MathLabs

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 2 of 4: Pair each subset with its complement
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aIaIc=∏i=1nai=Sn(∣I∣=k)a_Ia_{I^c}=\prod_{i=1}^na_i=S_n\qquad(|I|=k)
Detailed analysis

If ∣I∣=k|I|=k, then ∣Ic∣=n−k|I^c|=n-k and the products over II and IcI^c multiply to the product of all variables, namely Sn=a1a2⋯anS_n=a_1a_2\cdots a_n.