Olympiad problems

2000 Singapore Team Selection Test, 1

Find all functions $f : R \to R$ such for any $x, y \in R,$ $$(x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2)$$

2015 India National Olympiad, 3

Tags: functional equation , algebra , function , trigonometry

Find all real functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x^2+yf(x))=xf(x+y)$.

2016 Germany Team Selection Test, 2

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

2017 IMO Shortlist, A6

Tags: functional equation , algebra

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

1987 Austrian-Polish Competition, 3

Tags: functional , limit , functional equation , sequence

A function $f: R \to R$ satisfies $f (x + 1) = f (x) + 1$ for all $x$. Given $a \in R$, define the sequence $(x_n)$ recursively by $x_0 = a$ and $x_{n+1} = f (x_n)$ for $n \ge 0$. Suppose that, for some positive integer m, the difference $x_m - x_0 = k$ is an integer. Prove that the limit $\lim_{n\to \infty}\frac{x_n}{n}$ exists and determine its value.

2015 VTRMC, Problem 6

Tags: absolute value , functional equation , fe

Let $(a_1,b_1),\ldots,(a_n,b_n)$ be $n$ points in $\mathbb R^2$ (where $\mathbb R$ denotes the real numbers), and let $\epsilon>0$ be a positive number. Can we find a real-valued function $f(x,y)$ that satisfies the following three conditions? 1. $f(0,0)=1$; 2. $f(x,y)\ne0$ for only finitely many $(x,y)\in\mathbb R^2$; 3. $\sum_{r=1}^n\left|f(x+a_r,y+b_r)-f(x,y)\right|<\epsilon$ for every $(x,y)\in\mathbb R^2$. Justify your answer.

2009 District Olympiad, 4

Tags: functional equation , function , algebra

Fin the functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ such that: $$ \frac{f(x+y)+f(x)}{2x+f(y)} =\frac{2y+f(x)}{f(x+y)+f(y)} ,\quad\forall x,y\in\mathbb{N} . $$

2020 Peru Iberoamerican Team Selection Test, P2

Tags: functional equation , number theory

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ that satisfy the conditions: $i) f(f(x)) = xf(x) - x^2 + 2,\forall x\in\mathbb{Z}$ $ii) f$ takes all integer values

2008 Chile National Olympiad, 3

Tags: functional equation , algebra , function

Determine all strictly increasing functions $f : R \to R$ such that for all $x \ne y$ to hold $$\frac{2\left[f(y)-f\left(\frac{x+y}{2}\right) \right]}{f(x)-f(y)}=\frac{f(x)-f(y)}{2\left[f\left(\frac{x+y}{2}\right)-f(x) \right]}$$

1977 Germany Team Selection Test, 2

Tags: algebra , polynomial , functional equation

Determine the polynomials P of two variables so that: [b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$ [b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$ [b]c.)[/b] $P(1,0) =1.$

1999 Denmark MO - Mohr Contest, 3

Tags: functional equation , functional , algebra

A function $f$ satisfies $$f(x)+xf(1-x)=x$$ for all real numbers $x$. Determine the number $f (2)$. Find $f$ .

2003 Croatia National Olympiad, Problem 2

Tags: functional equation , algebra

A sequence $(a_n)_{n\ge0}$ satisfies $a_{m+n}+a_{m-n}=\frac12\left(a_{2m}+a_{2n}\right)$ for all integers $m,n$ with $m\ge n\ge0$. Given that $a_1=1$, find $a_{2003}$.

2015 District Olympiad, 4

Tags: algebra , functional equation , function

Let $ f: (0,\infty)\longrightarrow (0,\infty) $ a non-constant function having the property that $ f\left( x^y\right) = \left( f(x)\right)^{f(y)},\quad\forall x,y>0. $ Show that $ f(xy)=f(x)f(y) $ and $ f(x+y)=f(x)+f(y), $ for all $ x,y>0. $

2010 Contests, 1

Tags: functional equation , function , algebra , vieta

Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$.

2014 IMO Shortlist, A4

Tags: functional equation , function , algebra

Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$. [i]Proposed by Netherlands[/i]

2019 Federal Competition For Advanced Students, P2, 1

Tags: functional equation , algebra

Determine all functions $f: R\to R$, such that $f (2x + f (y)) = x + y + f (x)$ for all $x, y \in R$. (Gerhard Kirchner)

2017-IMOC, A5

Tags: algebra , functional equation , fe

Find all functions $f:\mathbb Z\to\mathbb Z$ such that $$f(mf(n+1))=f(m+1)f(n)+f(f(n))+1$$for all integer pairs $(m,n)$.

2020 Peru IMO TST, 6

Tags: functional equation , number theory

Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.

2017 Saudi Arabia IMO TST, 2

Tags: functional equation , algebra

Find all $f:\mathbb{R}\to\mathbb{R}$ satisfying: $$f(xf(y)-y)+f(xy-x)+f(x+y)=2xy,\quad\forall x,y\in\mathbb{R}.$$

1998 IMO Shortlist, 5

Tags: prime number , algebra , functional equation , number theory

Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$, \[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]

OMMC POTM, 2022 2

Tags: algebra , functional equation

Find all functions $f:\mathbb R \to \mathbb R$ (from the set of real numbers to itself) where$$f(x-y)+xf(x-1)+f(y)=x^2$$for all reals $x,y.$ Proposed by [b]cj13609517288[/b]

1993 French Mathematical Olympiad, Problem 3

Tags: function , algebra , functional equation , fe

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.

1995 North Macedonia National Olympiad, 5

Tags: functional , function , algebra , functional equation

Let $ a, b, c, d \in \mathbb {R}, $ $ b \neq0. $ Find the functions of the $ f: \mathbb{R} \to \mathbb{R} $ such that $ f (x + d \cdot f (y)) = ax + by + c, $ for all $ x, y \in \mathbb{R}. $

1968 IMO, 5

Tags: algebra , function , periodic function , functional equation

Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$. Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.

2023 IMC, 1

Tags: derivative , calculus , function , functional equation

Find all functions $f: \mathbb{R} \to \mathbb{R}$ that have a continuous second derivative and for which the equality $f(7x+1)=49f(x)$ holds for all $x \in \mathbb{R}$.