Olympiad problems

2000 Singapore Team Selection Test, 1

Find all functions $f : R \to R$ such for any $x, y \in R,$ $$(x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2)$$

1987 Austrian-Polish Competition, 3

Tags: functional , limit , functional equation , sequence

A function $f: R \to R$ satisfies $f (x + 1) = f (x) + 1$ for all $x$. Given $a \in R$, define the sequence $(x_n)$ recursively by $x_0 = a$ and $x_{n+1} = f (x_n)$ for $n \ge 0$. Suppose that, for some positive integer m, the difference $x_m - x_0 = k$ is an integer. Prove that the limit $\lim_{n\to \infty}\frac{x_n}{n}$ exists and determine its value.

1999 Denmark MO - Mohr Contest, 3

Tags: functional equation , functional , algebra

A function $f$ satisfies $$f(x)+xf(1-x)=x$$ for all real numbers $x$. Determine the number $f (2)$. Find $f$ .

1995 North Macedonia National Olympiad, 5

Tags: functional , function , algebra , functional equation

Let $ a, b, c, d \in \mathbb {R}, $ $ b \neq0. $ Find the functions of the $ f: \mathbb{R} \to \mathbb{R} $ such that $ f (x + d \cdot f (y)) = ax + by + c, $ for all $ x, y \in \mathbb{R}. $

1986 Tournament Of Towns, (116) 4

Tags: functional equation , functional , algebra , function , continuous , analysis

The function $F$ , defined on the entire real line, satisfies the following relation (for all $x$ ) : $F(x +1 )F(x) + F(x + 1 ) + 1 = 0$ . Prove that $F$ is not continuous. (A.I. Plotkin, Leningrad)

I Soros Olympiad 1994-95 (Rus + Ukr), 11.5

Tags: functional equation , functional , algebra

Is there a function $f(x)$ defined for all $x$ and such that for some $a$ and all $x$ holds the equality $$f(x) + f(2x^2 - 1) = 2x + a?$$

1993 Bulgaria National Olympiad, 1

Tags: algebra , functional equation , functional

Find all functions $f$ , defined and having values in the set of integer numbers, for which the following conditions are satisfied: (a) $f(1) = 1$; (b) for every two whole (integer) numbers $m$ and $n$, the following equality is satisfied: $$f(m+n)·(f(m)-f(n)) = f(m-n)·(f(m)+ f(n))$$

2024 Moldova EGMO TST, 11

Tags: functional

Find all functions $f$ from the positive integers to the positive integers such that such that for all integers $x, y$ we have $$2yf(f(x^2)+x)=f(x+1)f(2xy).$$

1979 Kurschak Competition, 2

Tags: functional , algebra , functional inequality , function

$f$ is a real-valued function defined on the reals such that $f(x) \le x$ and $f(x + y) \le f(x) + f(y)$ for all $x, y$. Prove that $f(x) = x$ for all $x$.

2022 Dutch BxMO TST, 1

Tags: functional , number theory

Find all functions $f : Z_{>0} \to Z_{>0}$ for which $f(n) | f(m) - n$ if and only if $n | m$ for all natural numbers $m$ and $n$.

2022 Dutch BxMO TST, 1

Tags: number theory , functional

Find all functions $f : Z_{>0} \to Z_{>0}$ for which $f(n) | f(m) - n$ if and only if $n | m$ for all natural numbers $m$ and $n$.

2011 Belarus Team Selection Test, 3

Tags: functional equation , algebra , functional

Find all functions $f: R \to R ,g: R \to R$ satisfying the following equality $f(f(x+y))=xf(y)+g(x)$ for all real $x$ and $y$. I. Gorodnin

1995 Austrian-Polish Competition, 4

Tags: polynomial , functional equation , algebra , functional

Determine all polynomials $P(x)$ with real coefficients such that $P(x)^2 + P\left(\frac{1}{x}\right)^2= P(x^2)P\left(\frac{1}{x^2}\right)$ for all $x$.

2004 Thailand Mathematical Olympiad, 2

Tags: functional equation , algebra , functional

Let $f : Q \to Q$ be a function satisfying the equation $f(x + y) = f(x) + f(y) + 2547$ for all rational numbers $x, y$. If $f(2004) = 2547$, find $f(2547)$.

2011 Indonesia TST, 1

Tags: functional , algebra , functional equation , functional equation in q

Let $Q^+$ denote the set of positive rationals. Determine all functions $f : Q^+ \to Q^+$ that satisfy both of these conditions: (i) $f(x)$ is an integer if and only if $x$ is an integer; (ii) $f(f(xf(y)) + x) = yf(x) + x$ for all $x, y \in Q^+$.

2009 Belarus Team Selection Test, 3

Tags: functional equation , algebra , functional

Find all real numbers $a$ for which there exists a function $f: R \to R$ asuch that $x + f(y) =a(y + f(x))$ for all real numbers $x,y\in R$. I.Voronovich

2000 Switzerland Team Selection Test, 12

Tags: functional equation , functional , algebra

Find all functions $f : R \to R$ such that for all real $x,y$, $f(f(x)+y) = f(x^2 -y)+4y f(x)$

2005 Slovenia Team Selection Test, 2

Tags: algebra , functional , functional equation

Find all functions $f : R^+ \to R^+$ such that $x^2(f(x)+ f(y)) = (x+y)f (f(x)y)$ for any $x,y > 0$.

2020 Dutch IMO TST, 3

Tags: functional equation , functional equation in z , functional , algebra

Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$ for all $x, y \in Z$

2005 iTest, 4

Tags: function , algebra , functional equation , functional

The function f is defined on the set of integers and satisfies $\bullet$ $f(n) = n - 2$, if $n \ge 2005$ $\bullet$ $f(n) = f(f(n+7))$, if $n < 2005$. Find $f(3)$.

2021 Ukraine National Mathematical Olympiad, 4

Tags: functional , algebra , functional equation

Find all the following functions $f:R\to R$ , which for arbitrary valid $x,y$ holds equality: $$f(xf(x+y))+f((x+y)f(y))=(x+y)^2$$ (Vadym Koval)

2017 QEDMO 15th, 4

Tags: functional equation , functional , algebra

Find all functions $f: R \to R$ for which the image $f ([a, b])$ for all real $a \le b$ is (not necessarily closed!) interval of length $b - a$.

2020 Estonia Team Selection Test, 3

Tags: functional equation , functional , algebra

Find all functions $f :R \to R$ such that for all real numbers $x$ and $y$ $$f(x^3+y^3)=f(x^3)+3x^3f(x)f(y)+3f(x)(f(y))^2+y^6f(y)$$

2010 Mathcenter Contest, 4

Tags: geometry , functional , convex

Let $P$ be a plane. Prove that there is no function $f :P\rightarrow P$ where, for any convex quadrilateral $ABCD$, the points $f(A)$, $f(B)$, $f(C)$, $f (D)$ are the vertices of a concave quadrilateral. [i](tatari/nightmare)[/i]

2022 Switzerland - Final Round, 3

Tags: algebra , functional equation , functional

Let $N$ be the set of positive integers. Find all functions $f : N \to N$ such that both $\bullet$ $f(f(m)f(n)) = mn$ $\bullet$ $f(2022a + 1) = 2022a + 1$ hold for all positive integers $m, n$ and $a$.