Com10atorics wrote: Find all functions $f:R\rightarrow R$ such that for all real numbers $x,y$: $f(x)f(y)+f(xy)\leq x+y$. Let $P(x,y)$ be the assertion $f(x)f(y)+f(xy)\le x+y$ $P(-1,-1)$ $\implies$ $f(-1)^2+f(1)\le -2$ and so $f(1)\le -2$ $P(1,1)$ $\implies$ $f(1)^2+f(1)\le 2$ and so $f(1)\in[-2,1]$ And so $f(1)=-2$ and first line gives $f(-1)=0$ $P(x,1)$ $\implies$ $-=f(x)\ge -x-1$ $P(-x,-1)$ $\implies$ $f(x)\le -x-1$ And so $\boxed{f(x)=-x-1\quad\forall x}$ which indeed fits