Olympiad problems

2019 Israel Olympic Revenge, P4

Call a function $\mathbb Z_{>0}\rightarrow \mathbb Z_{>0}$ $\emph{M-rugged}$ if it is unbounded and satisfies the following two conditions: $(1)$ If $f(n)|f(m)$ and $f(n)<f(m)$ then $n|m$. $(2)$ $|f(n+1)-f(n)|\leq M$. a. Find all $1-rugged$ functions. b. Determine if the number of $2-rugged$ functions is smaller than $2019$.

2013 Saudi Arabia BMO TST, 2

Tags: algebra , functional equation , functional

Find all functions $f : R \to R$ which satisfy for all $x, y \in R$ the relation $f(f(f(x) + y) + y) = x + y + f(y)$

1977 IMO Shortlist, 9

Tags: algebra , polynomial , functional equation

For which positive integers $n$ do there exist two polynomials $f$ and $g$ with integer coefficients of $n$ variables $x_1, x_2, \ldots , x_n$ such that the following equality is satisfied: \[\sum_{i=1}^n x_i f(x_1, x_2, \ldots , x_n) = g(x_1^2, x_2^2, \ldots , x_n^2) \ ? \]

The Golden Digits 2024, P2

Tags: algebra , functional equation

Let $n\in\mathbb{Z}$, $n\geq 2$. Find all functions $f:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}$ such that $$f(x_1+\dots +x_n)^2=\sum_{i=1}^nf(x_i) ^2+ 2\sum_{i<j}f(x_ix_j),$$ for all $x_1,\dots ,x_n\in\mathbb{R}_{>0}$. [i]Proposed by Andrei Vila[/i]

2025 Vietnam Team Selection Test, 1

Tags: function , algebra , fe , functional equation

Find all functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ such that $$\dfrac{f(x)f(y)}{f(xy)} = \dfrac{\left( \sqrt{f(x)} + \sqrt{f(y)} \right)^2}{f(x+y)}$$ holds for all positive rational numbers $x, y$.

2016 Thailand TSTST, 1

Tags: functional equation , function , algebra

Find all functions $f:\mathbb{Q}\to\mathbb{Q}$ such that $$f(xy)+f(x+y)=f(x)f(y)+f(x)+f(y)$$ for all $x,y\in\mathbb{Q}$.

2018 Iran Team Selection Test, 1

Tags: function , algebra , functional equation

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ that satisfy the following conditions: a. $x+f(y+f(x))=y+f(x+f(y)) \quad \forall x,y \in \mathbb{R}$ b. The set $I=\left\{\frac{f(x)-f(y)}{x-y}\mid x,y\in \mathbb{R},x\neq y \right\}$ is an interval. [i]Proposed by Navid Safaei[/i]

2019 European Mathematical Cup, 4

Tags: functional equation , algebra

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that $$f(x)+f(yf(x)+f(y))=f(x+2f(y))+xy$$for all $x,y\in \mathbb{R}$. [i]Proposed by Adrian Beker[/i]

2017 Bulgaria EGMO TST, 1

Tags: function , algorithm , induction , number theory , euclidean algorithm , algebra , functional equation

Let $\mathbb{Q^+}$ denote the set of positive rational numbers. Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$ that satisfy the conditions \[ f \left( \frac{x}{x+1}\right) = \frac{f(x)}{x+1} \qquad \text{and} \qquad f \left(\frac{1}{x}\right)=\frac{f(x)}{x^3}\] for all $x \in \mathbb{Q^+}.$

2010 Ukraine Team Selection Test, 3

Tags: function , algebra , functional equation

Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\] [i]Proposed by Japan[/i]

2021 Federal Competition For Advanced Students, P2, 4

Tags: algebra , functional equation , functional

Let $a$ be a real number. Determine all functions $f: R \to R$ with $f (f (x) + y) = f (x^2 - y) + af (x) y$ for all $x, y \in R$. (Walther Janous)

2016 Middle European Mathematical Olympiad, 4

Tags: functional equation , function , divisibility , algebra , number theory

Find all $f : \mathbb{N} \to \mathbb{N} $ such that $f(a) + f(b)$ divides $2(a + b - 1)$ for all $a, b \in \mathbb{N}$. Remark: $\mathbb{N} = \{ 1, 2, 3, \ldots \} $ denotes the set of the positive integers.

1993 French Mathematical Olympiad, Problem 3

Tags: functional equation , fe , function , algebra

Let $f$ be a function from $\mathbb Z$ to $\mathbb R$ which is bounded from above and satisfies $f(n)\le\frac12(f(n-1)+f(n+1))$ for all $n$. Show that $f$ is constant.

1998 Iran MO (3rd Round), 3

Tags: algebra , functional equation

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

2002 Canada National Olympiad, 5

Tags: function , modular arithmetic , functional equation , algebra

Let $\mathbb N = \{0,1,2,\ldots\}$. Determine all functions $f: \mathbb N \to \mathbb N$ such that \[ xf(y) + yf(x) = (x+y) f(x^2+y^2) \] for all $x$ and $y$ in $\mathbb N$.

2021 Science ON grade XII, 1

Tags: functional equation , integration , differentiable function , calculus

Find all differentiable functions $f, g:[0,\infty) \to \mathbb{R}$ and the real constant $k\geq 0$ such that \begin{align*} f(x) &=k+ \int_0^x \frac{g(t)}{f(t)}dt \\ g(x) &= -k-\int_0^x f(t)g(t) dt \end{align*} and $f(0)=k, f'(0)=-k^2/3$ and also $f(x)\neq 0$ for all $x\geq 0$.\\ \\ [i] (Nora Gavrea)[/i]

2016 India IMO Training Camp, 2

Tags: function , functional equation , algebra

Find all functions $f:\mathbb R\to\mathbb R$ such that $$f\left( x^2+xf(y)\right)=xf(x+y)$$ for all reals $x,y$.

2024 Belarusian National Olympiad, 10.3

Tags: algebra , functional equation

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for every $x,y \in \mathbb{R}$ the following equation holds:$$1+f(xy)=f(x+f(y))+(y-1)f(x-1)$$ [i]M. Zorka[/i]

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 $A$.

2012 Online Math Open Problems, 46

Tags: function , inequalities , algebra , functional equation

If $f$ is a function from the set of positive integers to itself such that $f(x) \leq x^2$ for all natural $x$, and $f\left( f(f(x)) f(f(y))\right) = xy$ for all naturals $x$ and $y$. Find the number of possible values of $f(30)$. [i]Author: Alex Zhu[/i]

2022 Peru MO (ONEM), 3

Tags: algebra , functional equation , functional

Let $R$ be the set of real numbers and $f : R \to R$ be a function that satisfies: $$f(xy) + y + f(x + f(y)) = (y + 1)f(x),$$ for all real numbers $x, y$. a) Determine the value of $f(0)$. b) Prove that $f(x) = 2-x$ for every real number $x$.

1997 Estonia National Olympiad, 2

Tags: functional , functional equation , function , algebra

A function $f$ satisfies the following condition for each $n\in N$: $f (1)+ f (2)+...+ f (n) = n^2 f (n)$. Find $f (1997)$ if $f (1) = 999$.

2019 USA TSTST, 1

Tags: functional equation

Find all binary operations $\diamondsuit: \mathbb R_{>0}\times \mathbb R_{>0}\to \mathbb R_{>0}$ (meaning $\diamondsuit$ takes pairs of positive real numbers to positive real numbers) such that for any real numbers $a, b, c > 0$, [list] [*] the equation $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ holds; and [*] if $a\ge 1$ then $a\,\diamondsuit\, a\ge 1$. [/list] [i]Evan Chen[/i]

2024 Abelkonkurransen Finale, 2b

Tags: real number , functional equation , involution , algebra

Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying \[xf(f(x)+y)=f(xy)+x^2\] for all $x,y \in \mathbb{R}$.

2017-IMOC, N3

Tags: algebra , fe , functional equation , number theory

Find all functions $f:\mathbb N\to\mathbb N_0$ such that for all $m,n\in\mathbb N$, \begin{align*} f(mn)&=f(m)f(n)\\ f(m+n)&=\min(f(m),f(n))\qquad\text{if }f(m)\ne f(n)\end{align*}