Olympiad problems

2015 Turkey Team Selection Test, 7

Find all the functions $f:R\to R$ such that \[f(x^2) + 4y^2f(y) = (f(x-y) + y^2)(f(x+y) + f(y))\] for every real $x,y$.

Fractal Edition 1, P3

Tags: functional equation

Find all functions $ f : \mathbb{R} \to \mathbb{R} $ that satisfy the following two conditions: \[ \left\{ \begin{array}{ll} \mbox{If } f(0) = 0, \mbox{ then } f(x) \neq 0 \mbox{ for any non-zero } x. \\ \\ f(x + y)f(y + z)f(z + x) = f(x + y + z)f(xy + yz + zx) - f(x)f(y)f(z) \quad \forall x, y, z \in \mathbb{R}. \end{array} \right. \]

2024 Dutch BxMO/EGMO TST, IMO TSTST, 3

Tags: functional equation , algebra

Find all pairs of positive integers $(a, b)$ such that $f(x)=x$ is the only function $f:\mathbb{R}\to \mathbb{R}$ that satisfies $$f^a(x)f^b(y)+f^b(x)f^a(y)=2xy$$ for all $x, y\in \mathbb{R}$.

2015 IMO Shortlist, A2

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 $x,y\in\mathbb{Z}$.

PEN K Problems, 6

Tags: functional equation , polynomial , function , algebra

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f^{(19)}(n)+97f(n)=98n+232.\]

2023 IMC, 3

Tags: polynomial , algebra , functional equation

Find all polynomials $P$ in two variables with real coefficients satisfying the identity $$P(x,y)P(z,t)=P(xz-yt,xt+yz).$$

2018 Irish Math Olympiad, 6

Tags: functional equation , algebra

Find all real-valued functions $f$ satisfying $f(2x + f(y)) + f(f(y)) = 4x + 8y$ for all real numbers $x$ and $y$.

1992 Poland - First Round, 4

Tags: functional equation , function , algebra

Determine all functions $f: R \longrightarrow R$ such that $f(x+y)-f(x-y)=f(x)*f(y)$ for $x,y \in R$

2008 IMO Shortlist, 4

Tags: inequality , function , functional equation , algebra

For an integer $ m$, denote by $ t(m)$ the unique number in $ \{1, 2, 3\}$ such that $ m \plus{} t(m)$ is a multiple of $ 3$. A function $ f: \mathbb{Z}\to\mathbb{Z}$ satisfies $ f( \minus{} 1) \equal{} 0$, $ f(0) \equal{} 1$, $ f(1) \equal{} \minus{} 1$ and $ f\left(2^{n} \plus{} m\right) \equal{} f\left(2^n \minus{} t(m)\right) \minus{} f(m)$ for all integers $ m$, $ n\ge 0$ with $ 2^n > m$. Prove that $ f(3p)\ge 0$ holds for all integers $ p\ge 0$. [i]Proposed by Gerhard Woeginger, Austria[/i]

2023 Dutch IMO TST, 4

Tags: functional equation , algebra , functional equation in r

Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.

2017 Estonia Team Selection Test, 6

Tags: algebra , functional equation

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

2018 India National Olympiad, 6

Tags: number theory , power of prime , functional equation

Let $\mathbb N$ denote set of all natural numbers and let $f:\mathbb{N}\to\mathbb{N}$ be a function such that $\text{(a)} f(mn)=f(m).f(n)$ for all $m,n \in\mathbb{N}$; $\text{(b)} m+n$ divides $f(m)+f(n)$ for all $m,n\in \mathbb N$. Prove that, there exists an odd natural number $k$ such that $f(n)= n^k$ for all $n$ in $\mathbb{N}$.

2016 Postal Coaching, 3

Tags: function , functional equation , algebra

Find all real numbers $a$ such that there exists a function $f:\mathbb R\to \mathbb R$ such that the following conditions are simultaneously satisfied: (a) $f(f(x))=xf(x)-ax,\;\forall x\in\mathbb{R};$ (b) $f$ is not a constant function; (c) $f$ takes the value $a$.

2018-IMOC, A3

Tags: functional equation , fe , algebra

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

2021 Thailand TSTST, 2

Tags: unbounded , functional equation , algebra , function

Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be such that $$f(x+f(y))^2\geq f(x)\left(f(x+f(y))+f(y)\right)$$ for all $x,y\in\mathbb{R}^+$. Show that $f$ is [i]unbounded[/i], i.e. for each $M\in\mathbb{R}^+$, there exists $x\in\mathbb{R}^+$ such that $f(x)>M$.

2010 Saudi Arabia IMO TST, 2

Tags: functional , functional equation , algebra

Find all functions $f,g : N \to N$ such that for all $m ,n \in N$ the following relation holds: $$f(m ) - f(n) = (m - n)(g(m) + g(n))$$. Note: $N = \{0,1,2,...\}$

2025 Nordic, 1

Tags: function , fe , algebra , functional equation

Let $n$ be a positive integer greater than $2$. Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying: $(f(x+y))^{n} = f(x^{n})+f(y^{n}),$ for all integers $x,y$

1996 Bosnia and Herzegovina Team Selection Test, 4

Tags: algebra , functional equation

Solve the functional equation $$f(x+y)+f(x-y)=2f(x)\cos{y}$$ where $x,y \in \mathbb{R}$ and $f : \mathbb{R} \rightarrow \mathbb{R}$

2013 Brazil National Olympiad, 3

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

Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that \[f(x+y) \left(f(x) + f(y)\right) = f(xy)\] for all non-zero reals $x, y$ such that $x+y \neq 0$.

2022 Romania National Olympiad, P4

Tags: functional equation , algebra , combinatorics

Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that $$|f(A)\cap f(B)|=|A\cap B|$$ whenever $A$ and $B$ are two distinct subsets of $X$. [i] (Sergiu Novac)[/i]

2025 Turkey Team Selection Test, 3

Tags: functional equation , algebra

Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for all $x,y \in \mathbb{R}-\{0\}$, $$ f(x) \neq 0 \text{ and } \frac{f(x)}{f(y)} + \frac{f(y)}{f(x)} - f \left( \frac{x}{y}-\frac{y}{x} \right) =2 $$

2015 Thailand TSTST, 2

Tags: algebra , functional equation

Let $\mathbb{N} = \{1, 2, 3, \dots\}$ and let $f : \mathbb{N}\to\mathbb{R}$. Prove that there is an infinite subset $A$ of $\mathbb{N}$ such that $f$ is increasing on $A$ or $f$ is decreasing on $A$.

2017 Czech And Slovak Olympiad III A, 3

Tags: functional , functional equation , algebra

Find all functions $f: R \to R$ such that for all real numbers $x, y$ holds $f(y - xy) = f(x)y + (x - 1)^2 f(y)$

2024 IFYM, Sozopol, 5

Tags: functional equation , algebra

The function $f: A \rightarrow A$ is such that $f(x) \leq x^2 \mbox{ and } f(x+y) \leq f(x) + f(y) + 2xy$ for any $x, y \in A$. a) If $A = \mathbb{R}$, find all functions satisfying the conditions. b) If $A = \mathbb{R}^{-}$, prove that there are infinitely many functions satisfying the conditions. [i](With $\mathbb{R}^{-}$ we denote the set of negative real numbers.)[/i]

MathLinks Contest 4th, 3.2

Tags: algebra , functional equation

Determine all functions $f : R \to R$ such that $f(x) \ge 0$ for all positive reals $x$, $f(0) = 0$ and for all reals $x, y$ $$f(x + y -xy) = f(x) + f(y) - f(xy).$$