Found problems: 1513
2021 Nordic, 2
Find all functions $f:R->R$ satisfying that for every $x$ (real number):
$f(x)(1+|f(x)|)\geq x \geq f(x(1+|x|))$
2019 BAMO, 4
Let $S$ be a finite set of nonzero real numbers, and let $f : S\to S$ be a function with the following property:
for each $x \in S$, either $f ( f (x)) = x+ f (x)$ or $f ( f (x)) = \frac{x+ f (x)}{2}$.
Prove that $f (x) = x$ for all $x \in S$.
2001 IMO Shortlist, 1
Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying
\[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\
1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\
+ f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\
+ f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases}
\]
for all nonnegative integers $ p$, $ q$, $ r$.
2011 Peru IMO TST, 1
Let $\Bbb{Z}^+$ denote the set of positive integers. Find all functions $f:\Bbb{Z}^+\to \Bbb{Z}^+$ that satisfy the following condition: for each positive integer $n,$ there exists a positive integer $k$ such that $$\sum_{i=1}^k f_i(n)=kn,$$ where $f_1(n)=f(n)$ and $f_{i+1}(n)=f(f_i(n)),$ for $i\geq 1. $
2016 Dutch IMO TST, 4
Find all funtions $f:\mathbb R\to\mathbb R$ such that: $$f(xy-1)+f(x)f(y)=2xy-1$$ for all $x,y\in \mathbb{R}$.
2022 Balkan MO Shortlist, A4
Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(0)\neq 0$ and
\[f(f(x)) + f(f(y)) = f(x + y)f(xy),\]
for all $x, y \in\mathbb{R}$.
2011 Greece Team Selection Test, 3
Find all functions $f,g: \mathbb{Q}\to \mathbb{Q}$ such that the following two conditions hold:
$$f(g(x)-g(y))=f(g(x))-y \ \ (1)$$
$$g(f(x)-f(y))=g(f(x))-y\ \ (2)$$
for all $x,y \in \mathbb{Q}$.
1993 Brazil National Olympiad, 5
Find at least one function $f: \mathbb R \rightarrow \mathbb R$ such that $f(0)=0$ and $f(2x+1) = 3f(x) + 5$ for any real $x$.
2017 European Mathematical Cup, 1
Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that the inequality $$f(x)+yf(f(x))\le x(1+f(y))$$
holds for all positive integers $x, y$.
Proposed by Adrian Beker.
2020 Iran Team Selection Test, 4
Given a function $g:[0,1] \to \mathbb{R}$ satisfying the property that for every non empty dissection of the trivial $[0,1]$ to subsets $A,B$ we have either $\exists x \in A; g(x) \in B$ or $\exists x \in B; g(x) \in A$ and we have furthermore $g(x)>x$ for $x \in [0,1]$. Prove that there exist infinite $x \in [0,1]$ with $g(x)=1$.
[i]Proposed by Ali Zamani [/i]
2007 Stars of Mathematics, 1
Prove that there exists just one function $ f:\mathbb{N}^2\longrightarrow\mathbb{N} $ which simultaneously satisfies:
$ \text{(1)}\quad f(m,n)=f(n,m),\quad\forall m,n\in\mathbb{N} $
$ \text{(2)}\quad f(n,n)=n,\quad\forall n\in\mathbb{N} $
$ \text{(3)}\quad n>m\implies (n-m)f(m,n)=nf(m,n-m), \quad\forall m,n\in\mathbb{N} $
2015 Thailand Mathematical Olympiad, 9
Determine all functions $f : R \to R$ satisfying $f(f(x) + 2y)= 6x + f(f(y) -x)$ for all real numbers $x,y$
2019 IMO, 1
Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$
[i]Proposed by Liam Baker, South Africa[/i]
2021 Romania National Olympiad, 4
Let $A$ be a finite set of non-negative integers. Determine all functions $f:\mathbb{Z}_{\ge 0} \to A$ such that \[f(|x-y|)=|f(x)-f(y)|\] for each $x,y\in\mathbb Z_{\ge 0}$.
[i]Andrei Bâra[/i]
2015 USA Team Selection Test, 1
Let $f : \mathbb Q \to \mathbb Q$ be a function such that for any $x,y \in \mathbb Q$, the number $f(x+y)-f(x)-f(y)$ is an integer. Decide whether it follows that there exists a constant $c$ such that $f(x) - cx$ is an integer for every rational number $x$.
[i]Proposed by Victor Wang[/i]
2015 VTRMC, Problem 6
Let $(a_1,b_1),\ldots,(a_n,b_n)$ be $n$ points in $\mathbb R^2$ (where $\mathbb R$ denotes the real numbers), and let $\epsilon>0$ be a positive number. Can we find a real-valued function $f(x,y)$ that satisfies the following three conditions?
1. $f(0,0)=1$;
2. $f(x,y)\ne0$ for only finitely many $(x,y)\in\mathbb R^2$;
3. $\sum_{r=1}^n\left|f(x+a_r,y+b_r)-f(x,y)\right|<\epsilon$ for every $(x,y)\in\mathbb R^2$.
Justify your answer.
2017 Romania EGMO TST, P3
Determine all functions $f:\mathbb R\to\mathbb R$ such that \[f(xy-1)+f(x)f(y)=2xy-1,\]for any real numbers $x{}$ and $y{}.$
2015 Estonia Team Selection Test, 5
Find all functions $f$ from reals to reals which satisfy $f (f(x) + f(y)) = f(x^2) + 2x^2 f(y) + (f(y))^2$ for all real numbers $x$ and $y$.
2015 Saudi Arabia GMO TST, 1
Find all functions $f : R \to R$ satisfying the following conditions
(a) $f(1) = 1$,
(b) $f(x + y) = f(x) + f(y)$, $\forall (x,y) \in R^2$
(c) $f\left(\frac{1}{x}\right) =\frac{ f(x)}{x^2 }$, $\forall x \in R -\{0\}$
Trần Nam Dũng
2021 Olimphíada, 6
Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f : \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for all $m, n \in \mathbb{Z}_{>0 }$:
$$f(mf(n)) + f(n) | mn + f(f(n)).$$
2015 Latvia Baltic Way TST, 2
It is known about the function $f : R \to R$ that
$\bullet$ $f(x) > f(y)$ for all real $x > y$
$\bullet$ $f(x) > x$ for all real $x$
$\bullet$ $f(2x - f (x)) = x$ for all real $x$.
Prove that $f(x) = x + f(0)$ for all real numbers $x$.
2022 Iran MO (3rd Round), 2
Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that for all $x,y\in\mathbb{N}$:
$$0\le y+f(x)-f^{f(y)}(x)\le1$$
that here
$$f^n(x)=\underbrace{f(f(\ldots(f}_{n}(x))\ldots)$$
2018 Saudi Arabia BMO TST, 4
Find all functions $f : Z \to Z$ such that $x f (2f (y) - x) + y^2 f (2x - f (y)) = \frac{(f (x))^2}{x} + f (y f (y))$ ,
for all $x, y \in Z$, $x \ne 0$.
1987 IMO, 1
Prove that there is no function $f$ from the set of non-negative integers into itself such that $f(f(n))=n+1987$ for all $n$.
2023 USAMO, 2
Let $\mathbb{R}^+$ be the set of positive real numbers. Find all functions $f \colon \mathbb{R}^+ \to \mathbb{R}^+$ such that, for all $x,y \in \mathbb{R}^+$,
$$f(xy+f(x))=xf(y)+2.$$