Olympiad problems

2015 VTRMC, Problem 6

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.

2017-IMOC, A5

Tags: algebra , functional equation , fe

Find all functions $f:\mathbb Z\to\mathbb Z$ such that $$f(mf(n+1))=f(m+1)f(n)+f(f(n))+1$$for all integer pairs $(m,n)$.

1993 French Mathematical Olympiad, Problem 3

Tags: function , algebra , functional equation , fe

Let $f$ be a function from $\mathbb Z$ to $\mathbb R$ which is bounded from above and satisfies $f(n)\le\frac12(f(n-1)+f(n+1))$ for all $n$. Show that $f$ is constant.

2024 Iran MO (3rd Round), 1

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

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 $a$.

2008 SEEMOUS, Problem 3

Tags: linear algebra , inequalities , functional equation , fe , matrix

Let $\mathcal M_n(\mathbb R)$ denote the set of all real $n\times n$ matrices. Find all surjective functions $f:\mathcal M_n(\mathbb R)\to\{0,1,\ldots,n\}$ which satisfy $$f(XY)\le\min\{f(X),f(Y)\}$$for all $X,Y\in\mathcal M_n(\mathbb R)$.

2017-IMOC, A2

Tags: algebra , number theory , functional equation , fe

Find all functions $f:\mathbb N\to\mathbb N$ such that \begin{align*} x+f(y)&\mid f(y+f(x))\\ f(x)-2017&\mid x-2017\end{align*}

2000 Brazil Team Selection Test, 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.

2000 Moldova National Olympiad, Problem 4

Tags: functional equation , polynomial , fe , algebra

Find all polynomials $P(x)$ with real coefficients that satisfy the relation $$1+P(x)=\frac{P(x-1)+P(x+1)}2.$$

2018-IMOC, N1

Tags: number theory , fe , functional equation

Find all functions $f:\mathbb N\to\mathbb N$ satisfying $$x+f^{f(x)}(y)\mid2(x+y)$$for all $x,y\in\mathbb N$.

2018-IMOC, N2

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

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 $x,y\in\mathbb N$.

2019 Brazil Team Selection Test, 5

Tags: algebra , wrapped , fe

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 $y$.

2010 Contests, 1

Tags: functional equation , algebra , 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]

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 $x\in\mathbb R$.

2008 VJIMC, Problem 2

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

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

1973 Bulgaria National Olympiad, Problem 4

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

2014 Contests, 1

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

Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[ xf(y) + f(xf(y)) - xf(f(y)) - f(xy) = 2x + f(y) - f(x+y)\] holds for all $x,y \in \mathbb{R}$.