Olympiad problems

2020 Canadian Junior Mathematical Olympiad, 3

Tags: algebra , Canada , P1

There are $n \ge 3$ distinct positive real numbers. Show that there are at most $n-2$ different integer power of three that can be written as the sum of three distinct elements from these $n$ numbers.

2022 Middle European Mathematical Olympiad, 1

Tags: functional equation , 2022 , P1 , Reals , Surjective , memo , MEMO 2022

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 $y$.

2020 DMO Stage 1, 1.

Tags: inequalities , DMO DAY 2 , P1

[b]Q[/b] Let $p,q,r$ be non negative reals such that $pqr=1$. Find the maximum value for the expression $$\sum_{cyc} p[r^{4}+q^{4}-p^{4}-p]$$ [i]Proposed by Aritra12[/i]

2020 Canada National Olympiad, 1

Tags: algebra , Canada , P1

There are $n \ge 3$ distinct positive real numbers. Show that there are at most $n-2$ different integer power of three that can be written as the sum of three distinct elements from these $n$ numbers.