Olympiad problems

2011 Abels Math Contest (Norwegian MO), 3b

Find all functions $f$ from the real numbers to the real numbers such that $f(xy) \le \frac12 \left(f(x) + f(y) \right)$ for all real numbers $x$ and $y$.

2012 Swedish Mathematical Competition, 1

Tags: algebra , functional equation , functional

The function $f$ satisfies the condition $$f (x + 1) = \frac{1 + f (x)}{1 - f (x)}$$ for all real $x$, for which the function is defined. Determine $f(2012)$, if we known that $f(1000)=2012$.

2015 Estonia Team Selection Test, 5

Tags: algebra , functional equation , functional

Find all functions $f$ from reals to reals which satisfy $f (f(x) + f(y)) = f(x^2) + 2x^2 f(y) + (f(y))^2$ for all real numbers $x$ and $y$.

2022 Saudi Arabia BMO + EGMO TST, 2.4

Tags: algebra , functional , functional equation

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 $f(2022)$.

1993 Tournament Of Towns, (377) 5

Tags: function , algebra , functional , functional equation

Does there exist a piecewise linear function $f$ defined on the segment [$-1,1]$ (including the ends) such that $f(f(x)) = -x$ for all x? (A function is called piecewise linear if its graph is the union of a finite set of points and intervals; it may be discontinuous).

2017 Switzerland - Final Round, 2

Tags: algebra , functional , functional equation

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

2013 Saudi Arabia GMO TST, 1

Tags: functional , algebra , functional equation

Find all functions $f : R \to R$ which satisfy $f \left(\frac{\sqrt3}{3} x\right) = \sqrt3 f(x) - \frac{2\sqrt3}{3} x$ and $f(x)f(y) = f(xy) + f \left(\frac{x}{y} \right) $ for all $x, y \in R$, with $y \ne 0$

2007 Estonia Team Selection Test, 5

Tags: functional , continuous , functional equation , algebra

Find all continuous functions $f: R \to R$ such that for all reals $x$ and $y$, $f(x+f(y)) = y+f(x+1)$.

2020 Greece Team Selection Test, 1

Tags: algebra , functional , functional equation

Let $R_+=(0,+\infty)$. Find all functions $f: R_+ \to R_+$ such that $f(xf(y))+f(yf(z))+f(zf(x))=xy+yz+zx$, for all $x,y,z \in R_+$. by Athanasios Kontogeorgis (aka socrates)

2008 Indonesia TST, 2

Tags: algebra , functional equation , functional

Find all functions $f : R \to R$ that satisfies the condition $$f(f(x - y)) = f(x)f(y) - f(x) + f(y) - xy$$ for all real numbers $x, y$.

2003 Austrian-Polish Competition, 1

Tags: polynomial , functional , algebra , functional equation

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

2014 Belarus Team Selection Test, 1

Tags: algebra , functional equation , functional

Find all functions$ f : R_+ \to R_+$ such that $f(f(x)+y)=x+f(y)$ , for all $x, y \in R_+$ (Folklore) [hide=PS]Using search terms [color=#f00]+ ''f(x+f(y))'' + ''f(x)+y[/color]'' I found the same problem [url=https://artofproblemsolving.com/community/c6h1122140p5167983]in Q[/url], [url=https://artofproblemsolving.com/community/c6h1597644p9926878]continuous in R[/url], [url=https://artofproblemsolving.com/community/c6h1065586p4628238]strictly monotone in R[/url] , [url=https://artofproblemsolving.com/community/c6h583742p3451211 ]without extra conditions in R[/url] [/hide]

1993 Czech And Slovak Olympiad IIIA, 5

Tags: functional equation , functional , algebra

Find all functions $f : Z \to Z$ such that $f(-1) = f(1)$ and $f(x)+ f(y) = f(x+2xy)+ f(y-2xy)$ for all $x,y \in Z$

2011 Indonesia TST, 1

Tags: algebra , functional equation , functional equation in q , functional

Let $Q^+$ denote the set of positive rationals. Determine all functions $f : Q^+ \to Q^+$ that satisfy both of these conditions: (i) $f(x)$ is an integer if and only if $x$ is an integer; (ii) $f(f(xf(y)) + x) = yf(x) + x$ for all $x, y \in Q^+$.

2007 Switzerland - Final Round, 5

Tags: functional , functional equation , algebra

Determine all functions $f : R_{\ge 0} \to R_{\ge 0}$ with the following properties: (a) $f(1) = 0$, (b) $f(x) > 0$ for all $x > 1$, (c) For all $x, y\ge 0$ with $x + y > 0$ holds $$f(xf(y))f(y) = f\left( \frac{xy}{x + y}\right)$$

2005 Slovenia Team Selection Test, 2

Tags: functional , functional equation , algebra

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

2015 Thailand Mathematical Olympiad, 9

Tags: functional , functional equation , algebra

Determine all functions $f : R \to R$ satisfying $f(f(x) + 2y)= 6x + f(f(y) -x)$ for all real numbers $x,y$

1994 Austrian-Polish Competition, 1

Tags: functional , functional inequality , functional equation , algebra

A function $f: R \to R$ satisfies the conditions: $f (x + 19) \le f (x) + 19$ and $f (x + 94) \ge f (x) + 94$ for all $x \in R$. Prove that $f (x + 1) = f (x) + 1$ for all $x \in R$.

1993 Austrian-Polish Competition, 8

Tags: polynomial , functional , functional equation , algebra

Determine all real polynomials $P(z)$ for which there exists a unique real polynomial $Q(x)$ satisfying the conditions $Q(0)= 0$, $x + Q(y + P(x))= y + Q(x + P(y))$ for all $x,y \in R$.

1995 Austrian-Polish Competition, 4

Tags: functional equation , functional , polynomial , algebra

Determine all polynomials $P(x)$ with real coefficients such that $P(x)^2 + P\left(\frac{1}{x}\right)^2= P(x^2)P\left(\frac{1}{x^2}\right)$ for all $x$.

2009 IMAC Arhimede, 4

Tags: algebra , functional equation , functional

Let $m,n \in Z, m\ne n, m \ne 0, n \ne 0$ . Find all $f: Z \to R$ such that $f(mx+ny)=mf(x)+nf(y)$ for all $x,y \in Z$ .

1986 Austrian-Polish Competition, 9

Tags: continuous , functional , functional equation , composition , algebra

Find all continuous monotonic functions $f : R \to R$ that satisfy $f (1) = 1$ and $f(f (x)) = f (x)^2$ for all $x \in R$.

1998 Switzerland Team Selection Test, 1

Tags: functional , periodic , function , algebra

A function $f : R -\{0\} \to R$ has the following properties: (i) $f(x)- f(y) = f(x)f\left(\frac{1}{y}\right)- f(y)f\left(\frac{1}{x}\right)$ for all $x,y \ne 0$, (ii) $f$ takes the value $\frac12$ at least once. Determine $f(-1)$. Prove that $f$ is a periodic function

2019 Swedish Mathematical Competition, 5

Tags: algebra , inequalities , functional , functional inequality , functional equation

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 $f (2019)$.

2014 Swedish Mathematical Competition, 3

Tags: algebra , functional equation , functional

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