Olympiad problems

2007 Canada National Olympiad, 3

Suppose that $ f$ is a real-valued function for which \[ f(xy)+f(y-x)\geq f(y+x)\] for all real numbers $ x$ and $ y$. a) Give a non-constant polynomial that satisfies the condition. b) Prove that $ f(x)\geq 0$ for all real $ x$.

2011 Abels Math Contest (Norwegian MO), 3b

Tags: functional , algebra , function , functional inequalities , inequalities

Find all functions $f$ from the real numbers to the real numbers such that $f(xy) \le \frac12 \left(f(x) + f(y) \right)$ for all real numbers $x$ and $y$.

2022 German National Olympiad, 6

Tags: function , algebra , functional equation , functional inequality , functional inequalities

Consider functions $f$ satisfying the following four conditions: (1) $f$ is real-valued and defined for all real numbers. (2) For any two real numbers $x$ and $y$ we have $f(xy)=f(x)f(y)$. (3) For any two real numbers $x$ and $y$ we have $f(x+y) \le 2(f(x)+f(y))$. (4) We have $f(2)=4$. Prove that: a) There is a function $f$ with $f(3)=9$ satisfying the four conditions. b) For any function $f$ satisfying the four conditions, we have $f(3) \le 9$.