MathDB

sum with permutations of 1,2.,...n

Source: 7th QEDMO problem 10 (14. - 17. 1. 2010) https://artofproblemsolving.com/community/c1512515_qedmo_200507

May 9, 2021

combinatorics

Problem Statement

Let

a_1, a_2, ..., a_n

be positive real numbers. Furthermore, let

S_n

denote the set of all permutations of set

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

. Prove that

\sum_{\pi \in S_n} \frac{1}{a_{\pi(1)}(a_{\pi(1)}+a_{\pi(2)})...(a_{\pi(1)}+a_{\pi(2)}+...+a_{\pi(n)})}=\frac{1}{a_1 a_2 ... a_n}