Question 19
(IRE) Let $a_1\ge\cdots\ge a_n\ge a_{n+1}=0$ be a sequence of real numbers. Prove that
$$\sqrt{\sum_{k=1}^n a_k}\le\sum_{k=1}^n\sqrt{k}(\sqrt{a_k}-\sqrt{a_{k+1}}).$$
$$\sqrt{\sum_{k=1}^n a_k}\le\sum_{k=1}^n\sqrt{k}(\sqrt{a_k}-\sqrt{a_{k+1}}).$$