Olympiad problems

2019 Dutch IMO TST, 4

Find all functions $f : Z \to Z$ satisfying $\bullet$ $ f(p) > 0$ for all prime numbers $p$, $\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.

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]

2006 Cezar Ivănescu, 3

Tags: function , bijectivity , find all functions , algebra

[b]a)[/b] Prove that the function $ f:\mathbb{Z}_{\ge 0}\longrightarrow\mathbb{Z}_{\ge 0} , $ given as $ f(n)=n+(-1)^n $ is bijective. [b]b)[/b] Find all surjective functions $ g:\mathbb{Z}_{\ge 0}\longrightarrow\mathbb{Z}_{\ge 0} $ that have the property that $ g(n)\ge n+(-1)^n , $ for any nonnegative integer.

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, 1

Tags: function , find all functions , algebra

Find the functions $ f:\mathbb{Z}\longrightarrow\mathbb{Z}_{\ge 0} $ that satisfy the following two conditions: $ \text{(a)} f(m+n)=f(n)+f(m)+2mn,\quad\forall m,n\in\mathbb{Z} $ $ \text{(b)} f(f(1))-f(1) $ is a perfect square [i]Marin Ionescu[/i]

2008 Grigore Moisil Intercounty, 3

Tags: function , algebra , functional equation , find all functions

Let be two nonzero real numbers $ a,b, $ and a function $ f:\mathbb{R}\longrightarrow [0,\infty ) $ satisfying the functional equation $$ f(x+a+b)+f(x)=f(x+a)+f(x+b) . $$ [b]1)[/b] Prove that $ f $ is periodic if $ a/b $ is rational. [b]2)[/b] If $ a/b $ is not rational, could $ f $ be nonperiodic?

2012 Bogdan Stan, 2

Tags: function , find all functions , calculus , integration

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]

2016 Romania National Olympiad, 4

Tags: real analysis , find all functions , differentiability , darboux , function

Find all functions, $ f:\mathbb{R}\longrightarrow\mathbb{R} , $ that have the properties that $ f^2 $ is differentiable and $ f=\left( f^2 \right)' . $

2019 Dutch IMO TST, 4

Tags: number theory , function , find all functions , prime number , functional equation

Find all functions $f : Z \to Z$ satisfying $\bullet$ $ f(p) > 0$ for all prime numbers $p$, $\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.

2007 Gheorghe Vranceanu, 3

Tags: function , find all functions , real analysis , continuity , ftc

Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that admit a primitive $ F $ defined as $ F(x)=\left\{\begin{matrix} f(x)/x, & x\neq 0 \\ 2007, & x=0 \end{matrix}\right. . $

2010 Laurențiu Panaitopol, Tulcea, 2

Tags: function , find all functions , algebra

Find the strictly monotone functions $ f:\{ 0\}\cup\mathbb{N}\longrightarrow\{ 0\}\cup\mathbb{N} $ that satisfy the following two properties: $ \text{(i)} f(2n)=n+f(n), $ for any nonnegative integers $ n. $ $ \text{(ii)} f(n) $ is a perfect square if and only if $ n $ is a perfect square.

2019 Danube Mathematical Competition, 2

Tags: function , find all functions , algebra , functional equation

Find all nondecreasing functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the relation $$ f\left( f\left( x^2 \right) +y+f(y) \right) =x^2+2f(y) , $$ for any real numbers $ x,y. $

2015 District Olympiad, 3

Tags: function , continuity , monotone functions , find all functions , real analysis

Find all continuous and nondecreasing functions $ f:[0,\infty)\longrightarrow\mathbb{R} $ that satisfy the inequality: $$ \int_0^{x+y} f(t) dt\le \int_0^x f(t) dt +\int_0^y f(t) dt,\quad\forall x,y\in [0,\infty) . $$

2011 Laurențiu Duican, 3

Tags: function , find all functions , real analysis

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]

2006 Petru Moroșan-Trident, 2

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

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]

2008 Grigore Moisil Intercounty, 1

Tags: function , find all functions , trigonometry , algebra , monotone

Find all monotonic functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ with the property that $$ (f(\sin x))^2-3f(x)=-2, $$ for any real numbers $ x. $ [i]Dorin Andrica[/i] and [i]Mihai Piticari[/i]

2019 Teodor Topan, 3

Tags: function , functional equation , find all functions , algebra

Let be a positive real number $ r, $ a natural number $ n, $ and a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ satisfying $ f(rxy)=(f(x)f(y))^n, $ for any real numbers $ x,y. $ [b]a)[/b] Give three distinct examples of what $ f $ could be if $ n=1. $ [b]b)[/b] For a fixed $ n\ge 2, $ find all possibilities of what $ f $ could be. [i]Bogdan Blaga[/i]

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: $$ \left\{ \begin{matrix} f(x,x+y)=f(x,y) ,& \quad \forall x,y\in\mathbb{R} \\f(x,y+z)=f(x,y) +f(x,z) ,& \quad \forall x,y\in\mathbb{R} \end{matrix} \right. $$

2001 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: functional equation , find all functions , algebra

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

2017 Romania National Olympiad, 4

Tags: function , algebra , find all functions

Let be two natural numbers $ b>a>0 $ and a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following property. $$ f\left( x^2+ay\right)\ge f\left( x^2+by\right) ,\quad\forall x,y\in\mathbb{R} $$ [b]a)[/b] Show that $ f(s)\le f(0)\le f(t) , $ for any real numbers $ s<0<t. $ [b]b)[/b] Prove that $ f $ is constant on the interval $ (0,\infty ) . $ [b]c)[/b] Give an example of a non-monotone such function.

2009 Romania National Olympiad, 3

Tags: function , find all functions , algebra

Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the relation $$ f\left( x^3+y^3 \right) =xf\left( y^2 \right) + yf\left( x^2 \right) , $$ for all real numbers $ x,y. $

2008 Grigore Moisil Intercounty, 4

Tags: function , algebra , functional equation , find all functions

Given two rational numbers $ a,b, $ find the functions $ f:\mathbb{Q}\longrightarrow\mathbb{Q} $ that verify $$ f(x+a+f(y))=f(x+b)+y, $$ for any rational $ x,y. $ [i]Vasile Pop[/i]

2017 Czech-Polish-Slovak Match, 3

Tags: functional equation , positive , find all functions , algebra

Find all functions ${f : (0, +\infty) \rightarrow R}$ satisfying $f(x) - f(x+ y) = f \left( \frac{x}{y}\right) f(x + y)$ for all $x, y > 0$. (Austria)

2003 Alexandru Myller, 4

Tags: function , real analysis , find all functions , functional equation

Find the differentiable functions $ f:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ that verify $ f(0)=0 $ and $$ f'(x)=1/3\cdot f'\left( x/3 \right) +2/3\cdot f'\left( 2x/3 \right) , $$ for any nonnegative real number $ x. $

2007 Grigore Moisil Intercounty, 1

Tags: calculus , integration , find all functions

Find all functions $ f:[0,1]\longrightarrow \mathbb{R} $ that are continuous and have the property that, for any continuous function $ g:[0,1]\longrightarrow [0,1] , $ the following equality holds. $$ \int_0^1 f\left( g(x) \right) dx =\int_0^1 g(x) dx $$