Olympiad problems

2006 Petru Moroșan-Trident, 2

Tags: function , calculus , derivative , real analysis , find all functions

Find the twice-differentiable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that have the property that $$ f'(x)+F(x)=2f(x)+x^2/2, $$ for any real numbers $ x; $ where $ F $ is a primitive of $ f. $ [i]Carmen Botea[/i]

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]

2001 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: functional equation , algebra , find all functions

Find all functions $f: R \to R$ such that, for any $x, y \in R$: $f\left( f\left( x \right)-y \right)\cdot f\left( x+f\left( y \right) \right)={{x}^{2}}-{{y}^{2}}$

2011 Laurențiu Duican, 3

Tags: real analysis , find all functions , function

Find the $ \mathcal{C}^1 $ class functions $ f:[0,2]\longrightarrow\mathbb{R} $ having the property that the application $ x\mapsto e^{-x} f(x) $ is nonincreasing on $ [0,1] , $ nondecreasing on $ [1,2] , $ and satisfying $$ \int_0^2 xf(x)dx=f(0)+f(2) . $$ [i]Cristinel Mortici[/i]

2007 Nicolae Coculescu, 4

Tags: algebra , function , find all functions

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