Olympiad problems

2012 Mathcenter Contest + Longlist, 5 sl13

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]

2012 QEDMO 11th, 11

Tags: functional equation , functional , algebra

Find all functions $f: R\to R$, such that $f (xf (y) + f (x)) = xy$ for all $x, y \in R $.

2020 Czech-Austrian-Polish-Slovak Match, 4

Tags: algebra , functional equation , functional

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)

1997 Rioplatense Mathematical Olympiad, Level 3, 6

Tags: functional , functional equation , number theory , algebra

Let $N$ be the set of positive integers. Determine if there is a function $f: N\to N$ such that $f(f(n))=2n$, for all $n$ belongs to $N$.

1992 Romania Team Selection Test, 1

Tags: functional , function , algebra

Suppose that$ f : N \to N$ is an increasing function such that $f(f(n)) = 3n$ for all $n$. Find $f(1992)$.

2016 Costa Rica - Final Round, F3

Tags: digit , functional equation , functional , algebra

Let $f: Z^+ \to Z^+ \cup \{0\}$ a function that meets the following conditions: a) $f (a b) = f (a) + f (b)$, b) $f (a) = 0$ provided that the digits of the unit of $a$ are $7$, c) $f (10) = 0$. Find $f (2016).$

2015 Saudi Arabia IMO TST, 1

Tags: algebra , functional , functional equation

Find all functions $f : R_{>0} \to R$ such that $f \left(\frac{x}{y}\right) = f(x) + f(y) - f(x)f(y)$ for all $x, y \in R_{>0}$. Here, $R_{>0}$ denotes the set of all positive real numbers. Nguyễn Duy Thái Sơn

2020 Estonia Team Selection Test, 3

Tags: algebra , functional , functional equation

Find all functions $f :R \to R$ such that for all real numbers $x$ and $y$ $$f(x^3+y^3)=f(x^3)+3x^3f(x)f(y)+3f(x)(f(y))^2+y^6f(y)$$

2005 iTest, 4

Tags: function , functional equation , functional , algebra

The function f is defined on the set of integers and satisfies $\bullet$ $f(n) = n - 2$, if $n \ge 2005$ $\bullet$ $f(n) = f(f(n+7))$, if $n < 2005$. Find $f(3)$.

2011 QEDMO 9th, 7

Tags: functional equation , functional , algebra

Find all functions $f: R\to R$, such that $f(xy + x + y) + f(xy-x-y)=2f (x) + 2f (y)$ for all $x, y \in R$.

2015 Saudi Arabia GMO TST, 1

Tags: algebra , functional , functional equation

Find all functions $f : R \to R$ satisfying the following conditions (a) $f(1) = 1$, (b) $f(x + y) = f(x) + f(y)$, $\forall (x,y) \in R^2$ (c) $f\left(\frac{1}{x}\right) =\frac{ f(x)}{x^2 }$, $\forall x \in R -\{0\}$ Trần Nam Dũng

1998 Belarusian National Olympiad, 8

Tags: algebra , functional equation , functional

a) Prove that for no real a such that $0 \le a <1$ there exists a function defined on the set of all positive numbers and taking values in the same set, satisfying for all positive $x$ the equality $$f\left(f(x)+\frac{1}{f(x)}\right)=x+a \,\,\,\,\,\,\, (*) $$ b) Prove that for any $a>1$ there are infinitely many functions defined on the set of all positive numbers, with values in the same set, satisfying ($*$) for all positive x.

2019 Greece Team Selection Test, 4

Tags: functional equation , functional , algebra

Find all functions $f:(0,\infty)\mapsto\mathbb{R}$ such that $\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}$ for every $x,y>0$.

2013 Saudi Arabia BMO TST, 2

Tags: algebra , functional equation , functional

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

2021 Federal Competition For Advanced Students, P2, 4

Tags: algebra , functional equation , functional

Let $a$ be a real number. Determine all functions $f: R \to R$ with $f (f (x) + y) = f (x^2 - y) + af (x) y$ for all $x, y \in R$. (Walther Janous)

2022 Peru MO (ONEM), 3

Tags: algebra , functional equation , functional

Let $R$ be the set of real numbers and $f : R \to R$ be a function that satisfies: $$f(xy) + y + f(x + f(y)) = (y + 1)f(x),$$ for all real numbers $x, y$. a) Determine the value of $f(0)$. b) Prove that $f(x) = 2-x$ for every real number $x$.

1997 Estonia National Olympiad, 2

Tags: functional , functional equation , function , algebra

A function $f$ satisfies the following condition for each $n\in N$: $f (1)+ f (2)+...+ f (n) = n^2 f (n)$. Find $f (1997)$ if $f (1) = 999$.

2010 Saudi Arabia IMO TST, 2

Tags: functional equation , functional , algebra

Find all functions $f,g : N \to N$ such that for all $m ,n \in N$ the following relation holds: $$f(m ) - f(n) = (m - n)(g(m) + g(n))$$. Note: $N = \{0,1,2,...\}$

2020 Dutch BxMO TST, 3

Tags: functional equation , functional , algebra

Find all functions $f: R \to R$ that satisfy $$f (x^2y) + 2f (y^2) =(x^2 + f (y)) \cdot f (y)$$ for all $x, y \in R$

2020 Durer Math Competition Finals, 5

Tags: algebra , functional equation , functional

Let $H = \{-2019,-2018, ...,-1, 0, 1, 2, ..., 2020\}$. Describe all functions $f : H \to H$ for which a) $x = f(x) - f(f(x))$ holds for every $x \in H$. b) $x = f(x) + f(f(x)) - f(f(f(x)))$ holds for every $x \in H$. c) $x = f(x) + 2f(f(x)) - 3f(f(f(x)))$ holds for every $x \in H$. PS. (a) + (b) for category E 1.5, (b) + (c) for category E+ 1.2

2015 Cuba MO, 1

Tags: algebra , functional , functional equation

Let $f$ be a function of the positive reals in the positive reals, such that $$f(x) \cdot f(y) - f(xy) = \frac{x}{y} + \frac{y}{x} \ \ for \ \ all \ \ x, y > 0 .$$ (a) Find $f(1)$. (b) Find $f(x)$.

2009 QEDMO 6th, 12

Tags: functional equation , functional , algebra

Find all functions $f: R\to R$, which satisfy the equation $f (xy + f (x)) = xf (y) + f (x)$.

1997 All-Russian Olympiad Regional Round, 11.8

Tags: algebra , functional , functional equation

For which $a$, there is a function $f: R \to R$, different from a constant, such that $$f(a(x + y)) = f(x) + f(y) ?$$

2003 Cuba MO, 4

Tags: number theory , functional

Let $f : N \to N$ such that $f(p) = 1$ for all p prime and $f(ab) =bf(a) + af(b)$ for all $a, b \in N$. Prove that if $n = p^{a_1}_1 p^{a_1}_2... p^{a_1}_k$ is the canonical distribution of $n$ and $p_i$ does not divide $a_i$ ($i = 1, 2, ..., k$) then $\frac{n}{gcd(n,f(n))}$ is square free (not divisible by a square greater than $1$).

2020 Dutch BxMO TST, 3

Tags: algebra , functional equation , functional

Find all functions $f: R \to R$ that satisfy $$f (x^2y) + 2f (y^2) =(x^2 + f (y)) \cdot f (y)$$ for all $x, y \in R$