Olympiad problems

2023 Taiwan TST Round 2, A

Find all functions $f : \mathbb{R} \to \mathbb{R}$, such that $$f\left(xy+f(y)\right)f(x)=x^2f(y)+f(xy)$$ for all $x,y \in \mathbb{R}$ [i]Proposed by chengbilly[/i]

2025 Greece National Olympiad, 3

Tags: algebra , function , function equation

Let $f(x):\mathbb {Q} \rightarrow \mathbb {Q}$ be a function satisfying $f(x+2y)+f(2x-y)=5f(x)+5f(y)$ Find all such functions.

KoMaL A Problems 2017/2018, A. 706

Tags: function equation , algebra , function

Find all positive integer $k$s for which such $f$ exists and unique: $f(mn)=f(n)f(m)$ for $n, m \in \mathbb{Z^+}$ $f^{n^k}(n)=n$ for all $n \in \mathbb{Z^+}$ for which $f^x (n)$ means the n times operation of function $f$(i.e. $f(f(...f(n))...)$)

2023 Iran Team Selection Test, 3

Tags: algebra , function equation , functional equation

Find all function $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that for every three real positive number $x,y,z$ : $$ x+f(y) , f(f(y)) + z , f(f(z))+f(x) $$ are length of three sides of a triangle and for every postive number $p$ , there is a triangle with these sides and perimeter $p$. [i]Proposed by Amirhossein Zolfaghari [/i]

1996 Czech and Slovak Match, 4

Tags: integer equation , function equation , algebra , number theory

Decide whether there exists a function $f : Z \rightarrow Z$ such that for each $k =0,1, ...,1996$ and for any integer $m$ the equation $f (x)+kx = m$ has at least one integral solution $x$.

2013 Middle European Mathematical Olympiad, 1

Tags: function , function equation , algebra

Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that \[ f( xf(x) + 2y) = f(x^2)+f(y)+x+y-1 \] holds for all $ x, y \in \mathbb{R}$.

2010 Indonesia TST, 3

Tags: function , algebra , function equation

Determine all real numbers $ a$ such that there is a function $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ x\plus{}f(y)\equal{}af(y\plus{}f(x))\] for all real numbers $ x$ and $ y$. [i]Hery Susanto, Malang[/i]

Taiwan TST 2015 Round 1, 2

Tags: function , algebra , function equation

Find all functions $f:\mathbb{Q}\rightarrow\mathbb{R} \setminus \{ 0 \}$ such that \[(f(x))^2f(2y)+(f(y))^2f(2x)=2f(x)f(y)f(x+y)\] for all $x,y\in\mathbb{Q}$

2010 Indonesia TST, 3

Tags: function , algebra , function equation

Determine all real numbers $ a$ such that there is a function $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ x\plus{}f(y)\equal{}af(y\plus{}f(x))\] for all real numbers $ x$ and $ y$. [i]Hery Susanto, Malang[/i]