Olympiad problems

2013 Saudi Arabia Pre-TST, 3.1

Let $f : R \to R$ be a function satisfying $f(f(x)) = 4x + 1$ for all real number $x$. Prove that the equation $f(x) = x$ has a unique solution.

2014 Costa Rica - Final Round, 5

Tags: algebra , functional equation , functional , recurrence relation

Let $f : N\to N$ such that $$f(1) = 0\,\, , \,\,f(3n) = 2f(n) + 2\,\, , \,\,f(3n-1) = 2f(n) + 1\,\, , \,\,f(3n-2) = 2f(n).$$ Determine the smallest value of $n$ so that $f (n) = 2014.$

2017 Costa Rica - Final Round, F1

Tags: algebra , functional equation , functional

Let $f: Z ^+ \to R$, such that $f (1) = 2018$ and $f (1) + f (2) + ...+ f (n) = n^2f (n)$, for all $n> 1$. Find the value $f (2017)$.

2002 Singapore MO Open, 4

Tags: functional equation , functional , algebra

Find all real-valued functions $f : Q \to R$ defined on the set of all rational numbers $Q$ satisfying the conditions $f(x + y) = f(x) + f(y) + 2xy$ for all $x, y$ in $Q$ and $f(1) = 2002.$ Justify your answers.

1990 Romania Team Selection Test, 3

Tags: polynomial , algebra , functional , functional equation

Find all polynomials $P(x)$ such that $2P(2x^2 -1) = P(x)^2 -1$ for all $x$.

2022 Dutch BxMO TST, 1

Tags: number theory , functional

Find all functions $f : Z_{>0} \to Z_{>0}$ for which $f(n) | f(m) - n$ if and only if $n | m$ for all natural numbers $m$ and $n$.

1996 Singapore Team Selection Test, 2

Tags: functional , functional equation , algebra

Prove that there is a function $f$ from the set of all natural numbers to itself such that for any natural number $n$, $f(f(n)) = n^2$.

2006 Switzerland - Final Round, 1

Tags: functional , functional equation , algebra

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

2000 Belarus Team Selection Test, 1.3

Tags: algebra , functional , functional equation

Does there exist a function $f : N\to N$ such that $f ( f (n-1)) = f (n+1)- f (n)$ for all $n \ge 2$?

2010 Saudi Arabia BMO TST, 4

Tags: sequence , number theory , functional , function , algebra

Let $f : N \to [0, \infty)$ be a function satisfying the following conditions: a) $f(4)=2$ b) $\frac{1}{f( 0 ) + f( 1)} + \frac{1}{f( 1 ) + f( 2 )} + ... + \frac{1}{f (n ) + f(n + 1) }= f ( n + 1)$ for all integers $n \ge 0$. Find $f(n)$ in closed form.

2008 Thailand Mathematical Olympiad, 6

Tags: inequalities , algebra , functional inequality , functional

Let $f : R \to R$ be a function satisfying the inequality $|f(x + y) -f(x) - f(y)| < 1$ for all reals $x, y$. Show that $\left| f\left( \frac{x}{2008 }\right) - \frac{f(x)}{2008} \right| < 1$ for all real numbers $x$.

2001 Estonia Team Selection Test, 3

Tags: functional , functional equation , algebra

Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.

1998 Estonia National Olympiad, 3

Tags: function , periodic , functional equation , functional , algebra

A function $f$ satisfies the conditions $f (x) \ne 0$ and $f (x+2) = f (x-1) f (x+5)$ for all real x. Show that $f (x+18) = f (x)$ for any real $x$.

2015 Latvia Baltic Way TST, 2

Tags: functional equation , functional , algebra

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 $x$.

2020 Costa Rica - Final Round, 4

Tags: algebra , functional equation , functional , function

Consider the function $ h$, defined for all positive real numbers, such that: $$10x -6h(x) = 4h \left(\frac{2020}{x}\right) $$ for all $x > 0$. Find $h(x)$ and the value of $h(4)$.

1999 Denmark MO - Mohr Contest, 3

Tags: algebra , functional , functional equation

A function $f$ satisfies $$f(x)+xf(1-x)=x$$ for all real numbers $x$. Determine the number $f (2)$. Find $f$ .

1963 Swedish Mathematical Competition., 4

Tags: functional equation , functional , algebra , analysis

Given the real number $k$, find all differentiable real-valued functions $f(x)$ defined on the reals such that $f(x+y) = f(x) + f(y) + f(kxy)$ for all $x, y$.

1965 Dutch Mathematical Olympiad, 5

Tags: algebra , functional , functional equation

The function ƒ. which is defined for all real numbers satisfies: $$f(x+y)+f(x-y)=2f(x)+2f(y)$$ Prove that $f(0) = 0$, $f(-x) = f(x)$, $f(2x) = 4 f (x)$, $$f(x + y + z) = f(x + y) + f(y + z) + f(z + x) -f(x) - f(y) -f(z).$$

2013 QEDMO 13th or 12th, 4

Tags: periodic , functional , functional equation , algebra

Let $a> 0$ and $f: R\to R$ a function such that $f (x) + f (x + 2a) + f (x + 3a) + f (x + 5a) = 1$ for all $x\in R$ . Show that $f$ is periodic, that is, that there is some $b> 0$ for which $f (x) = f (x + b)$ for every $x \in R$ holds. Find the smallest such $b$, which works for all these functions .

2018 Swedish Mathematical Competition, 2

Tags: algebra , functional equation , functional

Find all functions $f: R \to R$ that satisfy $f (x) + 2f (\sqrt[3]{1-x^3}) = x^3$ for all real $x$. (Here $\sqrt[3]{x}$ is defined all over $R$.)

1997 Singapore Team Selection Test, 3

Tags: algebra , functional equation , 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

2021 Final Mathematical Cup, 1

Tags: functional , functional equation , algebra

Let $N$ is the set of all positive integers. Determine all mappings $f: N-\{1\} \to N$ such that for every $n \ne m$ the following equation is true $$f(n)f(m)=f\left((nm)^{2021}\right)$$

1995 Grosman Memorial Mathematical Olympiad, 6

Tags: functional equation , functional , algebra

(a) Prove that there is a unique function $f : Q \to Q$ satisfying: (i) $f(q)= 1 + f\left(\frac{q}{1-2q}\right)$ for $0<q< \frac12$ (ii) $f(q)= 1 + f(q-1)$ for $1<q\le 2$ (iii) $f(q)f\left(\frac{1}{q}\right)=1$ for all $q\in Q^+$ (b) For this function $f$ , find all $r\in Q^+$ such that $f(r) = r$

MathLinks Contest 3rd, 1

Tags: functional , algebra

Find all functions $f : (0, +\infty) \to (0, +\infty)$ which are increasing on $[1, +\infty)$ and for all positive reals $a, b, c$ they fulfill the following relation $f(ab)f(bc)f(ca)=f(a^2b^2c^2)+f(a^2)+f(b^2)+f(c^2)$.

2009 Ukraine Team Selection Test, 4

Tags: functional equation , functional , algebra

Let $n$ be some positive integer. Find all functions $f:{{R}^{+}}\to R$ (i.e., functions defined by the set of all positive real numbers with real values) for which equality holds $f\left( {{x}^{n+1}}+ {{y}^{n+1}} \right)={{x}^{n}}f\left( x \right)+{{y}^{n}}f\left( y \right)$ for any positive real numbers $x, y$