Olympiad problems

2022 Korea -Final Round, P3

A function $g \colon \mathbb{R} \to \mathbb{R}$ is given such that its range is a finite set. Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfies $$2f(x+g(y))=f(2g(x)+y)+f(x+3g(y))$$ for all $x, y \in \mathbb{R}$.

2004 Spain Mathematical Olympiad, Problem 3

Tags: algebra , functional equation

Represent for $\mathbb {Z}$ the set of all integers. Find all of the functions ${f:}$ $ \mathbb{Z} \rightarrow \mathbb{Z}$ such that for any ${x,y}$ integers, they satisfy: ${f(x + f(y)) = f(x) - y.}$

1995 IMO Shortlist, 5

Tags: function , algebra , functional equation

Let $ \mathbb{R}$ be the set of real numbers. Does there exist a function $ f: \mathbb{R} \mapsto \mathbb{R}$ which simultaneously satisfies the following three conditions? [b](a)[/b] There is a positive number $ M$ such that $ \forall x:$ $ \minus{} M \leq f(x) \leq M.$ [b](b)[/b] The value of $f(1)$ is $1$. [b](c)[/b] If $ x \neq 0,$ then \[ f \left(x \plus{} \frac {1}{x^2} \right) \equal{} f(x) \plus{} \left[ f \left(\frac {1}{x} \right) \right]^2 \]

2024 Baltic Way, 2

Tags: functional equation , algebra

Let $\mathbb{R}^+$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^+\to\mathbb{R}^+$ such that \[ \frac{f(a)}{1+a+ca}+\frac{f(b)}{1+b+ab}+\frac{f(c)}{1+c+bc} = 1 \] for all $a,b,c \in \mathbb{R}^+$ that satisfy $abc=1$.

2018 Taiwan TST Round 3, 5

Tags: algebra , functional equation , function

Find all functions $ f: \mathbb{N} \to \mathbb{N} $ such that $$ f\left(x+yf\left(x\right)\right) = x+f\left(y\right)f\left(x\right) $$ holds for all $ x,y \in \mathbb{N} $

2020 IMEO, Problem 3

Tags: functional equation , algebra

Find all functions $f:\mathbb{R^+} \to \mathbb{R^+}$ such that for all positive real $x, y$ holds $$xf(x)+yf(y)=(x+y)f\left(\frac{x^2+y^2}{x+y}\right)$$. [i]Fedir Yudin[/i]

2021 Flanders Math Olympiad, 4

Tags: inequalities , functional , functional equation , algebra

(a) Prove that for every $x \in R$ holds that $$-1 \le \frac{x}{x^2 + x + 1} \le \frac 13$$ (b) Determine all functions $f : R \to R$ for which for every $x \in R$ holds that $$f \left( \frac{x}{x^2 + x + 1} \right) = \frac{x^2}{x^4 + x^2 + 1}$$

2023 Israel TST, P1

Tags: algebra , functional equation , function

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all $x, y\in \mathbb{R}$ the following holds: \[f(x)+f(y)=f(xy)+f(f(x)+f(y))\]

2020 IMO Shortlist, A6

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 $b$

2022 Kosovo National Mathematical Olympiad, 2

Tags: function , functional equation

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$, $$f(f(x-y)-yf(x))=xf(y).$$

2016 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: function , functional equation , algebra

Find all functions $f : \mathbb{Q} \rightarrow \mathbb{R}$ such that: $a)$ $f(1)+2>0$ $b)$ $f(x+y)-xf(y)-yf(x)=f(x)f(y)+f(x)+f(y)+xy$, $\forall x,y \in \mathbb{Q}$ $c)$ $f(x)=3f(x+1)+2x+5$, $\forall x \in \mathbb{Q}$

2000 Mongolian Mathematical Olympiad, Problem 4

Tags: functional equation , fe , functional inequality , algebra

Suppose that a function $f:\mathbb R\to\mathbb R$ satisfies the following conditions: (i) $\left|f(a)-f(b)\right|\le|a-b|$ for all $a,b\in\mathbb R$; (ii) $f(f(f(0)))=0$. Prove that $f(0)=0$.

2021 Iran MO (3rd Round), 3

Tags: functional equation , algebra , polynomial , function

Polynomial $P$ with non-negative real coefficients and function $f:\mathbb{R}^+\to \mathbb{R}^+$ are given such that for all $x, y\in \mathbb{R}^+$ we have $$f(x+P(x)f(y)) = (y+1)f(x)$$ (a) Prove that $P$ has degree at most 1. (b) Find all function $f$ and non-constant polynomials $P$ satisfying the equality.

2013 QEDMO 13th or 12th, 4

Tags: periodic , functional , functional equation , algebra

Let $a> 0$ and $f: R\to R$ a function such that $f (x) + f (x + 2a) + f (x + 3a) + f (x + 5a) = 1$ for all $x\in R$ . Show that $f$ is periodic, that is, that there is some $b> 0$ for which $f (x) = f (x + b)$ for every $x \in R$ holds. Find the smallest such $b$, which works for all these functions .

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

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.

2007 District Olympiad, 3

Tags: algebra , number theory , functional equation , function

Find all functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ that satisfy the following relation: $$ f(x)^2+y\vdots x^2+f(y) ,\quad\forall x,y\in\mathbb{N} . $$

1987 IMO Longlists, 6

Tags: function , algebra , functional equation

Let f be a function that satisfies the following conditions: $(i)$ If $x > y$ and $f(y) - y \geq v \geq f(x) - x$, then $f(z) = v + z$, for some number $z$ between $x$ and $y$. $(ii)$ The equation $f(x) = 0$ has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the other solutions; $(iii)$ $f(0) = 1$. $(iv)$ $f(1987) \leq 1988$. $(v)$ $f(x)f(y) = f(xf(y) + yf(x) - xy)$. Find $f(1987)$. [i]Proposed by Australia.[/i]

1998 Brazil National Olympiad, 2

Tags: function , algebra , functional equation

Find all functions $f : \mathbb N \to \mathbb N$ satisfying, for all $x \in \mathbb N$, \[ f(2f(x)) = x + 1998 . \]

2017 Estonia Team Selection Test, 6

Tags: algebra , functional equation

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

2022 Kazakhstan National Olympiad, 3

Tags: functional equation , algebra

Given $m\in\mathbb{N}$. Find all functions $f:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ such that $$f(f(x)+y)-f(x)=\left( \frac{f(y)}{y}-1\right)x+f^m(y)$$ holds for all $x,y\in\mathbb{R^{+}}.$ ($f^m(x) =$ $f$ applies $m$ times.)

1989 Romania Team Selection Test, 1

Tags: functional , functional equation , algebra

Let $F$ be the set of all functions $f : N \to N$ which satisfy $f(f(x))-2 f(x)+x = 0$ for all $x \in N$. Determine the set $A =\{ f(1989) | f \in F\}$.

2003 Austrian-Polish Competition, 1

Tags: polynomial , functional , algebra , functional equation

Find all real polynomials $p(x) $ such that $p(x-1)p(x+1)= p(x^2-1)$.

1990 Putnam, B1

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

Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, \[ \left( f(x) \right)^2 = \displaystyle\int_0^x \left[ \left( f(t) \right)^2 + \left( f'(t) \right)^2 \right] \, \mathrm{d}t + 1990. \]

2022 Romania National Olympiad, P4

Tags: algebra , combinatorics , functional equation

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]