Question 26
(CZS) (a) Prove that $(a_1+a_2+\cdots+a_k)^2\le k(a_1^2+\cdots+a_k^2)$, where $k\ge1$ is a natural number and $a_1,\ldots,a_k$ are arbitrary real numbers.
(b) If real numbers $a_1,\ldots,a_n$ satisfy
$$a_1+a_2+\cdots+a_n\ge\sqrt{(n-1)(a_1^2+\cdots+a_n^2)},$$
show that they are all nonnegative.
(b) If real numbers $a_1,\ldots,a_n$ satisfy
$$a_1+a_2+\cdots+a_n\ge\sqrt{(n-1)(a_1^2+\cdots+a_n^2)},$$
show that they are all nonnegative.