Olympiad problems

2012 Bogdan Stan, 2

Find the continuous functions $ f:\left[ 0,\frac{1}{3} \right] \longrightarrow (0,\infty ) $ that satisfy the functional relation $$ 54\int_0^{1/3} f(x)dx +32\int_0^{1/3} \frac{dx}{\sqrt{x+f(x)}} =21. $$ [i]Cristinel Mortici[/i]

2019 Dutch IMO TST, 3

Tags: functional inequality , find all functions , algebra

Find all functions $f : Z \to Z$ satisfying the following two conditions: (i) for all integers $x$ we have $f(f(x)) = x$, (ii) for all integers $x$ and y such that $x + y$ is odd, we have $f(x) + f(y) \ge x + y$.

2007 Nicolae Coculescu, 4

Tags: function , algebra , find all functions

Prove that there exists a nonconstant function $ f:\mathbb{R}^2\longrightarrow\mathbb{R} $ verifying the following system of relations: $$ \left\{ \begin{matrix} f(x,x+y)=f(x,y) ,& \quad \forall x,y\in\mathbb{R} \\f(x,y+z)=f(x,y) +f(x,z) ,& \quad \forall x,y\in\mathbb{R} \end{matrix} \right. $$

2011 N.N. Mihăileanu Individual, 2

Tags: find all functions , polynom

Let $ 0 $ be a root for a polynom $ f\in\mathbb{R}[X] $ that has the property that $ f(X^2-X+1) =f^2(X)-f(X)+1. $ Determine this polynom. [i]Nelu Chichirim[/i]

2012 Centers of Excellency of Suceava, 2

Tags: function , find all functions , algebra

Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify, for any nonzero real number $ x $ the relation $$ xf(x/a)-f(a/x)=b, $$ where $ a\neq 0,b $ are two real numbers. [i]Dan Popescu[/i]