathbb{R} \\) such that the sequence defined by\n\\[ a_{n+1} = 2^{n} - 3a_{n}, \\quad n = 0, 1, 2, \\cdots \\]\nis increasing."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 9147,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all \\( t \\) such that \\( x-t \\) is a factor of \\( 10x^2 + 21x - 10 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{2}"
} | {
"index": 9148,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all \\( x \\in [1,2) \\) such that for any positive integer \\( n \\), the value of \\( \\left\\lfloor 2^n x \\right\\rfloor \\mod 4 \\) is either 1 or 2."
}
] | math | {
"style": "rule",
"ground_truth": "4/3"
} | {
"index": 9149,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all angles $\\theta,$ $0 \\le \\theta \\le 2 \\pi,$ with the following property: For all real numbers $x,$ $0 \\le x \\le 1,$\n\\[x^2 \\cos \\theta - x(1 - x) + (1 - x)^2 \\sin \\theta > 0.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{\\pi}{12}, \\frac{5 \\pi}{12} \\right)"
} | {
"index": 9150,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all complex numbers $z$ such that\n\\[z^2 = -77 - 36i.\\]Enter all complex numbers, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "2 - 9i, -2 + 9i"
} | {
"index": 9151,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all composite positive integers \\(m\\) such that, whenever the product of two positive integers \\(a\\) and \\(b\\) is \\(m\\), their sum is a power of $2$ .\n\n*Proposed by Harun Khan*"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 9152,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all five-digit numbers \\(\\overline{abcde}\\) that are divisible by 9, and \\(\\overline{ace} - \\overline{bda} = 760\\)."
}
] | math | {
"style": "rule",
"ground_truth": "81828"
} | {
"index": 9153,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all four-digit numbers that are 9 times larger than their reversed counterparts."
}
] | math | {
"style": "rule",
"ground_truth": "9801"
} | {
"index": 9154,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $ f : \\mathbb{R} \\rightarrow \\mathbb{R} $ such that\n\\[ f( xf(x) + 2y) = f(x^2)+f(y)+x+y-1 \\]\nholds for all $ x, y \\in \\mathbb{R}$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x + 1"
} | {
"index": 9155,
"solution": "\nTo solve the functional equation\n\n\\[\nf(xf(x) + 2y) = f(x^2) + f(y) + x + y - 1\n\\]\n\nfor all real numbers \\( x \\) and \\( y \\), we need to find all possible functions \\( f : \\mathbb{R} \\rightarrow \\mathbb{R} \\) that satisfy this condition.\n\n**Step 1: Initial Evaluatio... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $ f: (0, \\infty) \\mapsto (0, \\infty)$ (so $ f$ is a function from the positive real numbers) such that\n\\[ \\frac {\\left( f(w) \\right)^2 \\plus{} \\left( f(x) \\right)^2}{f(y^2) \\plus{} f(z^2) } \\equal{} \\frac {w^2 \\plus{} x^2}{y^2 \\plus{} z^2}\n\\... | math | {
"style": "rule",
"ground_truth": "f(x) = x \\text{ or } f(x) = \\frac{1}{x}"
} | {
"index": 9156,
"solution": "\nTo find all functions \\( f: (0, \\infty) \\to (0, \\infty) \\) satisfying the given functional equation:\n\n\\[\n\\frac {\\left( f(w) \\right)^2 + \\left( f(x) \\right)^2}{f(y^2) + f(z^2) } = \\frac {w^2 + x^2}{y^2 + z^2}\n\\]\n\nfor all positive real numbers \\( w, x, y, z \\) such... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $ f: \\mathbb{Q}^{\\plus{}} \\mapsto \\mathbb{Q}^{\\plus{}}$ such that: \n\\[ f(x) \\plus{} f(y) \\plus{} 2xy f(xy) \\equal{} \\frac {f(xy)}{f(x\\plus{}y)}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{x^2}"
} | {
"index": 9157,
"solution": "\nLet \\( f: \\mathbb{Q}^{+} \\to \\mathbb{Q}^{+} \\) be a function such that:\n\\[ f(x) + f(y) + 2xy f(xy) = \\frac{f(xy)}{f(x+y)} \\]\nfor all \\( x, y \\in \\mathbb{Q}^{+} \\).\n\nFirst, we denote the assertion of the given functional equation as \\( P(x, y) \\).\n\n1. From \\( P(1,... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $ f: \\mathbb{R}^{ \\plus{} }\\to\\mathbb{R}^{ \\plus{} }$ satisfying $ f\\left(x \\plus{} f\\left(y\\right)\\right) \\equal{} f\\left(x \\plus{} y\\right) \\plus{} f\\left(y\\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \\mathbb{R}^{ \\plus... | math | {
"style": "rule",
"ground_truth": "f(x) = 2x"
} | {
"index": 9158,
"solution": "\nTo find all functions \\( f: \\mathbb{R}^{+} \\to \\mathbb{R}^{+} \\) satisfying the given functional equation:\n\n\\[\nf(x + f(y)) = f(x + y) + f(y)\n\\]\n\nfor all positive real numbers \\( x \\) and \\( y \\), we will proceed as follows.\n\n### Step 1: Exploring the Functional Equ... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $ f:\\mathbb{R}\\rightarrow\\mathbb{R} $ such that for all $x,y\\in{{\\mathbb{R}}}$ holds\n$f(x^2)+f(2y^2)=(f(x+y)+f(y))(f(x-y)+f(y))$"
}
] | math | {
"style": "rule",
"ground_truth": "$f(x) = \\frac{1}{2},f(x) = 0,f(x) = x^2$"
} | {
"index": 9159,
"solution": "\nTo solve the functional equation \n\n\\[\nf(x^2) + f(2y^2) = (f(x+y) + f(y))(f(x-y) + f(y))\n\\]\n\nfor all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\), we will analyze the equation under specific substitutions and deduce the form of \\( f(x) \\).\n\n### Step 1: Substitution an... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f : \\mathbb{N}\\rightarrow \\mathbb{N}$ satisfying following condition:\n\\[f(n+1)>f(f(n)), \\quad \\forall n \\in \\mathbb{N}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "{f(n)=n}"
} | {
"index": 9160,
"solution": "\nWe are tasked with finding all functions \\( f: \\mathbb{N} \\rightarrow \\mathbb{N} \\) that satisfy the condition:\n\\[\nf(n+1) > f(f(n)), \\quad \\forall n \\in \\mathbb{N}.\n\\]\n\nTo solve this problem, let us first analyze the condition given:\n\\[\nf(n+1) > f(f(n)).\n\\]\nThis... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f : \\mathbb{R} \\to \\mathbb{R}$ such that\n\\[f(x(x + f(y))) = (x + y)f(x),\\]\nfor all $x, y \\in\\mathbb{R}$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = 0 \\text{ and } f(x) = x."
} | {
"index": 9161,
"solution": "\nLet's consider the functional equation \\( f(x(x + f(y))) = (x + y)f(x) \\) for all \\( x, y \\in \\mathbb{R} \\).\n\n### Step 1: Test simple functions\nFirst, let's test the simplest potential solutions.\n\n1. **\\( f(x) = 0 \\):**\n - Substituting \\( f(x) = 0 \\) into the equati... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f$ defined on all real numbers and taking real values such that \\[f(f(y)) + f(x - y) = f(xf(y) - x),\\] for all real numbers $x, y.$"
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = 0"
} | {
"index": 9162,
"solution": " \nTo solve the given functional equation, we need to find all functions \\( f : \\mathbb{R} \\rightarrow \\mathbb{R} \\) such that:\n\\[\nf(f(y)) + f(x - y) = f(xf(y) - x)\n\\]\nholds for all real numbers \\( x \\) and \\( y \\).\n\n### Step-by-... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f$ defined on the non-negative reals and taking non-negative real values such that: $f(2)=0,f(x)\\ne0$ for $0\\le x<2$, and $f(xf(y))f(y)=f(x+y)$ for all $x,y$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = \\begin{cases}\n\\frac{2}{2 - x}, & 0 \\leq x < 2, \\\\\n0, & x \\geq 2.\n\\end{cases}"
} | {
"index": 9163,
"solution": "\nWe need to find all functions \\( f: [0, \\infty) \\to [0, \\infty) \\) that satisfy the following conditions:\n1. \\( f(2) = 0 \\).\n2. \\( f(x) \\neq 0 \\) for \\( 0 \\leq x < 2 \\).\n3. \\( f(xf(y))f(y) = f(x+y) \\) for all \\( x, y \\geq 0 \\).\n\nLet's begin by analyzing these c... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy: $f(xf(y))=yf(x)$ for all $x,y$; and $f(x)\\to0$ as $x\\to\\infty$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x)=\\frac1x"
} | {
"index": 9164,
"solution": "\nTo tackle this problem, we want to find all functions \\( f: \\mathbb{R}^+ \\to \\mathbb{R}^+ \\) that satisfy:\n\n1. \\( f(xf(y)) = yf(x) \\) for all \\( x, y \\in \\mathbb{R}^+ \\).\n2. \\( \\lim_{x \\to \\infty} f(x) = 0 \\).\n\n### Step-by-step Solution:\n\n1. **Substitute Specia... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f$ from the reals to the reals such that \\[ \\left(f(x)+f(z)\\right)\\left(f(y)+f(t)\\right)=f(xy-zt)+f(xt+yz) \\] for all real $x,y,z,t$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = 0, \\quad f(x) = \\frac{1}{2}, \\quad f(x) = x^2."
} | {
"index": 9165,
"solution": "\nTo solve the given functional equation for all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\):\n\n\\[\n(f(x) + f(z))(f(y) + f(t)) = f(xy - zt) + f(xt + yz),\n\\]\n\nwe start by analyzing specific cases to deduce possible forms for \\( f(x) \\).\n\n1. **Testing the Zero Function:**... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f$ from the set $\\mathbb{R}$ of real numbers into $\\mathbb{R}$ which satisfy for all $x, y, z \\in \\mathbb{R}$ the identity \\[f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = 0 \\text{ and } f(x) = x^2"
} | {
"index": 9166,
"solution": "\nWe need to find all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) satisfying the given functional equation for all \\( x, y, z \\in \\mathbb{R} \\):\n\n\\[\nf(f(x) + f(y) + f(z)) = f(f(x) - f(y)) + f(2xy + f(z)) + 2f(xz - yz).\n\\]\n\nTo solve this, we'll explore potentia... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f: (0, \\infty) \\to (0, \\infty)$ such that\n\\begin{align*}\nf(y(f(x))^3 + x) = x^3f(y) + f(x) \n\\end{align*}\nfor all $x, y>0$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x"
} | {
"index": 9167,
"solution": "\nWe are given the functional equation for functions \\( f: (0, \\infty) \\to (0, \\infty) \\) defined by:\n\n\\[\nf(y(f(x))^3 + x) = x^3f(y) + f(x) \n\\]\n\nfor all \\( x, y > 0 \\). We need to find all such functions \\( f \\).\n\n1. **Initial Observation:**\n We consider the speci... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f: R \\to R$ such that, for any $x, y \\in R$:\n$f\\left( f\\left( x \\right)-y \\right)\\cdot f\\left( x+f\\left( y \\right) \\right)={{x}^{2}}-{{y}^{2}}$"
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x"
} | {
"index": 9168,
"solution": "\nConsider the functional equation that must be satisfied for all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\):\n\n\\[\nf\\left( f(x) - y \\right) \\cdot f\\left( x + f(y) \\right) = x^2 - y^2\n\\]\n\n### Step 1: Analyze the structure\n\n1. **Initial Observation**: The equation is... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f: \\mathbb R \\to \\mathbb R$ such that \\[ f( xf(x) + f(y) ) = f^2(x) + y \\] for all $x,y\\in \\mathbb R$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x \\text{ or } f(x) = -x"
} | {
"index": 9169,
"solution": "\nTo find all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) satisfying the given functional equation:\n\n\\[\nf(xf(x) + f(y)) = f^2(x) + y \\quad \\text{for all } x, y \\in \\mathbb{R},\n\\]\n\nwe proceed with the following steps.\n\n### Step 1: Analyzing the Functional Equation\n\... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f: \\mathbb R \\to \\mathbb R$ such that for any $x,y \\in \\mathbb R$, the multiset $\\{(f(xf(y)+1),f(yf(x)-1)\\}$ is identical to the multiset $\\{xf(f(y))+1,yf(f(x))-1\\}$.\n\n[i]Note:[/i] The multiset $\\{a,b\\}$ is identical to the multiset $\\{c,d\\}$ ... | math | {
"style": "rule",
"ground_truth": "f(x) \\equiv x \\text{ or } f(x) \\equiv -x"
} | {
"index": 9170,
"solution": "\nLet \\( f: \\mathbb{R} \\to \\mathbb{R} \\) be a function such that for any \\( x, y \\in \\mathbb{R} \\), the multiset \\( \\{ f(xf(y) + 1), f(yf(x) - 1) \\} \\) is identical to the multiset \\( \\{ xf(f(y)) + 1, yf(f(x)) - 1 \\} \\).\n\nWe aim to find all such functions \\( f \\).\... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ such that\n$$f(x^2 + y) \\ge (\\frac{1}{x} + 1)f(y)$$\nholds for all $x \\in \\mathbb{R} \\setminus \\{0\\}$ and all $y \\in \\mathbb{R}$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = 0"
} | {
"index": 9171,
"solution": "\nTo find all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) that satisfy the given inequality:\n\n\\[\nf(x^2 + y) \\ge \\left(\\frac{1}{x} + 1\\right)f(y)\n\\]\n\nfor all \\( x \\in \\mathbb{R} \\setminus \\{0\\} \\) and \\( y \\in \\mathbb{R} \\), we'll start by analyzing ... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ such that\n$$f(xy) = f(x)f(y) + f(f(x + y))$$\nholds for all $x, y \\in \\mathbb{R}$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = 0 \\text{ and } f(x) = x - 1."
} | {
"index": 9172,
"solution": "\nWe are tasked with finding all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) satisfying the equation:\n\n\\[\nf(xy) = f(x)f(y) + f(f(x+y))\n\\]\n\nfor all \\( x, y \\in \\mathbb{R} \\).\n\n### Step 1: Consideration of Simple Cases\n\nFirst, let us consider the case where ... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ such that $$f(f(x)+y)+xf(y)=f(xy+y)+f(x)$$ for reals $x, y$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x \\text{ or } f(x) = 0"
} | {
"index": 9173,
"solution": "\nLet \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) be a function satisfying the functional equation:\n\n\\[\nf(f(x) + y) + x f(y) = f(xy + y) + f(x)\n\\]\n\nfor all real numbers \\( x \\) and \\( y \\).\n\n### Step 1: Initial Substitution\n\nStart by substituting \\( y = 0 \\) into ... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f: \\mathbb{R}^+ \\to \\mathbb{R}^+$ such that \n$$(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),$$\nfor all positive real numbers $x, y, z$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x"
} | {
"index": 9174,
"solution": "\nTo solve this functional equation problem, we want to find all functions \\( f: \\mathbb{R}^+ \\to \\mathbb{R}^+ \\) such that:\n\n\\[\n(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x)\n\\]\n\nfor all positive real numbers \\(x, y, z\\). Our goal is to prove that the function satisfies ... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+}$ such that\n$$f(xf(x + y)) = yf(x) + 1$$\nholds for all $x, y \\in \\mathbb{R}^{+}$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = \\frac{1}{x}"
} | {
"index": 9175,
"solution": "\nTo solve the functional equation \\( f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+} \\) given by\n\n\\[\nf(xf(x + y)) = yf(x) + 1\n\\]\n\nfor all \\( x, y \\in \\mathbb{R}^{+} \\), we aim to determine a function \\( f \\) that satisfies this relationship.\n\n### Step 1: Assume a fun... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+}$, such that $f(x+f(x)+f(y))=2f(x)+y$ for all positive reals $x,y$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x"
} | {
"index": 9176,
"solution": "\nTo find all functions \\( f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+} \\) such that \n\n\\[\nf(x+f(x)+f(y)) = 2f(x) + y \n\\]\n\nfor all positive reals \\(x, y\\), we proceed with the following steps:\n\n**Step 1: Analyze the functional equation**\n\nLet's fix \\( y = y_0 \\) in ... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f:(0,\\infty)\\rightarrow (0,\\infty)$ such that for any $x,y\\in (0,\\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \\left(f(f(x^2)) + f(f(y^2))\\right).$$"
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = \\frac{1}{x}"
} | {
"index": 9177,
"solution": "\nLet's find all functions \\( f: (0, \\infty) \\rightarrow (0, \\infty) \\) that satisfy the functional equation:\n\n\\[\nxf(x^2)f(f(y)) + f(yf(x)) = f(xy) \\left( f(f(x^2)) + f(f(y^2)) \\right).\n\\]\n\nTo solve this problem, consider the possibility \\( f(x) = \\frac{1}{x} \\). We w... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb {Z}\\to\\mathbb Z$, satisfy that for any integer ${a}$, ${b}$, ${c}$,\n$$2f(a^2+b^2+c^2)-2f(ab+bc+ca)=f(a-b)^2+f(b-c)^2+f(c-a)^2$$"
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = 0 \\text{ or } f(x) = x"
} | {
"index": 9178,
"solution": "\nWe are given the functional equation for \\( f: \\mathbb{Z} \\to \\mathbb{Z} \\):\n\\[\n2f(a^2 + b^2 + c^2) - 2f(ab + bc + ca) = f(a - b)^2 + f(b - c)^2 + f(c - a)^2\n\\]\nfor any integers \\( a, b, \\) and \\( c \\).\n\nTo find all such functions \\( f \\), we proceed as follows:\n\... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{N}\\rightarrow \\mathbb{N}$ such that the inequality $$f(x)+yf(f(x))\\le x(1+f(y))$$\nholds for all positive integers $x, y$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x"
} | {
"index": 9179,
"solution": "\nLet's analyze the problem by working with the given inequality:\n\n\\[\nf(x) + y f(f(x)) \\le x(1 + f(y))\n\\]\n\nfor all positive integers \\(x, y\\).\n\nTo find all functions \\(f: \\mathbb{N} \\rightarrow \\mathbb{N}\\) satisfying this inequality, we will first test some small val... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{R} \\rightarrow \\mathbb{R}$ that satisfy the conditions\n\\[f(1+xy)-f(x+y)=f(x)f(y) \\quad \\text{for all } x,y \\in \\mathbb{R},\\]\nand $f(-1) \\neq 0$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x - 1"
} | {
"index": 9180,
"solution": "\nTo find all functions \\( f:\\mathbb{R} \\rightarrow \\mathbb{R} \\) satisfying the functional equation\n\n\\[\nf(1+xy) - f(x+y) = f(x)f(y)\n\\]\n\nfor all \\( x, y \\in \\mathbb{R} \\), and also given that \\( f(-1) \\neq 0 \\), we proceed as follows:\n\n### Step 1: Investigate poss... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{R} \\rightarrow \\mathbb{R}$, such that $$f(xy+f(x^2))=xf(x+y)$$ for all reals $x, y$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = 0 \\text{ and } f(x) = x"
} | {
"index": 9181,
"solution": "To find the functions \\( f : \\mathbb{R} \\rightarrow \\mathbb{R} \\) that satisfy the functional equation:\n\\[\nf(xy + f(x^2)) = x f(x + y),\n\\]\nfor all real numbers \\( x \\) and \\( y \\), we will proceed with the following steps:\n\n### Step 1: Explore Simple Solutions\nFirst, ... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{R} \\to \\mathbb{R}$ such that for any two real numbers $x,y$ holds\n$$f(xf(y)+2y)=f(xy)+xf(y)+f(f(y)).$$"
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = 2x \\text{ and } f(x) = 0"
} | {
"index": 9182,
"solution": "\nTo solve the functional equation problem, we need to identify all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that satisfy the given functional equation for any real numbers \\( x, y \\):\n\n\\[\nf(xf(y) + 2y) = f(xy) + xf(y) + f(f(y)).\n\\]\n\nWe will explore potential solutio... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{R}\\rightarrow\\mathbb{R}$ such that $$f(x^2y)=f(xy)+yf(f(x)+y)$$ for all real numbers $x$ and $y$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = 0"
} | {
"index": 9183,
"solution": "\nTo find all functions \\( f:\\mathbb{R}\\rightarrow\\mathbb{R} \\) that satisfy the functional equation\n\n\\[\nf(x^2y) = f(xy) + yf(f(x) + y)\n\\]\n\nfor all real numbers \\( x \\) and \\( y \\), let's proceed as follows:\n\n1. **Initial Substitution and Simplification:**\n\n Subs... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{R}\\rightarrow\\mathbb{R}$ such that $f(0)\\neq 0$ and for all $x,y\\in\\mathbb{R}$,\n \\[ f(x+y)^2 = 2f(x)f(y) + \\max \\left\\{ f(x^2+y^2), f(x^2)+f(y^2) \\right\\}. \\]"
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = -1 \\text{ and } f(x) = x - 1."
} | {
"index": 9184,
"solution": "\nTo find all functions \\( f:\\mathbb{R}\\rightarrow\\mathbb{R} \\) satisfying the given functional equation, we start with the condition:\n\\[ f(0)\\neq 0, \\]\nand the functional equation for all \\( x, y \\in \\mathbb{R} \\):\n\\[ f(x+y)^2 = 2f(x)f(y) + \\max \\left\\{ f(x^2+y^2), ... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{R}\\rightarrow\\mathbb{R}$ that satisfy \\[f(x^2-y)+2yf(x)=f(f(x))+f(y)\\] for all $x,y\\in\\mathbb{R}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\[ f(x) = -x^2, \\quad f(x) = 0, \\quad f(x) = x^2 \\]"
} | {
"index": 9185,
"solution": "Plugging in $y$ as $0:$ \\begin{equation}\nf(x^2)=f(f(x))+f(0) \\text{ } (1)\n\\end{equation}\nPlugging in $x, y$ as $0:$ \\[f(0)=f(f(0))+f(0)\\] or \\[f(f(0))=0\\] Plugging in $x$ as $0:$ \\[f(-y)+2yf(0)=f(f(0))+f(y),\\] but since $f(f(0))=0,$ \\begin{equation}\nf(-y)+2yf(0)=f(y) \\... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{R}\\to \\mathbb{R}$ such that \n$$f(x)+f(yf(x)+f(y))=f(x+2f(y))+xy$$for all $x,y\\in \\mathbb{R}$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x + 1"
} | {
"index": 9186,
"solution": "\nLet \\( f: \\mathbb{R} \\to \\mathbb{R} \\) be a function that satisfies the equation:\n\\[\nf(x) + f(yf(x) + f(y)) = f(x + 2f(y)) + xy\n\\]\nfor all \\( x, y \\in \\mathbb{R} \\).\n\nWe need to find all such functions \\( f \\).\n\n### Step 1: Check for Simple Solutions\nAssume \\( ... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{R}\\to\\mathbb{R}$ such that $f(x)f(y)+f(x+y)=xy$ for all real numbers $x$ and $y$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x - 1 \\text{ or } f(x) = -x - 1."
} | {
"index": 9187,
"solution": "\nWe are tasked with finding all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that for all real numbers \\( x \\) and \\( y \\), the following functional equation holds:\n\n\\[\nf(x)f(y) + f(x+y) = xy.\n\\]\n\n**Step 1: Plug-in Specific Values**\n\nFirst, let's test for simpl... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{R}\\to\\mathbb{R}$ which satisfy the following conditions: $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2.$"
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x"
} | {
"index": 9188,
"solution": "\nLet \\( f: \\mathbb{R} \\to \\mathbb{R} \\) be a function satisfying the following conditions:\n1. \\( f(x+1) = f(x) + 1 \\).\n2. \\( f(x^2) = f(x)^2 \\).\n\nWe aim to find all such functions.\n\n### Step 1: Analyze the First Condition\n\nThe first condition \\( f(x+1) = f(x) + 1 \\)... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{R}\\to\\mathbb{R}$ which satisfy the following equality for all $x,y\\in\\mathbb{R}$ \\[f(x)f(y)-f(x-1)-f(y+1)=f(xy)+2x-2y-4.\\][i]"
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x^2 + 1"
} | {
"index": 9189,
"solution": "\nTo determine all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that satisfy the given functional equation for all \\( x, y \\in \\mathbb{R} \\):\n\n\\[\nf(x)f(y) - f(x-1) - f(y+1) = f(xy) + 2x - 2y - 4,\n\\]\n\nwe proceed as follows.\n\n### Step 1: Substitute Special Values\n\n1.... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{R}^+ \\rightarrow \\mathbb{R}^+$, such that $$f(x^{2023}+f(x)f(y))=x^{2023}+yf(x)$$ for all $x, y>0$."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x"
} | {
"index": 9190,
"solution": "\nTo solve the functional equation for functions \\( f: \\mathbb{R}^+ \\rightarrow \\mathbb{R}^+ \\) such that\n\n\\[\nf(x^{2023} + f(x)f(y)) = x^{2023} + yf(x)\n\\]\n\nfor all \\( x, y > 0 \\), we will proceed with the following steps:\n\n### Step 1: Initial Substitution\nSubstitute \... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions $f : \\mathbb{Z}^+ \\to \\mathbb{Z}^+$ (where $\\mathbb{Z}^+$ is the set of positive integers) such that $f(n!) = f(n)!$ for all positive integers $n$ and such that $m - n$ divides $f(m) - f(n)$ for all distinct positive integers $m$ , $n$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\[\n\\boxed{f(n)=1, f(n)=2, f(n)=n}\n\\]"
} | {
"index": 9191,
"solution": "By the first condition we have $f(1)=f(1!)=f(1)!$ and $f(2)=f(2!)=f(2)!$ , so $f(1)=1$ or $2$ and similarly for $f(2)$ . By the second condition, we have \\[n\\cdot n!=(n+1)!-n! \\mid f(n+1)!-f(n)! \\qquad \\qquad (1)\\] for all positive integers $n$ .\nSuppose that for some $n \\geq ... |
DeepScaleR | [
{
"role": "user",
"content": "Find all functions \\( f: \\mathbb{Q} \\rightarrow \\{-1, 1\\} \\) such that for all distinct \\( x, y \\in \\mathbb{Q} \\) satisfying \\( xy = 1 \\) or \\( x + y \\in \\{0, 1\\} \\), we have \\( f(x) f(y) = -1 \\).\n\nIntermediate question: Let \\( f \\) be a function having t... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 9192,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all integer $n$ such that the following property holds: for any positive real numbers $a,b,c,x,y,z$, with $max(a,b,c,x,y,z)=a$ , $a+b+c=x+y+z$ and $abc=xyz$, the inequality $$a^n+b^n+c^n \\ge x^n+y^n+z^n$$ holds."
}
] | math | {
"style": "rule",
"ground_truth": "n \\ge 0"
} | {
"index": 9193,
"solution": "\nWe are given the conditions \\( \\max(a, b, c, x, y, z) = a \\), \\( a + b + c = x + y + z \\), and \\( abc = xyz \\). We need to find all integer \\( n \\) such that the inequality\n\\[\na^n + b^n + c^n \\ge x^n + y^n + z^n\n\\]\nholds for any positive real numbers \\( a, b, c, x, y... |
DeepScaleR | [
{
"role": "user",
"content": "Find all integer values of $a$ so that the polynomial\n\\[x^3 + 3x^2 + ax + 7 = 0\\]has at least one integer root. Enter all possible values of $a,$ separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "-71, -27, -11, 9"
} | {
"index": 9194,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all integer values of the parameter \\(a\\) for which the system\n\\[\n\\begin{cases}\nx - 2y = y^2 + 2, \\\\\nax - 2y = y^2 + x^2 + 0.25a^2\n\\end{cases}\n\\]\nhas at least one solution. In the answer, indicate the sum of the found values of the parameter \\(a\\)."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 9195,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all integers $A$ if it is known that $A^{6}$ is an eight-digit number composed of the digits $0, 1, 2, 2, 2, 3, 4, 4$."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 9196,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find all integers $\\,a,b,c\\,$ with $\\,1<a<b<c\\,$ such that \\[ (a-1)(b-1)(c-1) \\] is a divisor of $abc-1.$"
}
] | math | {
"style": "rule",
"ground_truth": "(2, 4, 8) \\text{ and } (3, 5, 15)"
} | {
"index": 9197,
"solution": "\nWe are tasked with finding all integers \\( a, b, c \\) with \\( 1 < a < b < c \\) such that \n\n\\[\n(a-1)(b-1)(c-1)\n\\]\n\nis a divisor of \n\n\\[\nabc - 1.\n\\]\n\nLet's first express \\( abc - 1 \\) in terms of potential divisors' expressions:\n\n1. We want \\((a-1)(b-1)(c-1) \\... |
DeepScaleR | [
{
"role": "user",
"content": "Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$."
}
] | math | {
"style": "rule",
"ground_truth": "(m, n) = (0, 0) \\text{ and } (m, n) = (-2, 2)"
} | {
"index": 9198,
"solution": "\nWe are tasked with finding all integer pairs \\((m, n)\\) such that:\n\n\\[\nm^5 - n^5 = 16mn\n\\]\n\n**Step 1: Algebraic Manipulation**\n\nWe begin by rewriting the given equation as:\n\n\\[\nm^5 - n^5 - 16mn = 0\n\\]\n\n**Step 2: Factorization**\n\nUsing the identity for the differ... |
DeepScaleR | [
{
"role": "user",
"content": "Find all integers $m$ such that $m^{2}+6 m+28$ is a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "6,-12"
} | {
"index": 9199,
"solution": "We must have $m^{2}+6 m+28=n^{2}$, where $n$ is an integer. Rewrite this as $(m+3)^{2}+19=$ $n^{2} \\Rightarrow n^{2}-(m+3)^{2}=19 \\Rightarrow(n-m-3)(n+m+3)=19$. Let $a=n-m-3$ and $b=n+m+3$, so we want $a b=19$. This leaves only 4 cases: - $a=1, b=19$. Solve the system $n-m-3=1$ and $... |
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