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Bosnia and Herzegovina TST 1999 Day 2 Problem 2

Source: Bosnia and Herzegovina Team Selection Test 1999

September 20, 2018

setProductSumnumber theorycombinatorics

Problem Statement

For any nonempty set

S

, we define

\sigma(S)

and

\pi(S)

as sum and product of all elements from set

S

, respectively. Prove that

a)

\sum \limits_{} \frac{1}{\pi(S)} =n

b)

\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

S

of set

\{1,2,...,n\}