Problem

Source: Bulgarian TST 2007 for Balkan MO and ARO, I day Problem 2

Tags: inequalities, function, algebra proposed, algebra



Find all $a\in\mathbb{R}$ for which there exists a non-constant function $f: (0,1]\rightarrow\mathbb{R}$ such that \[a+f(x+y-xy)+f(x)f(y)\leq f(x)+f(y)\] for all $x,y\in(0,1].$