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Part of 1999 Bosnia and Herzegovina Team Selection Test

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Bosnia and Herzegovina TST 1999 Day 2 Problem 2

Source: Bosnia and Herzegovina Team Selection Test 1999

9/20/2018
For any nonempty set SS, we define σ(S)\sigma(S) and π(S)\pi(S) as sum and product of all elements from set SS, respectively. Prove that a)a) ∑1π(S)=n\sum \limits_{} \frac{1}{\pi(S)} =n b)b) ∑σ(S)π(S)=(n2+2n)−(1+12+13+...+1n)(n+1)\sum \limits_{} \frac{\sigma(S)}{\pi(S)} =(n^2+2n)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)(n+1) where ∑\sum denotes sum by all nonempty subsets SS of set {1,2,...,n}\{1,2,...,n\}
setProductSumnumber theorycombinatorics