Olympiad problems

1999 Mongolian Mathematical Olympiad, Problem 2

Tags: functional equation , fe

Find all functions $f:\mathbb R\to\mathbb R$ such that (i) $f(0)=1$; (ii) $f(x+f(y))=f(x+y)+1$ for all real $x,y$; (iii) there is a rational non-integer $x_0$ such that $f(x_0)$ is an integer.

2001 Moldova National Olympiad, Problem 3

Tags: polynomial , functional equation , fe , algebra

Find all polynomials $P(x)$ with real coefficieints such that $P\left(x^2\right)=P(x)P(x-1)$ for all $x\in\mathbb R$.

2014 BMT Spring, 6

Tags: fe , functional equation , algebra

Find $f(2)$ given that $f$ is a real-valued function that satisfies the equation $$4f(x)+\left(\frac23\right)(x^2+2)f\left(x-\frac2x\right)=x^3+1.$$

2000 Slovenia National Olympiad, Problem 2

Tags: functional equation , fe

Find all functions $f:\mathbb R\to\mathbb R$ such that for all $x,y\in\mathbb R$, $$f(x-f(y))=1-x-y.$$

2015 VTRMC, Problem 6

Tags: functional equation , fe , absolute value

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.

2010 IMO Shortlist, 1

Tags: algebra , functional equation , fe

Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$ [i]Proposed by Pierre Bornsztein, France[/i]

1973 Bulgaria National Olympiad, Problem 4

Tags: trigonometry , function , fe , functional equation , algebra

Find all functions $f(x)$ defined in the range $\left(-\frac\pi2,\frac\pi2\right)$ that are differentiable at $0$ and satisfy $$f(x)=\frac12\left(1+\frac1{\cos x}\right)f\left(\frac x2\right)$$ for every $x$ in the range $\left(-\frac\pi2,\frac\pi2\right)$. [i]L. Davidov[/i]

2022 Indonesia TST, A

Tags: function , algebra , fe , inequalities

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying \[ f(a^2) - f(b^2) \leq (f(a)+b)(a-f(b)) \] for all $a,b \in \mathbb{R}$.

2019 Brazil Team Selection Test, 5

Tags: wrapped , fe , algebra

Determine all the functions $f : \mathbb{R} \to \mathbb{R}$ such that \[ f(x^2 + f(y)) = f(f(x)) + f(y^2) + 2f(xy) \] for all real numbers $x$ and $y$.

1998 Slovenia National Olympiad, Problem 2

Tags: polynomial , fe , functional equation

Find all polynomials $p$ with real coefficients such that for all real $x$ $$(x-8)p(2x)=8(x-1)p(x).$$

2024 Iran MO (3rd Round), 1

Tags: algebra , complex number , fe , function , functional equation

Suppose that $T\in \mathbb N$ is given. Find all functions $f:\mathbb Z \to \mathbb C$ such that, for all $m\in \mathbb Z$ we have $f(m+T)=f(m)$ and: $$\forall a,b,c \in \mathbb Z: f(a)\overline{f(a+b)f(a+c)}f(a+b+c)=1.$$ Where $\overline{a}$ is the complex conjugate of $a$.

1994 French Mathematical Olympiad, Problem 5

Tags: fe , functional equation , algebra

Assume $f:\mathbb N_0\to\mathbb N_0$ is a function such that $f(1)>0$ and, for any nonnegative integers $m$ and $n$, $$f\left(m^2+n^2\right)=f(m)^2+f(n)^2.$$(a) Calculate $f(k)$ for $0\le k\le12$. (b) Calculate $f(n)$ for any natural number $n$.

2008 VJIMC, Problem 2

Tags: fe , functional equation , recurrence relation , algebra , sequence

Find all functions $f:(0,\infty)\to(0,\infty)$ such that $$f(f(f(x)))+4f(f(x))+f(x)=6x.$$

2018-IMOC, N2

Tags: number theory , least common multiple , greatest common divisor , fe , functional equation

Find all functions $f:\mathbb N\to\mathbb N$ satisfying $$\operatorname{lcm}(f(x),y)\gcd(f(x),f(y))=f(x)f(f(y))$$ for all $x,y\in\mathbb N$.

2010 IMO, 1

Tags: algebra , functional equation , fe

Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$ [i]Proposed by Pierre Bornsztein, France[/i]

1999 Mongolian Mathematical Olympiad, Problem 6

Tags: functional-geometry , fe , functional equation , geometry

Let $f$ be a map of the plane into itself with the property that if $d(A,B)=1$, then $d(f(A),f(B))=1$, where $d(X,Y)$ denotes the distance between points $X$ and $Y$. Prove that for any positive integer $n$, $d(A,B)=n$ implies $d(f(A),f(B))=n$.