Olympiad problems

2007 Nicolae Coculescu, 1

Find all functions $ f:\mathbb{Q}\longrightarrow\mathbb{R} $ satisfying the equation $$ f(x+y)+f(x-y)=f(x)+f(y) +f(f(x+y)) , $$ for any rational numbers $ x,y. $ [i]Mihai Onucu Drîmbe[/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 $f(x) + f(y) \ge x + y$.

2012 Bogdan Stan, 3

Tags: function , find all functions , real analyssi

Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the equality $$ \int_a^b f(x)dx=f(b)-f(a), $$ for any real numbers $ a,b. $ [i]Cosmin Nitu[/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]

2009 Romania National Olympiad, 1

Tags: function , definite integration , calculus , real analysis , find all functions , integration , inequalities

Find all functions $ f\in\mathcal{C}^1 [0,1] $ that satisfy $ f(1)=-1/6 $ and $$ \int_0^1 \left( f'(x) \right)^2 dx\le 2\int_0^1 f(x)dx. $$