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Problem 2

Let a1,a2,…,ana_1,a_2,\ldots,a_n be positive real numbers, and let SrS_r be the sum of all products of rr of them. Prove that SkSn−k≥(nk) ⁣2SnS_kS_{n-k}\ge\binom{n}{k}^{\!2}S_n for k=1,2,…,n−1k=1,2,\ldots,n-1.
Step 1 of 4: Index the elementary symmetric sums by subsets
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Sr=∑∣I∣=raI,aI=∏i∈IaiS_r=\sum_{|I|=r}a_I,\qquad a_I=\prod_{i\in I}a_i
Detailed analysis

For a subset I⊆{1,…,n}I\subseteq\{1,\ldots,n\} write aI=∏i∈Iaia_I=\prod_{i\in I}a_i. Then Sr=∑∣I∣=raIS_r=\sum_{|I|=r}a_I. There are (nk)\binom nk subsets of size kk.