Olympiad problems

2001 Kazakhstan National Olympiad, 4

Find all functions $ f: \mathbb {R} \rightarrow \mathbb {R} $ satisfying the equality $ f (x ^ 2-y ^ 2) = (x-y) (f (x) + f (y)) $ for any $ x, y \in \mathbb {R} $.

1997 Estonia National Olympiad, 2

Tags: function , algebra , functional , functional equation

A function $f$ satisfies the following condition for each $n\in N$: $f (1)+ f (2)+...+ f (n) = n^2 f (n)$. Find $f (1997)$ if $f (1) = 999$.

2005 Thailand Mathematical Olympiad, 14

Tags: algebra , functional equation , functional

A function $f : Z \to Z$ is given so that $f(m + n) = f(m) + f(n) + 2mn - 2548$ for all positive integers $m, n$. Given that $f(2548) = -2548$, find the value of $f(2)$.

2020 Costa Rica - Final Round, 4

Tags: algebra , functional equation , function , functional

Consider the function $ h$, defined for all positive real numbers, such that: $$10x -6h(x) = 4h \left(\frac{2020}{x}\right) $$ for all $x > 0$. Find $h(x)$ and the value of $h(4)$.

2022 Saudi Arabia IMO TST, 3

Tags: algebra , functional equation , functional

Find all non-constant functions $f : Q^+ \to Q^+$ satisfying the equation $$f(ab + bc + ca) =f(a)f(b) +f(b)f(c)+f(c)f(a)$$ for all $a, b,c \in Q^+$ .

1991 Greece National Olympiad, 1

Tags: function , algebra , functional

Prove that there is no function $f: \mathbb{Z}\to\mathbb{Z}$ such that $f(f(x))=x+1$, for all $x\in\mathbb{Z}$.

2013 Grand Duchy of Lithuania, 1

Tags: functional , increasing , integer

Let $f : R \to R$ and $g : R \to R$ be strictly increasing linear functions such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that $f(x) - g(x)$ is an integer for any $x \in R$.

2011 Saudi Arabia IMO TST, 3

Tags: functional , functional equation , algebra

Find all functions $f : R \to R$ such that $$2f(x) =f(x+y)+f(x+2y)$$, for all $x \in R$ and for all $y \ge 0$.

2010 Thailand Mathematical Olympiad, 6

Tags: functional equation , functional , algebra

Let $f : R \to R$ be a function satisfying the functional equation $f(3x + y) + f(3x-y) = f(x + y) + f(x - y) + 16f(x)$ for all reals $x, y$. Show that $f$ is even, that is, $f(-x) = f(x)$ for all reals $x$

2011 Mathcenter Contest + Longlist, 7 sl9

Tags: functional equation , algebra , functional

Find the function $\displaystyle{f : \mathbb{R}-\left\{ 0\,\right\} \rightarrow \mathbb{R} }$ such that $$f(x)+f(1-\frac{1}{x}) = \frac{1}{x},\,\,\, \forall x \in \mathbb{R}- \{ 0, 1\,\}$$ [i](-InnoXenT-)[/i]

2011 QEDMO 10th, 9

Tags: algebra , functional equation , functional

Let $X = Q-\{-1,0,1\}$. We consider the function $f: X\to X$ given by $f (x) = x -\frac{1}{x} .$ Is there an $a \in X$ such that for every natural number n there is a $y \in X$ with $f (f (...( f (y)) ...)) = a$ where $f$ occurs exactly $n$ times on the left side?

2016 Thailand Mathematical Olympiad, 9

Tags: functional equation , functional , algebra

A real number $a \ne 0$ is given. Determine all functions $f : R \to R$ satisfying $f(x)f(y) + f(x + y) = axy$ for all real numbers $x, y$.

2010 Belarus Team Selection Test, 2.4

Tags: functional equation , algebra , functional equation in q , functional

Find all functions $f, g : Q \to Q$ satisfying the following equality $f(x + g(y)) = g(x) + 2 y + f(y)$ for all $x, y \in Q$. (I. Voronovich)