Olympiad problems

2021 JHMT HS, 4

There is a unique differentiable function $f$ from $\mathbb{R}$ to $\mathbb{R}$ satisfying $f(x) + (f(x))^3 = x + x^7$ for all real $x.$ The derivative of $f(x)$ at $x = 2$ can be expressed as a common fraction $a/b.$ Compute $a + b.$

1998 AMC 12/AHSME, 17

Tags: function , algebra , functional equation , geometry , 3d geometry

Let $ f(x)$ be a function with the two properties: [list=a] [*] for any two real numbers $ x$ and $ y$, $ f(x \plus{} y) \equal{} x \plus{} f(y)$, and [*] $ f(0) \equal{} 2$ [/list] What is the value of $ f(1998)$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 1996\qquad \textbf{(D)}\ 1998\qquad \textbf{(E)}\ 2000$

2018 India IMO Training Camp, 3

Tags: algebra , functional equation

Find all functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that $$f(x)f\left(yf(x)-1\right)=x^2f(y)-f(x),$$for all $x,y \in \mathbb{R}$.

2022-IMOC, A5

Tags: algebra , functional equation

Find all functions $f:\mathbb R\to \mathbb R$ such that \begin{align*} \left (x \left (f(x)-\dfrac{f(y)+f(z)}{2} \right) +y \left (f(y)-\dfrac{f(z)+f(x)}{2} \right ) +z\left (f(z)- \dfrac{f(x)+f(y)}{2} \right) \right )f(x+y+z)= \\ f(x^3)+f(y^3)+f(z^3)-3f(xyz) \end{align*} for all $x,y,z\in \mathbb R.$

1976 Bulgaria National Olympiad, Problem 2

Tags: fe , functional equation , algebra , polynomial

Find all polynomials $p(x)$ satisfying the condition: $$p(x^2-2x)=p(x-2)^2.$$

1986 Traian Lălescu, 2.1

Tags: calculus , integration , function , real analysis , algebra , functional equation , primitive

Let be a nonnegative integer $ n. $ Find all continuous functions $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ for which the following equation holds: $$ (1+n)\int_0^x f(t) dt =nxf(x) ,\quad\forall x>0. $$

2007 Cuba MO, 4

Tags: functional , functional equation , algebra

Find all functions $f : R_+ \to R_+$ such that $$x^2(f(x)+f(y)) = (x+y)f(f(x)y)$$ for all positive real $x, y$.

2024 Macedonian Mathematical Olympiad, Problem 3

Tags: algebra , functional equation , function

Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfy the equation $$f(f(x+y))=f(x+y)+f(x)f(y)-xy,$$ for any two real numbers $x$ and $y$.

2023 Korea Summer Program Practice Test, P2

Tags: functional equation , function , algebra

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

2008 IMO Shortlist, 4

Tags: function , algebra , functional equation , inequality

For an integer $ m$, denote by $ t(m)$ the unique number in $ \{1, 2, 3\}$ such that $ m \plus{} t(m)$ is a multiple of $ 3$. A function $ f: \mathbb{Z}\to\mathbb{Z}$ satisfies $ f( \minus{} 1) \equal{} 0$, $ f(0) \equal{} 1$, $ f(1) \equal{} \minus{} 1$ and $ f\left(2^{n} \plus{} m\right) \equal{} f\left(2^n \minus{} t(m)\right) \minus{} f(m)$ for all integers $ m$, $ n\ge 0$ with $ 2^n > m$. Prove that $ f(3p)\ge 0$ holds for all integers $ p\ge 0$. [i]Proposed by Gerhard Woeginger, Austria[/i]

2017 Czech And Slovak Olympiad III A, 3

Tags: functional equation , functional , algebra

Find all functions $f: R \to R$ such that for all real numbers $x, y$ holds $f(y - xy) = f(x)y + (x - 1)^2 f(y)$

2020 Iranian Our MO, 6

Tags: functional equation , algebra , polynomial

Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ and plynomials $P(x),Q(x),R(x)$ with positive real coefficients such that $Q(-1)=-1$ and for all positive reals $x,y$:$$f(\frac{x}{y}+R(y))=\frac{f(x)}{Q(y)}+P(y).$$ [i]Proposed by Alireza Danaie, Ali Mirazaie Anari[/i] [b]Rated 2[/b]

2019 Dutch IMO TST, 4

Tags: number theory , function , find all functions , prime number , functional equation

Find all functions $f : Z \to Z$ satisfying $\bullet$ $ f(p) > 0$ for all prime numbers $p$, $\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.

2002 IMO, 5

Tags: functional equation , algebra

Find all functions $f$ from the reals to the reals such that \[ \left(f(x)+f(z)\right)\left(f(y)+f(t)\right)=f(xy-zt)+f(xt+yz) \] for all real $x,y,z,t$.

1990 Romania Team Selection Test, 3

Tags: polynomial , algebra , functional , functional equation

Find all polynomials $P(x)$ such that $2P(2x^2 -1) = P(x)^2 -1$ for all $x$.

2024 Taiwan TST Round 2, A

Tags: functional equation , algebra

Let $\mathbb{R}_+$ be the set of positive real numbers. Find all functions $f\colon \mathbb{R}_+ \to \mathbb{R}_+$ such that \[f(xy + x + y) + f \left( \frac1x \right) f\left( \frac1y \right) = 1\] for every $x$, $y\in \mathbb{R}_+$. [i]Proposed by Li4 and Untro368.[/i]

2005 India National Olympiad, 6

Tags: function , functional equation , algebra

Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that \[ f(x^2 + yf(z)) = xf(x) + zf(y) , \] for all $x, y, z \in \mathbb{R}$.

2025 District Olympiad, P3

Tags: function , complex number , functional equation , algebra

Determine all functions $f:\mathbb{C}\rightarrow\mathbb{C}$ such that $$|wf(z)+zf(w)|=2|zw|$$ for all $w,z\in\mathbb{C}$.

2020 IMO Shortlist, A8

Tags: functional equation , algebra

Let $R^+$ be the set of positive real numbers. Determine all functions $f:R^+$ $\rightarrow$ $R^+$ such that for all positive real numbers $x$ and $y:$ \[f(x+f(xy))+y=f(x)f(y)+1\] [i]Ukraine[/i]

2019 ELMO Shortlist, A4

Tags: algebra , functional equation

Find all nondecreasing functions $f:\mathbb R\to \mathbb R$ such that, for all $x,y\in \mathbb R$, $$f(f(x))+f(y)=f(x+f(y))+1.$$ [i]Proposed by Carl Schildkraut[/i]

1983 IMO Shortlist, 11

Tags: algebra , functional equation , continuous function

Let $f : [0, 1] \to \mathbb R$ be continuous and satisfy: \[ \begin{cases}bf(2x) = f(x), &\mbox{ if } 0 \leq x \leq 1/2,\\ f(x) = b + (1 - b)f(2x - 1), &\mbox{ if } 1/2 \leq x \leq 1,\end{cases}\] where $b = \frac{1+c}{2+c}$, $c > 0$. Show that $0 < f(x)-x < c$ for every $x, 0 < x < 1.$

1995 China National Olympiad, 2

Tags: function , algebra , functional equation

Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions: (1) $f(1)=1$; (2) $\forall n\in \mathbb{N}$, $3f(n) f(2n+1) =f(2n) ( 1+3f(n) )$; (3) $\forall n\in \mathbb{N}$, $f(2n) < 6 f(n)$. Find all solutions of equation $f(k) +f(l)=293$, where $k<l$. ($\mathbb{N}$ denotes the set of all natural numbers).

2014 Federal Competition For Advanced Students, P2, 2

Tags: functional equation , algebra

Let $S$ be the set of all real numbers greater than or equal to $1$. Determine all functions$ f: S \to S$, so that for all real numbers $x ,y \in S$ with $x^2 -y^2 \in S$ the condition $f (x^2 -y^2) = f (xy)$ is fulfilled.

2010 Brazil National Olympiad, 1

Tags: function , vieta , algebra , functional equation

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

2012 Balkan MO, 4

Tags: function , inequalities , functional equation

Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold: (i) $f(n!)=f(n)!$ for every positive integer $n$, (ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.