Olympiad problems

1986 Traian Lălescu, 2.2

We know that the function $ f: \left[ 0,\frac{\pi }{2}\right]\longrightarrow [a,b], f(x)=\sqrt[n]{\cos x } +\sqrt[n]{\sin x} , $ is surjective for a given natural number $ n\ge 2. $ Determine the numbers $ a,b, $ and the monotony of $ f. $

2011 Miklós Schweitzer, 4

Tags: abstract algebra , surjective , group theory

Let G, H be two finite groups, and let $\varphi, \psi: G \to H$ be two surjective but non-injective homomorphisms. Prove that there exists an element of G that is not the identity element of G but whose images under $\varphi$ and $\psi$ are the same.

2003 Belarusian National Olympiad, 7

Tags: surjective , algebra , function

Does there exist a surjective function $f:R \to R$ such that the expression $f(x + y) - f(x) - f(y)$ takes exactly two values $0$ and $1$ for various real $x$ and $y$ ? (E. Barabanov)

2014 APMO, 3

Tags: divisibility , hensel , surjective , quadratic residue

Find all positive integers $n$ such that for any integer $k$ there exists an integer $a$ for which $a^3+a-k$ is divisible by $n$. [i]Warut Suksompong, Thailand[/i]

2022 Middle European Mathematical Olympiad, 1

Tags: functional equation , real , surjective

Find all functions $f: \mathbb R \to \mathbb R$ such that $$f(x+f(x+y))=x+f(f(x)+y)$$ holds for all real numbers $x$ and $y$.

2014 Contests, 3

Tags: quadratic residue , divisibility , hensel , surjective

Find all positive integers $n$ such that for any integer $k$ there exists an integer $a$ for which $a^3+a-k$ is divisible by $n$. [i]Warut Suksompong, Thailand[/i]