Kvant Math Problem 565
Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 2m06s Source on kvant.digital Problem Let $a_1$, $a_2$, $\ldots$, $a_n$ be distinct positive numbers. Denote by $b_k$ the arithmetic mean of all possible products of $k$ of the given numbers ($k=1$, 2, $\ldots$, $n$): $$\begin{align*} b_1&=\dfrac{a_1+a_2+\ldots+a_n}n,\ b_2&=\dfrac{a_1a_2+a_1a_3+\ldots+a_{n-1}a_n}{\dfrac{n(n-1)}2},\ {.}&,{\ldots},{\ldots},{\ldots},{\ldots},{\ldots},{\ldots},{\ldots},{\ldots},{\ldots}\ b_n&=a_1a_2\ldots a_n. \end{align*}$$ Prove the inequalities: $$b_1\ge\sqrt{b_2};$$ $$b_k^2\ge b_{k-1}b_{k+1}\quad(k=2{,}~\ldots{,}~n-1);$$ $$\sqrt[\scriptstyle k~]{b_k}\ge\sqrt[\scriptstyle k+1~]{b_{k+1}}\quad(k=2{,}~\ldots{,}~n-1).$$ M. Rosenberg Exploration For each $k$, the quantity $b_k$ is the average...