MathDB

4

Part of 2007 South East Mathematical Olympiad

Problems(2)

sequence problem

Source: China south east mathematical Olympiad 2007 problem 4

7/12/2013
A sequence of positive integers with nn terms satisfies ∑i=1nai=2007\sum_{i=1}^{n} a_i=2007. Find the least positive integer nn such that there exist some consecutive terms in the sequence with their sum equal to 3030.
pigeonhole principlesymmetrycombinatorics unsolvedcombinatorics
inequality abc=1

Source: China south east mathematical Olympiad 2007 problem 4

7/12/2013
Let aa,bb,cc be positive real numbers satisfying abc=1abc=1. Prove that inequality aka+b+bkb+c+ckc+a≥32\dfrac{a^k}{a+b}+ \dfrac{b^k}{b+c}+\dfrac{c^k}{c+a}\ge \dfrac{3}{2} holds for all integer kk (k≥2k \ge 2).
inequalitiesinequalities unsolved