Olympiad problems

2019 IFYM, Sozopol, 7

The function $f: \mathbb{R}\rightarrow \mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\forall$ $x\in \mathbb{R}$ and $f(x)=x(x-1)$ for $\forall$ $x\in (0,1]$. Find the greatest real number $m$, for which the inequality $f(x)\geq -\frac{8}{9}$ is true for $\forall$ $x\in (-\infty , m]$.

2020 Final Mathematical Cup, 1

Tags: functional equation , algebra , functional equation in r

Find all such functions $f:\mathbb{R} \to \mathbb{R}$ that for any real $x,y$ the following equation is true. $$f(f(x)+y)+1=f(x^2+y)+2f(x)+2y$$

2008 Canada National Olympiad, 2

Tags: algebra , function , functional equation

Determine all functions $ f$ defined on the set of rational numbers that take rational values for which \[ f(2f(x) \plus{} f(y)) \equal{} 2x \plus{} y, \] for each $ x$ and $ y$.

1964 Dutch Mathematical Olympiad, 4

Tags: functional equation , algebra , function

The function $ƒ$ is defined at $[0,1]$, and $f\{f(x)\} = ƒ(x)$. $\exists _{c\in [0,1]} \left[f(c) =\frac12 \right]$ Determine $f\left(\frac12 \right).$ $\forall _{t\in [0,1]}\exists _{s\in [0,1]}[f(s) = t]$. Determine $f$. Prove that the function $g$, with $g(x) = x$,$0 \le x \le k$, $g(x) = k$, $k \le x \le 1$ satisfies the relation $g\{g(x)\} = g(x)$.

2018 ELMO Shortlist, 1

Tags: functional equation , algebra , function

Let $f:\mathbb{R}\to\mathbb{R}$ be a bijective function. Does there always exist an infinite number of functions $g:\mathbb{R}\to\mathbb{R}$ such that $f(g(x))=g(f(x))$ for all $x\in\mathbb{R}$? [i]Proposed by Daniel Liu[/i]

2021-IMOC, N8

Tags: functional equation , algebra

Find all integer-valued polynomials $$f, g : \mathbb{N} \rightarrow \mathbb{N} \text{ such that} \; \forall \; x \in \mathbb{N}, \tau (f(x)) = g(x)$$ holds for all positive integer $x$, where $\tau (x)$ is the number of positive factors of $x$ [i]Proposed By - ckliao914[/i]

1947 Putnam, A2

Tags: functional equation , algebra , function

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 $$f(x) =f(1)^{x^{2}}.$$

STEMS 2021-22 Math Cat A-B, A3 B1

Tags: functional equation , algebra

Find all functions $f :\mathbb{N} \rightarrow \mathbb{N}$ such that $f(m + f(n)f(m)) = nf(m) + m$ holds for all $m,n \in \mathbb{N}$.

2023 Federal Competition For Advanced Students, P2, 1

Tags: functional equation , algebra

Given is a nonzero real number $\alpha$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy$$ for all $x, y \in \mathbb{R}$.

Gheorghe Țițeica 2025, P3

Tags: algebra , functional equation

Find all functions $ f: \mathbb{R}^{ \plus{} }\to\mathbb{R}^{ \plus{} }$ satisfying $ f\left(x \plus{} f\left(y\right)\right) \equal{} f\left(x \plus{} y\right) \plus{} f\left(y\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \mathbb{R}^{ \plus{} }$ denotes the set of all positive reals. [i]Proposed by Paisan Nakmahachalasint, Thailand[/i]

2021 Korea - Final Round, P6

Tags: functional equation , function

Find all functions $f,g: \mathbb{R} \to \mathbb{R}$ such that satisfies $$f(x^2-g(y))=g(x)^2-y$$ for all $x,y \in \mathbb{R}$

2020 Kazakhstan National Olympiad, 2

Tags: algebra , functional equation

Find all functions $ f: \mathbb {R} ^ + \to \mathbb {R} ^ + $ such that for any $ x, y \in \mathbb {R} ^ + $ the following equality holds: \[f (x) f (y) = f \left (\frac {xy} {x f (x) + y} \right). \] $ \mathbb {R} ^ + $ denotes the set of positive real numbers.

2024 IFYM, Sozopol, 3

Tags: algebra , functional equation

Find all functions $ f:\mathbb{Z} \to \mathbb{Z} $ such that \[ f(x + f(y) - 2y) + f(f(y)) = f(x) \] for all integers $ x $ and $ y $.

2019 Romanian Masters In Mathematics, 5

Tags: functional equation , fe , algebra

Determine all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying \[f(x + yf(x)) + f(xy) = f(x) + f(2019y),\] for all real numbers $x$ and $y$.

1998 Belarusian National Olympiad, 8

Tags: algebra , functional , functional equation

a) Prove that for no real a such that $0 \le a <1$ there exists a function defined on the set of all positive numbers and taking values in the same set, satisfying for all positive $x$ the equality $$f\left(f(x)+\frac{1}{f(x)}\right)=x+a \,\,\,\,\,\,\, (*) $$ b) Prove that for any $a>1$ there are infinitely many functions defined on the set of all positive numbers, with values in the same set, satisfying ($*$) for all positive x.

2020-IMOC, A2

Tags: functional equation , algebra , function

Find all function $f:\mathbb{R}^+$ $\rightarrow \mathbb{R}^+$ such that: $f(f(x) + y)f(x) = f(xy + 1) \forall x, y \in \mathbb{R}^+$

2017 Ukraine Team Selection Test, 7

Tags: functional equation , number theory , sum of digits , polynomial

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

2015 Baltic Way, 5

Tags: functional equation , algebra

Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying the equation \[|x|f(y)+yf(x)=f(xy)+f(x^2)+f(f(y))\] for all real numbers $x$ and $y$.

2009 IMO, 5

Tags: triangle inequality , induction , functional equation , function

Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths \[ a, f(b) \text{ and } f(b \plus{} f(a) \minus{} 1).\] (A triangle is non-degenerate if its vertices are not collinear.) [i]Proposed by Bruno Le Floch, France[/i]

2021 Peru IMO TST, P3

Tags: functional equation , algebra

Suppose the function $f:[1,+\infty)\to[1,+\infty)$ satisfies the following two conditions: (i) $f(f(x))=x^2$ for any $x\geq 1$; (ii) $f(x)\leq x^2+2021x$ for any $x\geq 1$. 1. Prove that $x<f(x)<x^2$ for any $x\geq 1$. 2. Prove that there exists a function $f$ satisfies the above two conditions and the following one: (iii) There are no real constants $c$ and $A$, such that $0<c<1$, and $\frac{f(x)}{x^2}<c$ for any $x>A$.

2004 India IMO Training Camp, 3

Tags: algebra , system of equations , trigonometry , functional equation , function

Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that \[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.

2018 ELMO Shortlist, 1

Tags: functional equation , function , algebra

Let $f:\mathbb{R}\to\mathbb{R}$ be a bijective function. Does there always exist an infinite number of functions $g:\mathbb{R}\to\mathbb{R}$ such that $f(g(x))=g(f(x))$ for all $x\in\mathbb{R}$? [i]Proposed by Daniel Liu[/i]

2013 QEDMO 13th or 12th, 4

Tags: functional , algebra , periodic , functional equation

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 .

2020 Thailand TSTST, 1

Tags: functional equation , algebra

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(\max \left\{ x, y \right\} + \min \left\{ f(x), f(y) \right\}) = x+y $$ for all $x,y \in \mathbb{R}$.

2013 Chile National Olympiad, 4

Tags: functional equation , functional , number theory , algebra

Consider a function f defined on the positive integers that meets the following conditions: $$f(1) = 1 \, , \,\, f(2n) = 2f(n) \, , \,\, nf(2n + 1) = (2n + 1)(f(n) + n) $$ for all $n \ge 1$. a) Prove that $f(n)$ is an integer for all $n$. b) Find all positive integers $m$ less than $2013$ that satisfy the equation $f(m) = 2m$.