# Functional inequalities (1)

Problem 1. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that

$\displaystyle \frac{1}{2}f(xy)+\frac{1}{2}f(xz)-f(x)f(yz)\geq\frac{1}{4}\,\,\forall x,y,z\in\mathbb{R}.$

Problem 2. Let $f:(0;+\infty)\to (0;+\infty)$ be a function such that

$f(2x)\geq x+f(f(x))\,\,\forall x\in (0;+\infty).$ Prove that $f(x)\geq x\,\,\forall x\in (0;+\infty).$

Problem 3. Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that

$f(x+19)-19\leq f(x)\leq f(x+94)-94\,\,\forall x\in\mathbb{R}.$ Prove that $f(x+1)=f(x)+1\,\,\forall x\in\mathbb{R}.$

Problem 4. Find all functions $f:[1;+\infty)\to [1;+\infty)$ such that

$f(x)\leq 2x+2\,\,\text{and}\,\, xf(x+1)=f^2(x)-1\,\,\forall x\in [1;+\infty).$

Problem 5. Find all functions $f:\mathbb{N}\to \mathbb{N}$ such that

$mf(n)+nf(m)=(m+n)f(m^2+n^2)\,\,\forall m,n\in \mathbb{N}.$

Problem 6. Find all injective mappings $f:\mathbb{N}^*\to\mathbb{N}^*$ such that for all positive integers $n$ the following relation holds: $f(f(n)) \leq \dfrac {n+f(n)}{2}.$

Problem 7. Find all surjective mappings $f:\mathbb{N}^*\to\mathbb{N}^*$ such that for all positive integers $n$ the following relation holds: $f(n) \geq n+(-1)^n.$

Problem 8. Determine all functions $f: \mathbb{N}^*\to \mathbb{N}^*$ such that for every positive integer $n$ we have:

$2n+2001\leq f(f(n))+f(n)\leq 2n+2002.$