Olympiad problems

1984 Austrian-Polish Competition, 9

Find all functions $f: Q \to R$ satisfying $f (x + y) = f (x)f (y) - f(xy) + 1$ for all $x,y \in Q$

OIFMAT I 2010, 1

Tags: functional , number theory

Let $ f (n) $ be a function that fulfills the following properties: $\bullet$ For each natural $ n $, $ f (n) $ is an integer greater than or equal to $ 0 $. $\bullet$ $f (n) = 2010 $, if $ n $ ends in $ 7 $. For example, $ f (137) = 2010 $. $\bullet$ If $ a $ is a divisor of $ b $, then: $ f \left(\frac {b} {a} \right) = | f (b) -f (a) | $. Find $ \displaystyle f (2009 ^ {2009 ^ {2009}}) $ and justify your answer.

1993 Bulgaria National Olympiad, 1

Tags: functional , functional equation , algebra

Find all functions $f$ , defined and having values in the set of integer numbers, for which the following conditions are satisfied: (a) $f(1) = 1$; (b) for every two whole (integer) numbers $m$ and $n$, the following equality is satisfied: $$f(m+n)·(f(m)-f(n)) = f(m-n)·(f(m)+ f(n))$$

VI Soros Olympiad 1999 - 2000 (Russia), 10.3

Tags: functional equation , functional , algebra

Find all functions $f$ that map the set of real numbers into the set of real numbers, satisfying the following conditions: 1) $|f(x)|\ge 1$, 2) $f(x+y)=\frac{f(x)+f(y)}{1+f(x)f(y)}$ of all real values of $x $ and $y$.

1998 Italy TST, 1

Tags: functional equation , functional , function , algebra

A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying $\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.

2020 Greece National Olympiad, 1

Tags: algebra , polynomial , functional equation , functional , polynomial equation

Find all non constant polynomials $P(x),Q(x)$ with real coefficients such that: $P((Q(x))^3)=xP(x)(Q(x))^3$

2016 Estonia Team Selection Test, 3

Tags: algebra , functional equation , functional

Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y \in R$.

2016 Saudi Arabia GMO TST, 2

Tags: algebra , functional , functional equation

Find all functions $f : Z \to Z$ such that $f (2m + f (m) + f (m)f (n)) = nf (m) + m$ for any integers $m, n$

2015 Saudi Arabia BMO TST, 1

Tags: algebra , functional equation , functional , function , increasing

Find all strictly increasing functions $f : Z \to R$ such that for any $m, n \in Z$ there exists a $k \in Z$ such that $f(k) = f(m) - f(n)$. Nguyễn Duy Thái Sơn

1995 Singapore Team Selection Test, 1

Tags: functional equation , functional , algebra

Let $N =\{1, 2, 3, ...\}$ be the set of all natural numbers and $f : N\to N$ be a function. Suppose $f(1) = 1$, $f(2n) = f(n)$ and $f(2n + 1) = f(2n) + 1$ for all natural numbers $n$. (i) Calculate the maximum value $M$ of $f(n)$ for $n \in N$ with $1 \le n \le 1994$. (ii) Find all $n \in N$, with 1 \le n \le 1994, such that $f(n) = M$.

1994 Swedish Mathematical Competition, 6

Tags: number theory , functional , 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 $b = 4^5$?

1994 Austrian-Polish Competition, 8

Tags: functional equation , functional , algebra

Given real numbers $a, b$, find all functions $f: R \to R$ satisfying $f(x,y) = af (x,z) + bf(y,z)$ for all $x,y,z \in R$.

2005 Thailand Mathematical Olympiad, 3

Tags: functional , functional equation , algebra

Does there exist a function $f : Z^+ \to Z^+$ such that $f(f(n)) = 2n$ for all positive integers $n$? Justify your answer, and if the answer is yes, give an explicit construction.

2016 Belarus Team Selection Test, 2

Tags: functional equation , functional , algebra

Find all real numbers $a$ such that exists function $\mathbb {R} \rightarrow \mathbb {R} $ satisfying the following conditions: 1) $f(f(x)) =xf(x)-ax$ for all real $x$ 2) $f$ is not constant 3) $f$ takes the value $a$

2021 Swedish Mathematical Competition, 4

Tags: algebra , functional , functional inequality

Give examples of a function $f : R \to R$ that satisfies $0 < f(x) < f(x + f(x)) <\sqrt2 x$, for all positive $x$, and show that there is no function $f : R \to R$ that satisfies $x < f(x + f(x)) <\sqrt2 f(x)$, for all positive $x$.

2020 Swedish Mathematical Competition, 3

Tags: bounded , algebra , functional equation , functional , isl

Determine all bounded functions $f: R \to R$, such that $f (f (x) + y) = f (x) + f (y)$, for all real $x, y$.

2007 Thailand Mathematical Olympiad, 9

Tags: algebra , functional equation , functional

Let $f : R \to R$ be a function satisfying the equation $f(x^2 + x + 3) + 2f(x^2 - 3x + 5) =6x^2 - 10x + 17$ for all real numbers $x$. What is the value of $f(85)$?

2014 Grand Duchy of Lithuania, 1

Tags: algebra , functional , functional equation

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

1994 All-Russian Olympiad Regional Round, 11.6

Tags: algebra , functional

Find all functions satisfying the equality $$(x-1)f \left(\dfrac{x+1}{x-1}\right)- f(x) = x$$ for all $x \ne 1$.

2018 Estonia Team Selection Test, 4

Tags: algebra , functional , functional equation

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

2004 Cuba MO, 8

Tags: algebra , functional , functional equation

Determine all functions $f : R_+ \to R_+$ such that: a) $f(xf(y))f(y) = f(x + y)$ for $x, y \ge 0$ b) $f(2) = 0$ c) $f(x) \ne 0$ for $0 \le x < 2$.

2011 Denmark MO - Mohr Contest, 4

Tags: algebra , functional

A function $f$ is given by $f(x) = x^2 - 2x$ . Prove that there exists a number a which satisfies $f(f(a)) = a$ without satisfying $f(a) = a$ .

2020 Grand Duchy of Lithuania, 1

Tags: algebra , functional , functional equation

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

1989 Chile National Olympiad, 6

Tags: algebra , function , functional equation , functional

The function $f$, with domain on the set of non-negative integers, is defined by the following : $\bullet$ $f (0) = 2$ $\bullet$ $(f (n + 1) -1)^2 + (f (n)-1) ^2 = 2f (n) f (n + 1) + 4$, taking $f (n)$ the largest possible value. Determine $f (n)$.

2018 Abels Math Contest (Norwegian MO) Final, 3b

Tags: function , functional equation , functional , algebra

Find all real functions $f$ defined on the real numbers except zero, satisfying $f(2019) = 1$ and $f(x)f(y)+ f\left(\frac{2019}{x}\right) f\left(\frac{2019}{y}\right) =2f(xy)$ for all $x,y \ne 0$