MathDB

10

Part of 2010 QEDMO 7th

Problems(1)

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

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

5/9/2021
Let a1,a2,...,ana_1, a_2, ..., a_n be positive real numbers. Furthermore, let SnS_n denote the set of all permutations of set {1,2,...,n}\{1, 2, ..., n\}. Prove that
∑π∈Sn1aπ(1)(aπ(1)+aπ(2))...(aπ(1)+aπ(2)+...+aπ(n))=1a1a2...an\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}
combinatorics