Olympiad problems

2019 Swedish Mathematical Competition, 5

Let $f$ be a function that is defined for all positive integers and whose values are positive integers. For $f$ it also holds that $f (n + 1)> f (n)$ and $f (f (n)) = 3n$, for each positive integer $n$. Calculate $f (2019)$.

1990 Swedish Mathematical Competition, 5

Tags: functional , algebra , functional equation

Find all monotonic positive functions $f(x)$ defined on the positive reals such that $f(xy) f\left( \frac{f(y)}{x}\right) = 1$ for all $x, y$.

2010 Contests, 1

Tags: functional equation , functional , algebra

Find all functions $ f : R \to R$ that satisfies $$xf(y) - yf(x)= f\left(\frac{y}{x}\right)$$ for all $x, y \in R$.

VI Soros Olympiad 1999 - 2000 (Russia), 10.1

Tags: functional , functional equation , algebra

Find all real functions of a real numbers, such that for any $x$, $y$ and $z$ holds the equality $$ f(x)f(y)f(z)-f(xyz)=xy+yz+xz+x+y+z.$$

2021 Flanders Math Olympiad, 4

Tags: algebra , inequalities , functional , functional equation

(a) Prove that for every $x \in R$ holds that $$-1 \le \frac{x}{x^2 + x + 1} \le \frac 13$$ (b) Determine all functions $f : R \to R$ for which for every $x \in R$ holds that $$f \left( \frac{x}{x^2 + x + 1} \right) = \frac{x^2}{x^4 + x^2 + 1}$$

2019 Switzerland - Final Round, 6

Tags: functional , functional equation , algebra

Show that there exists no function $f : Z \to Z$ such that for all $m, n \in Z$ $$f(m + f(n)) = f(m) - n.$$

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

Tags: functional , functional equation , function , 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 $x,y \ne 0$

1989 Swedish Mathematical Competition, 2

Tags: functional equation , algebra , continuous , functional

Find all continuous functions $f$ such that $f(x)+ f(x^2) = 0$ for all real numbers $x$.

2011 Belarus Team Selection Test, 1

Tags: functional equation , trigonometry , functional , algebra

Find all real $a$ such that there exists a function $f: R \to R$ satisfying the equation $f(\sin x )+ a f(\cos x) = \cos 2x$ for all real $x$. I.Voronovich

1981 Poland - Second Round, 3

Tags: functional , algebra , continuous

Prove that there is no continuous function $ f: \mathbb{R} \to \mathbb{R} $ satisfying the condition $ f(f(x)) = - x $ for every $ x $.

2021 Final Mathematical Cup, 1

Tags: functional equation , functional , 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 $$f(n)f(m)=f\left((nm)^{2021}\right)$$

2013 Costa Rica - Final Round, F2

Tags: algebra , functional , functional equation

Find all functions $f:R -\{0,2\} \to R$ that satisfy for all $x \ne 0,2$ $$f(x) \cdot \left(f\left(\sqrt[3]{\frac{2+x}{2-x}}\right) \right)^2=\frac{x^3}{4}$$

2019 Regional Olympiad of Mexico Center Zone, 2

Tags: algebra , functional , functional inequality , inequalities

Find all functions $ f: \mathbb {R} \rightarrow \mathbb {R} $ such that $ f (x + y) \le f (xy) $ for every pair of real $ x $, $ y$.

2011 Swedish Mathematical Competition, 6

Tags: functional , functional equation , algebra

How many functions $f:\mathbb N \to \mathbb N$ are there such that $f(0)=2011$, $f(1) = 111$, and $$f\left(\max \{x + y + 2, xy\}\right) = \min \{f (x + y), f (xy + 2)\}$$ for all non-negative integers $x$, $y$?

2012 QEDMO 11th, 11

Tags: algebra , functional , functional equation

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

2019 Saudi Arabia IMO TST, 1

Tags: algebra , functional , functional inequality

Find all functions $f : Z^+ \to Z^+$ such that $n^3 - n^2 \le f(n) \cdot (f(f(n)))^2 \le n^3 + n^2$ for every $n$ in positive integers

2008 Indonesia TST, 4

Tags: functional , number theory , functional equation in n , functional equation

Find all pairs of positive integer $\alpha$ and function $f : N \to N_0$ that satisfies (i) $f(mn^2) = f(mn) + \alpha f(n)$ for all positive integers $m, n$. (ii) If $n$ is a positive integer and $p$ is a prime number with $p|n$, then $f(p) \ne 0$ and $f(p)|f(n)$.

2022 Saudi Arabia BMO + EGMO TST, 2.4

Tags: functional , functional equation , algebra

Consider the function $f : R^+ \to R^+$ and satisfying $$f(x + 2y + f(x + y)) = f(2x) + f(3y), \,\, \forall \,\, x, y > 0.$$ 1. Find all functions $f(x)$ that satisfy the given condition. 2. Suppose that $f(4\sin^4x)f(4\cos^4x) \ge f^2(1)$ for all $x \in \left(0\frac{\pi}{2}\right) $. Find the minimum value of $f(2022)$.

2012 Belarus Team Selection Test, 3

Tags: functional , functional equation , algebra

Find all functions $f : Q \to Q$, such that $$f(x + f (y + f(z))) = y + f(x + z)$$ for all $x ,y ,z \in Q$ . (I. Voronovich)

2006 QEDMO 2nd, 13

Tags: algebra , functional , functional equation

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for any two reals $x$ and $y$, we have $f\left( f\left( x+y\right) \right) +xy=f\left( x+y\right) +f\left( x\right) f\left( y\right) $.

1999 Chile National Olympiad, 7

Tags: functional equation , functional , algebra

Let $f$ be a function defined on the set of positive integers , and with values in the same set, which satisfies: $\bullet$ $f (n + f (n)) = 1$ for all $n\ge 1$. $\bullet$ $f (1998) = 2$ Find the lowest possible value of the sum $f (1) + f (2) +... + f (1999)$, and find the formula of $f$ for which this minimum is satisfied,

2016 Estonia Team Selection Test, 3

Tags: functional , functional equation , algebra

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

2010 Indonesia TST, 1

Tags: functional equation , functional , algebra

Find all functions $ f : R \to R$ that satisfies $$xf(y) - yf(x)= f\left(\frac{y}{x}\right)$$ for all $x, y \in R$.

2010 Bundeswettbewerb Mathematik, 4

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

In the following, let $N_0$ denotes the set of non-negative integers. Find all polynomials $P(x)$ that fulfill the following two properties: (1) All coefficients of $P(x)$ are from $N_0$. (2) Exists a function $f : N_0 \to N_0$ such as $f (f (f (n))) = P (n)$ for all $n \in N_0$.

2020 Greece National Olympiad, 1

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

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