Olympiad problems

2019 China Team Selection Test, 5

Determine all functions $f: \mathbb{Q} \to \mathbb{Q}$ such that $$f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}$$ for all $x,y \in \mathbb{Q}$.

2024 Pan-African, 5

Tags: algebra , functional equation

Let $ \mathbb{R} $ denote the set of real numbers. Find all functions $ f: \mathbb{R} \to \mathbb{R} $ such that \[ f(x^2) - y f(y) = f(x+y)(f(x) - y) \] for all real numbers $ x $ and $ y $.

2016 Peru IMO TST, 6

Tags: functional equation

Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.

2010 IMO Shortlist, 1

Tags: fe , functional equation , algebra

Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$ [i]Proposed by Pierre Bornsztein, France[/i]

2020 Thailand Mathematical Olympiad, 3

Tags: algebra , functional equation

Suppose that $f : \mathbb{R}^+\to\mathbb R$ satisfies the equation $$f(a+b+c+d) = f(a)+f(b)+f(c)+f(d)$$ for all $a,b,c,d$ that are the four sides of some tangential quadrilateral. Show that $f(x+y)=f(x)+f(y)$ for all $x,y\in\mathbb{R}^+$.

2012 Germany Team Selection Test, 3

Tags: algebra , function , functional equation

Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$. [i]Proposed by Japan[/i]

2014 EGMO, 6

Tags: function , quadratic , functional equation , algebra

Determine all functions $f:\mathbb R\rightarrow\mathbb R$ satisfying the condition \[f(y^2+2xf(y)+f(x)^2)=(y+f(x))(x+f(y))\] for all real numbers $x$ and $y$.

1999 Mongolian Mathematical Olympiad, Problem 1

Tags: fe , functional equation , algebra

Suppose that a function $f:\mathbb R\to\mathbb R$ is such that for any real $h$ there exist at most $19990509$ different values of $x$ for which $f(x)\ne f(x+h)$. Prove that there is a set of at most $9995256$ real numbers such that $f$ is constant outside of this set.

2025 Euler Olympiad, Round 2, 3

Tags: trigonometry , functional equation , algebra

Find all functions $ f : \mathbb{R} \to \mathbb{R} $ such that the following two conditions hold: [b]1. [/b] For all real numbers $a$ and $b$ satisfying $a^2 + b^2 = 1$, We have $f(x) + f(y) \geq f(ax + by)$ for all real numbers $x, y$. [b]2.[/b] For all real numbers $x$ and $y$, there exist real numbers $a$ and $b$, such that $a^2 + b^2 = 1$ and $f(x) + f(y) = f(ax + by)$. [i]Proposed by Zaza Melikidze, Georgia[/i]

2010 Saudi Arabia IMO TST, 3

Tags: functional , function , functional equation , algebra

Let $f : N \to N$ be a strictly increasing function such that $f(f(n))= 3n$, for all $n \in N$. Find $f(2010)$. Note: $N = \{0,1,2,...\}$

2003 USA Team Selection Test, 4

Tags: parameterization , strong induction , induction , function , functional equation , algebra

Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that \[ f(m+n)f(m-n) = f(m^2) \] for $m,n \in \mathbb{N}$.

2014 Contests, 2

Tags: functional equation , function , algebra

Does there exist a function $f: \mathbb R \to \mathbb R $ satisfying the following conditions: (i) for each real $y$ there is a real $x$ such that $f(x)=y$ , and (ii) $f(f(x)) = (x - 1)f(x) + 2$ for all real $x$ ? [i]Proposed by Igor I. Voronovich, Belarus[/i]

1996 Czech And Slovak Olympiad IIIA, 5

Tags: functional , algebra , functional equation

For which integers $k$ does there exist a function $f : N \to Z$ such that $f(1995) =1996$ and $f(xy) = f(x)+ f(y)+k f(gcd(x,y))$ for all $x,y \in N$?

2001 Czech And Slovak Olympiad IIIA, 6

Tags: algebra , function , functional equation

Let be given natural numbers $a_1,a_2,...,a_n$ and a function $f : Z \to R$ such that $f(x) = 1$ for all integers $x < 0$ and $f(x) = 1- f(x-a_1)f(x-a_2)... f(x-a_n)$ for all integers $x \ge 0$. Prove that there exist natural numbers $s$ and $t$ such that for all integers $x > s$ it holds that $f(x+t) = f(x)$.

2017 Switzerland - Final Round, 2

Tags: algebra , functional , functional equation

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

2016 Thailand Mathematical Olympiad, 3

Tags: algebra , functional , functional equation

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

2017 Taiwan TST Round 1, 1

Tags: algebra , polynomial , functional equation

Find all polynomials $P$ with real coefficients which satisfy \[P(x)P(x+1)=P(x^2-x+3) \quad \forall x \in \mathbb{R}\]

2024 European Mathematical Cup, 4

Tags: algebra , function , functional equation

Find all functions $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $f(x+yf(x)) = xf(1+y)$ for all x, y positive reals.

2020 Dutch IMO TST, 3

Tags: algebra , functional equation in z , functional , functional equation

Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$ for all $x, y \in Z$

1977 IMO Shortlist, 1

Tags: algebra , functional inequality , functional equation

Find all functions $f : \mathbb{N}\rightarrow \mathbb{N}$ satisfying following condition: \[f(n+1)>f(f(n)), \quad \forall n \in \mathbb{N}.\]

2015 IMO Shortlist, A4

Tags: algebra , dorlir ahmeti , imo proposal , functional equation

Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]for all real numbers $x$ and $y$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2005 Korea National Olympiad, 4

Tags: function , derivative , calculus , functional equation , induction , algebra

Find all $f: \mathbb R \to\mathbb R$ such that for all real numbers $x$, $f(x) \geq 0$ and for all real numbers $x$ and $y$, \[ f(x+y)+f(x-y)-2f(x)-2y^2=0. \]

2005 Thailand Mathematical Olympiad, 15

Tags: functional equation , algebra , functional

A function $f : R \to R$ satisfy the functional equation $f(x + 2y) + 2f(y - 2x) = 3x -4y + 6$ for all reals $x, y$. Compute $f(2548)$.

2015 Saudi Arabia IMO TST, 1

Tags: functional equation , functional , algebra

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

Russian TST 2017, P2

Tags: algebra , functional equation

Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$