Problems

Prove that for $a,b,c>0$, the following inequality is valid: ${\left(\frac{a+b+c}{3}\right)}^2\ge \frac{ab+bc+ca}{3}$.

Prove that amongst any 7 different numbers it is always possible to choose two of them, $x$ and $y$, so that the following inequality was true: $$0<\frac{x-y}{1+xy}<\frac{1}{\sqrt{3}}.$$

Is it true that if $a$ is a positive number, then $a^2\ge a$? What about $a^2+1\ge a$?

Show for positive $a$ and $b$ that $a^2+b^2\ge 2ab$.

Is it true that if $b$ is a positive number, then $b^3+b^2\ge b$? What about $b^3+1\ge b$?

Show that if $a$ is positive, then $1+a\ge 2\sqrt{a}$.

Let $k$ be a natural number, prove the following inequality. $$\frac{1}{k^2}>\frac{1}{k}-\frac{1}{k+1}.$$

Show that if $a$ is a positive number, then $a^3+2\ge 2a\sqrt{a}$.

The numbers $a$, $b$ and $c$ are positive. By completing the square, show that $$\frac{a^2}{4}+b^2+c^2\ge ab-ac+2bc.$$

Let $m$ and $n$ be natural numbers such that $m>n$. Show that: $$\frac{1}{n^2}+\frac{1}{(n+1{)}^2}+\frac{1}{(n+2{)}^2}+\cdots +\frac{1}{m^2}>\frac{1}{n}-\frac{1}{m}.$$