Prove that for \(a, b, c > 0\), the following inequality is valid: \(\left(\frac{a+b+c}{3}\right)^2 \ge \frac{ab+bc+ca}{3}\).
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. \[\frac1{k^2} > \frac1{k} - \frac1{k+1}.\]
Show that if \(a\) is a positive number, then \(a^3+2 \ge 2a \sqrt{a}\).