Olympiad problems

1977 Germany Team Selection Test, 2

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] $P(1,0) =1.$

2019 ELMO Shortlist, A2

Tags: algebra , functional equation

Find all functions $f:\mathbb Z\to \mathbb Z$ such that for all surjective functions $g:\mathbb Z\to \mathbb Z$, $f+g$ is also surjective. (A function $g$ is surjective over $\mathbb Z$ if for all integers $y$, there exists an integer $x$ such that $g(x)=y$.) [i]Proposed by Sean Li[/i]

2012 China Team Selection Test, 3

Tags: function , modular arithmetic , algebra , functional equation , number theory

Given an integer $n\ge 2$, a function $f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}$ is called [i]good[/i], if for any integer $k,1\le k\le n-1$ there exists an integer $j(k)$ such that for every integer $m$ we have \[f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. \] Find the number of [i]good[/i] functions.

2001 Estonia Team Selection Test, 3

Tags: functional , functional equation , algebra

Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.

2023 IMC, 1

Tags: calculus , derivative , functional equation , function

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 $x \in \mathbb{R}$.

2019 239 Open Mathematical Olympiad, 6

Tags: algebra , functional equation

Find all functions $f : (0, +\infty) \to \mathbb{R}$ satisfying the following conditions: $(i)$ $f(x) + f(\frac{1}{x}) = 1$ for all $x> 0$; $(ii)$ $f(xy + x + y) = f(x)f(y)$ for all $x, y> 0$.

1990 IMO Longlists, 93

Tags: function , number theory , algebra , functional equation

Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that \[ f(xf(y)) \equal{} \frac {f(x)}{y} \] for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.

2018 Czech-Polish-Slovak Match, 1

Tags: function , functional equation , algebra

Determine all functions $f : \mathbb R \to \mathbb R$ such that for all real numbers $x$ and $y$, $$f(x^2 + xy) = f(x)f(y) + yf(x) + xf(x+y).$$ [i]Proposed by Walther Janous, Austria[/i]

2023 Dutch IMO TST, 4

Tags: functional equation , functional equation in r , algebra

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 $x, y \in \mathbb{Q^+}$.

2024 Taiwan Mathematics Olympiad, 3

Tags: algebra , functional equation

Find all functions $f$ from real numbers to real numbers such that $$2f((x+y)^2)=f(x+y)+(f(x))^2+(4y-1)f(x)-2y+4y^2$$ holds for all real numbers $x$ and $y$.

2008 Grigore Moisil Intercounty, 3

Tags: function , algebra , functional equation , find all functions

Let be two nonzero real numbers $ a,b, $ and a function $ f:\mathbb{R}\longrightarrow [0,\infty ) $ satisfying the functional equation $$ f(x+a+b)+f(x)=f(x+a)+f(x+b) . $$ [b]1)[/b] Prove that $ f $ is periodic if $ a/b $ is rational. [b]2)[/b] If $ a/b $ is not rational, could $ f $ be nonperiodic?

1947 Putnam, A2

Tags: function , algebra , functional equation

A real valued continuous function $f$ satisfies for all real $x$ and $y$ the functional equation $$ f(\sqrt{x^2 +y^2 })= f(x)f(y).$$ Prove that $$f(x) =f(1)^{x^{2}}.$$

2017 Latvia Baltic Way TST, 3

Tags: algebra , functional equation

Find all functions $f (x) : Z \to Z$ defined on integers, take integer values, and for all $x,y \in Z$ satisfy $$f(x+y)+f(xy)=f(x)f(y)+1$$

2024 Korea Winter Program Practice Test, Q7

Tags: functional equation , algebra

Find all functions $f, g: \mathbb{R} \rightarrow \mathbb{R} $ satisfying the following conditions: [list][*] $f$ is not a constant function and if $x \le y$ then $f(x)\le f(y)$ [*] For all real number $x$, $f(g(x))=g(f(x))=0$ [*] For all real numbers $x$ and $y$, $f(x)+f(y)+g(x)+g(y)=f(x+y)+g(x+y)$ [*] For all real numbers $x$ and $y$, $f(x)+f(y)+f(g(x)+g(y))=f(x+y)$ [/list]

2017 Thailand Mathematical Olympiad, 3

Tags: functional equation , functional , algebra

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

1999 Nordic, 1

Tags: functional equation , algebra

The function $f$ is defined for non-negative integers and satisfies the condition $f(n) = f(f(n + 11))$, if $n \le 1999$ and $f(n) = n - 5$, if $n > 1999$. Find all solutions of the equation $f(n) = 1999$.

2013 IMO Shortlist, A3

Tags: function , functional equation , algebra

Let $\mathbb Q_{>0}$ be the set of all positive rational numbers. Let $f:\mathbb Q_{>0}\to\mathbb R$ be a function satisfying the following three conditions: (i) for all $x,y\in\mathbb Q_{>0}$, we have $f(x)f(y)\geq f(xy)$; (ii) for all $x,y\in\mathbb Q_{>0}$, we have $f(x+y)\geq f(x)+f(y)$; (iii) there exists a rational number $a>1$ such that $f(a)=a$. Prove that $f(x)=x$ for all $x\in\mathbb Q_{>0}$. [i]Proposed by Bulgaria[/i]

2015 Mexico National Olympiad, 3

Tags: function , algebra , functional equation

Let $\mathbb{N} =\{1, 2, 3, ...\}$ be the set of positive integers. Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be a function that gives a positive integer value, to every positive integer. Suppose that $f$ satisfies the following conditions: $f(1)=1$ $f(a+b+ab)=a+b+f(ab)$ Find the value of $f(2015)$ Proposed by Jose Antonio Gomez Ortega

2021 Peru EGMO TST, 6

Tags: algebra , function , functional inequality , functional equation

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

2021 Saudi Arabia IMO TST, 6

Tags: algebra , functional equation , iteration

Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ satisfying \[f^{a^{2} + b^{2}}(a+b) = af(a) +bf(b)\] for all integers $a$ and $b$

2017 Canadian Mathematical Olympiad Qualification, 3

Tags: algebra , functional equation , function , difference of squares , special factorizations

Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following equation for all $x, y \in \mathbb{R}$. $$(x+y)f(x-y) = f(x^2-y^2).$$

2023 European Mathematical Cup, 4

Tags: functional equation , function , ivan novak orz

Let $f\colon\mathbb{N}\rightarrow\mathbb{N}$ be a function such that for all positive integers $x$ and $y$, the number $f(x)+y$ is a perfect square if and only if $x+f(y)$ is a perfect square. Prove that $f$ is injective. [i]Remark.[/i] A function $f\colon\mathbb{N}\rightarrow\mathbb{N}$ is injective if for all pairs $(x,y)$ of distinct positive integers, $f(x)\neq f(y)$ holds. [i]Ivan Novak[/i]

2025 China Team Selection Test, 24

Tags: number theory , functional equation , algebra

Find all functions $f:\mathbb Z\to\mathbb Z$ such that $f$ is unbounded and \[2f(m)f(n)-f(n-m)-1\] is a perfect square for all integer $m,n.$

2008 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: sequence , function , algebra , functional equation

Determine is there a function $a: \mathbb{N} \rightarrow \mathbb{N}$ such that: $i)$ $a(0)=0$ $ii)$ $a(n)=n-a(a(n))$, $\forall n \in$ $ \mathbb{N}$. If exists prove: $a)$ $a(k)\geq a(k-1)$ $b)$ Does not exist positive integer $k$ such that $a(k-1)=a(k)=a(k+1)$.

2012 QEDMO 11th, 11

Tags: functional equation , functional , algebra

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