Found problems: 1513
2017-IMOC, N5
Find all functions $f:\mathbb N\to\mathbb N$ such that
$$f(x)+f(y)\mid x^2-y^2$$holds for all $x,y\in\mathbb N$.
1963 Swedish Mathematical Competition., 4
Given the real number $k$, find all differentiable real-valued functions $f(x)$ defined on the reals such that $f(x+y) = f(x) + f(y) + f(kxy)$ for all $x, y$.
2024 Rioplatense Mathematical Olympiad, 5
Let $S = \{2, 3, 4, \dots\}$ be the set of positive integers greater than 1. Find all functions $f : S \to S$ that satisfy
\[
\text{gcd}(a, f(b)) \cdot \text{lcm}(f(a), b) = f(ab)
\]
for all pairs of integers $a, b \in S$.
Clarification: $\text{gcd}(a,b)$ is the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ is the least common multiple of $a$ and $b$.
2023 District Olympiad, P4
Determine all strictly increasing functions $f:\mathbb{N}_0\to\mathbb{N}_0$ which satisfy \[f(x)\cdot f(y)\mid (1+2x)\cdot f(y)+(1+2y)\cdot f(x)\]for all non-negative integers $x{}$ and $y{}$.
2007 France Team Selection Test, 2
Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$:
\[f(x-y+f(y))=f(x)+f(y).\]
1965 Dutch Mathematical Olympiad, 5
The function ƒ. which is defined for all real numbers satisfies:
$$f(x+y)+f(x-y)=2f(x)+2f(y)$$
Prove that $f(0) = 0$, $f(-x) = f(x)$, $f(2x) = 4 f (x)$, $$f(x + y + z) = f(x + y) + f(y + z) + f(z + x) -f(x) - f(y) -f(z).$$
2023 India IMO Training Camp, 2
Let $g:\mathbb{N}\to \mathbb{N}$ be a bijective function and suppose that $f:\mathbb{N}\to \mathbb{N}$ is a function such that:
[list]
[*] For all naturals $x$, $$\underbrace{f(\cdots (f}_{x^{2023}\;f\text{'s}}(x)))=x. $$
[*] For all naturals $x,y$ such that $x|y$, we have $f(x)|g(y)$.
[/list]
Prove that $f(x)=x$.
[i]Proposed by Pulkit Sinha[/i]
2016 IFYM, Sozopol, 3
Let $f: \mathbb{R}^2\rightarrow \mathbb{R}$ be a function for which for arbitrary $x,y,z\in \mathbb{R}$ we have that
$f(x,y)+f(y,z)+f(z,x)=0$.
Prove that there exist function $g:\mathbb{R}\rightarrow \mathbb{R}$ for which: $f(x,y)=g(x)-g(y),\, \forall x,y\in \mathbb{R}$.
1987 Traian Lălescu, 1.2
Let $ A $ be a subset of $ \mathbb{R} $ and let be a function $ f:A\longrightarrow\mathbb{R} $ satisfying
$$ f(x)-f(y)=(y-x)f(x)f(y),\quad\forall x,y\in A. $$
[b]a)[/b] Show that if $ A=\mathbb{R}, $ then $ f=0. $
[b]b)[/b] Find $ f, $ provided that $ A=\mathbb{R}\setminus\{1\} . $
2013 QEDMO 13th or 12th, 4
Let $a> 0$ and $f: R\to R$ a function such that $f (x) + f (x + 2a) + f (x + 3a) + f (x + 5a) = 1$ for all $x\in R$ . Show that $f$ is periodic, that is, that there is some $b> 0$ for which $f (x) = f (x + b)$ for every $x \in R$ holds. Find the smallest such $b$, which works for all these functions .
2024 Moldova Team Selection Test, 9
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$, such that $$f(xy+f(x^2))=xf(x+y)$$ for all reals $x, y$.
2017 Morocco TST-, 6
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
2018 Swedish Mathematical Competition, 2
Find all functions $f: R \to R$ that satisfy $f (x) + 2f (\sqrt[3]{1-x^3}) = x^3$ for all real $x$.
(Here $\sqrt[3]{x}$ is defined all over $R$.)
2023 Sinapore MO Open, P4
Find all functions $f: \mathbb{Z} \to \mathbb{Z}$, such that $$f(x+y)((f(x) - f(y))^2+f(xy))=f(x^3)+f(y^3)$$ for all integers $x, y$.
1985 IMO Longlists, 33
A sequence of polynomials $P_m(x, y, z), m = 0, 1, 2, \cdots$, in $x, y$, and $z$ is defined by $P_0(x, y, z) = 1$ and by
\[P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)\]
for $m > 0$. Prove that each $P_m(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z.$
2015 Postal Coaching, Problem 2
Find all functions $f: \mathbb{Q} \to \mathbb{R}$ such that $f(xy)=f(x)f(y)+f(x+y)-1$ for all rationals $x,y$
2005 India IMO Training Camp, 2
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]
2019 Pan-African Shortlist, A3
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$
f\left(x^2\right) - yf(y) = f(x + y) (f(x) - y)
$$
for all real numbers $x$ and $y$.
2017 Thailand TSTST, 1
Find all functions $f : Z \to Z$ satisfying $f(m + n) + f(mn -1) = f(m)f(n) + 2$ for all $m, n \in Z$.
1969 IMO Longlists, 8
Find all functions $f$ defined for all $x$ that satisfy the condition $xf(y) + yf(x) = (x + y)f(x)f(y),$ for all $x$ and $y.$ Prove that exactly two of them are continuous.
1997 Singapore Team Selection Test, 3
Let $f : R \to R$ be a function from the set $R$ of real numbers to itself. Find all such functions $f$ satisfying the two properties:
(a) $f(x + f(y)) = y + f(x)$ for all $x, y \in R$,
(b) the set $\{ \frac{f(x)}{x} :x$ is a nonzero real number $\}$ is finite
2021 Final Mathematical Cup, 1
Let $N$ is the set of all positive integers. Determine all mappings $f: N-\{1\} \to N$ such that for every $n \ne m$ the following equation is true $$f(n)f(m)=f\left((nm)^{2021}\right)$$
2013 Macedonian Team Selection Test, Problem 3
Denote by $\mathbb{Z}^{*}$ the set of all nonzero integers and denote by $\mathbb{N}_{0}$ the set of all nonnegative integers. Find all functions $f:\mathbb{Z}^{*} \rightarrow \mathbb{N}_{0}$ such that:
$(1)$ For all $a,b \in \mathbb{Z}^{*}$ such that $a+b \in \mathbb{Z}^{*}$ we have $f(a+b) \geq $ [b]min[/b] $\left \{ f(a),f(b) \right \}$.
$(2)$ For all $a, b \in \mathbb{Z}^{*}$ we have $f(ab) = f(a)+f(b)$.
2018 Baltic Way, 4
Find all functions $f:[0, \infty) \to [0,\infty)$, such that for any positive integer $n$ and and for any non-negative real numbers $x_1,x_2,\dotsc,x_n$
\[f(x_1^2+\dotsc+x_n^2)=f(x_1)^2+\dots+f(x_n)^2.\]
2025 Spain Mathematical Olympiad, 6
Let $\mathbb{R}_{\neq 0}$ be the set of nonzero real numbers. Find all functions $f:\mathbb{R}_{\neq 0}\rightarrow\mathbb{R}_{\neq 0}$ such that, for all $x,y\in\mathbb{R}_{\neq 0}$, \[(x-y)f(y^2)+f\left(xy\,f\left(\frac{x^2}{y}\right)\right)=f(y^2f(y)).\]