MathDB

12.5

Part of 2006 Moldova National Olympiad

Problems(1)

maybe easy

Source:

11/5/2011
Let a1,a2,...,an a_{1},a_{2},...,a_{n} be real positive numbers and k>m,k,m k>m, k,m natural numbers. Prove that (n−1)(a1m+a2m+...+anm)≤a2k+a3k+...+anka1k−m+a1k+a3k+...+anka2k−m+...+a1k+a2k+...+an−1kank−m(n-1)(a_{1}^m +a_{2}^m+...+a_{n}^m)\leq\frac{a_{2}^k+a_{3}^k+...+a_{n}^k}{a_{1}^{k-m}}+\frac{a_{1}^k+a_{3}^k+...+a_{n}^k}{a_2^{k-m}}+...+\frac{a_{1}^k+a_{2}^k+...+a_{n-1}^k}{a_{n}^{k-m}}
inequalitiesinequalities proposed