Olympiad problems

2018 Saudi Arabia BMO TST, 2

Find all functions $f : R \to R$ such that $f( 2x^3 + f (y)) = y + 2x^2 f (x)$ for all real numbers $x, y$.

2012 Mathcenter Contest + Longlist, 5 sl13

Tags: algebra , functional , inequalities

Define $f : \mathbb{R}^+ \rightarrow \mathbb{R}$ as the strictly increasing function such that $$f(\sqrt{xy})=\frac{f(x)+f(y)}{2}$$ for all positive real numbers $x,y$. Prove that there are some positive real numbers $a$ where $f(a)<0$. [i] (PP-nine) [/i]

1976 Chisinau City MO, 129

Tags: algebra , function , functional , periodic

The function $f (x)$ satisfies the relation $f(x+\pi)=\frac{f(x)}{3f(x) -1}$ for any real number $x$. Prove that the function $f (x)$ is periodic.

2016 Latvia Baltic Way TST, 4

Tags: functional equation , functional , algebra

Find all functions $f : R \to R$ defined for real numbers, take real values and for all real $x$ and $y$ the equality holds: $$f(2^x+2y) =2^yf(f(x))f(y).$$

2015 Latvia Baltic Way TST, 2

Tags: functional , algebra , functional equation

It is known about the function $f : R \to R$ that $\bullet$ $f(x) > f(y)$ for all real $x > y$ $\bullet$ $f(x) > x$ for all real $x$ $\bullet$ $f(2x - f (x)) = x$ for all real $x$. Prove that $f(x) = x + f(0)$ for all real numbers $x$.

1989 All Soviet Union Mathematical Olympiad, 509

Tags: functional , functional equation

$N$ is the set of positive integers. Does there exist a function $f: N \to N$ such that $f(n+1) = f( f(n) ) + f( f(n+2) )$ for all $n$?

1999 Austrian-Polish Competition, 3

Tags: algebra , functional , functional equation , system of equations

Given an integer $n \ge 2$, find all sustems of $n$ functions$ f_1,..., f_n : R \to R$ such that for all $x,y \in R$ $$\begin{cases} f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0 \\ f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0 \\ ... \\ f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0 \end {cases}$$

2021-IMOC qualification, A2

Tags: polynomial , algebra , functional equation , functional

Find all functions $f:R \to R$, such that $f(x)+f(y)=f(x+y)$, and there exists non-constant polynomials $P(x)$, $Q(x)$ such that $P(x)f(Q(x))=f(P(x)Q(x))$

2019 Saudi Arabia Pre-TST + Training Tests, 1.3

Tags: algebra , functional

Find all functions $f : R^+ \to R^+$ such that $f(3 (f (xy))^2 + (xy)^2) = (xf (y) + yf (x))^2$ for any $x, y > 0$.

2003 Austrian-Polish Competition, 1

Tags: algebra , polynomial , functional , functional equation

Find all real polynomials $p(x) $ such that $p(x-1)p(x+1)= p(x^2-1)$.

2020 Czech-Austrian-Polish-Slovak Match, 4

Tags: functional , functional equation , algebra

Let $a$ be a given real number. Find all functions $f : R \to R$ such that $(x+y)(f(x)-f(y))=a(x-y)f(x+y)$ holds for all $x,y \in R$. (Walther Janous, Austria)

2015 Costa Rica - Final Round, 5

Tags: inequalities , algebra , functional inequality , functional

Let $f: N^+ \to N^+$ be a function that satisfies that $$kf(n) \le f (kn) \le kf(n)+ k- 1, \,\, \forall k,n \in N^+$$ Prove that $$f(a) + f(b) \le f (a + b) \le f(a) + f(b) + 1, \,\, \forall a, b \in N^+$$

1997 Singapore Team Selection Test, 3

Tags: functional equation , algebra , functional

Let $f : R \to R$ be a function from the set $R$ of real numbers to itself. Find all such functions $f$ satisfying the two properties: (a) $f(x + f(y)) = y + f(x)$ for all $x, y \in R$, (b) the set $\{ \frac{f(x)}{x} :x$ is a nonzero real number $\}$ is finite

2009 Belarus Team Selection Test, 1

Tags: algebra , functional , functional equation

Find all functions $f: R \to R$ and $g:R \to R$ such that $f(x-f(y))=xf(y)-yf(x)+g(x)$ for all real numbers $x,y$. I.Voronovich

2006 Switzerland - Final Round, 1

Tags: algebra , functional , functional equation

Find all functions $f : R \to R$ such that for all $x, y \in R$ holds $$yf(2x) - xf(2y) = 8xy(x^2 - y^2).$$

2013 Saudi Arabia BMO TST, 2

Tags: functional , functional equation , algebra

Find all functions $f : R \to R$ which satisfy for all $x, y \in R$ the relation $f(f(f(x) + y) + y) = x + y + f(y)$

1994 Swedish Mathematical Competition, 6

Tags: functional , number theory , functional equation

Let $N$ be the set of non-negative integers. The function $f:N\to N$ satisfies $f(a+b) = f(f(a)+b)$ for all $a, b$ and $f(a+b) = f(a)+f(b)$ for $a+b < 10$. Also $f(10) = 1$. How many three digit numbers $n$ satisfy $f(n) = f(N)$, where $N$ is the "tower" $2, 3, 4, 5$, in other words, it is $2^a$, where $a = 3^b$, where $b = 4^5$?

2018 Saudi Arabia BMO TST, 2

2012 Mathcenter Contest + Longlist, 5 sl13

1976 Chisinau City MO, 129

2016 Latvia Baltic Way TST, 4

2015 Latvia Baltic Way TST, 2

1989 All Soviet Union Mathematical Olympiad, 509

1999 Austrian-Polish Competition, 3

2021-IMOC qualification, A2

2019 Saudi Arabia Pre-TST + Training Tests, 1.3

2003 Austrian-Polish Competition, 1

2020 Czech-Austrian-Polish-Slovak Match, 4

2015 Costa Rica - Final Round, 5

1997 Singapore Team Selection Test, 3

2009 Belarus Team Selection Test, 1

2006 Switzerland - Final Round, 1

2013 Saudi Arabia BMO TST, 2

1994 Swedish Mathematical Competition, 6

2013 Saudi Arabia Pre-TST, 3.1

2021 Dutch IMO TST, 3

2013 Costa Rica - Final Round, F1

2008 Switzerland - Final Round, 2

1998 Belarus Team Selection Test, 1

2010 Indonesia TST, 1

1996 Austrian-Polish Competition, 8

2015 Saudi Arabia Pre-TST, 2.2