Olympiad problems

1976 Bulgaria National Olympiad, Problem 2

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

2005 Federal Math Competition of S&M, Problem 3

Tags: functional equation , fe , algebra , polynomial

Determine all polynomials $p$ with real coefficients for which $p(0)=0$ and $$f(f(n))+n=4f(n)\qquad\text{for all }n\in\mathbb N,$$where $f(n)=\lfloor p(n)\rfloor$.

2001 Moldova National Olympiad, Problem 3

Tags: functional equation , fe , algebra , polynomial

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 SEEMOUS, Problem 3

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

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 $X,Y\in\mathcal M_n(\mathbb R)$.

2018-IMOC, A3

Tags: fe , functional equation , algebra

Find all functions $f:\mathbb R\to\mathbb R$ such that for reals $x,y$, $$f(xf(y)+y)=yf(x)+f(y).$$

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]

2017-IMOC, N5

Tags: number theory , fe , functional equation

Find all functions $f:\mathbb N\to\mathbb N$ such that $$f(x)+f(y)\mid x^2-y^2$$holds for all $x,y\in\mathbb N$.

1998 Slovenia National Olympiad, Problem 2

Tags: fe , functional equation , polynomial

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

2018 China Team Selection Test, 4

Tags: function , algebra , functional equation , fe

Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.

2019 Romanian Master of Mathematics, 5

Tags: fe , algebra , functional equation

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 $y$.

2000 Slovenia National Olympiad, Problem 2

Tags: functional equation , fe

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

2014 BMT Spring, 8

Tags: functional equation , fe , algebra

Suppose an integer-valued function $f$ satisfies $$\sum_{k=1}^{2n+1}f(k)=\ln|2n+1|-4\ln|2n-1|\enspace\text{and}\enspace\sum_{k=0}^{2n}f(k)=4e^n-e^{n-1}$$ for all non-negative integers $n$. Determine $\sum_{n=0}^\infty\frac{f(n)}{2^n}$.

2025 Nordic, 1

Tags: algebra , function , fe , functional equation

Let $n$ be a positive integer greater than $2$. Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying: $(f(x+y))^{n} = f(x^{n})+f(y^{n}),$ for all integers $x,y$

2023 Princeton University Math Competition, 1

Tags: fe , algebra

1. Given $n \geq 1$, let $A_{n}$ denote the set of the first $n$ positive integers. We say that a bijection $f: A_{n} \rightarrow A_{n}$ has a hump at $m \in A_{n} \backslash\{1, n\}$ if $f(m)>f(m+1)$ and $f(m)>f(m-1)$. We say that $f$ has a hump at 1 if $f(1)>f(2)$, and $f$ has a hump at $n$ if $f(n)>f(n-1)$. Let $P_{n}$ be the probability that a bijection $f: A_{n} \rightarrow A_{n}$, when selected uniformly at random, has exactly one hump. For how many positive integers $n \leq 2020$ is $P_{n}$ expressible as a unit fraction?

1988 Bulgaria National Olympiad, Problem 6

Tags: fe , functional equation , algebra , polynomial

Find all polynomials $p(x)$ satisfying $p(x^3+1)=p(x+1)^3$ for all $x$.

2000 Moldova National Olympiad, Problem 4

Tags: functional equation , fe , polynomial , algebra

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

2011 VJIMC, Problem 4

Tags: fe , functional equation , irreducibility , algebra , polynomial

Find all $\mathbb Q$-linear maps $\Phi:\mathbb Q[x]\to\mathbb Q[x]$ such that for any irreducible polynomial $p\in\mathbb Q[x]$ the polynomial $\Phi(p)$ is also irreducible.

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

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 $y$.

1997 Croatia National Olympiad, Problem 3

Tags: fe , number theory , algebra , functional equation

Function $f$ is defined on the positive integers by $f(1)=1$, $f(2)=2$ and $$f(n+2)=f(n+2-f(n+1))+f(n+1-f(n))\enspace\text{for }n\ge1.$$ (a) Prove that $f(n+1)-f(n)\in\{0,1\}$ for each $n\ge1$. (b) Show that if $f(n)$ is odd then $f(n+1)=f(n)+1$. (c) For each positive integer $k$ find all $n$ for which $f(n)=2^{k-1}+1$.

2005 VJIMC, Problem 3

Tags: fe , algebra , polynomial , functional equation

Find all reals $\lambda$ for which there is a nonzero polynomial $P$ with real coefficients such that $$\frac{P(1)+P(3)+P(5)+\ldots+P(2n-1)}n=\lambda P(n)$$for all positive integers $n$, and find all such polynomials for $\lambda=2$.

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.

1987 IMO, 1

Tags: function , functional equation , fe , algebra

Prove that there is no function $f$ from the set of non-negative integers into itself such that $f(f(n))=n+1987$ for all $n$.

2010 Contests, 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 $d(f(A),f(B))=n$.