MathDB
China 2010 quiz1 Problem 1

Source:

September 5, 2010
inductionstrong inductioninequalities unsolvedinequalities

Problem Statement

Assume real numbers ai,bi (i=0,1,⋯ ,2n)a_i,b_i\,(i=0,1,\cdots,2n) satisfy the following conditions: (1) for i=0,1,⋯ ,2n−1i=0,1,\cdots,2n-1, we have ai+ai+1≥0a_i+a_{i+1}\geq 0; (2) for j=0,1,⋯ ,n−1j=0,1,\cdots,n-1, we have a2j+1≤0a_{2j+1}\leq 0; (2) for any integer p,qp,q, 0≤p≤q≤n0\leq p\leq q\leq n, we have ∑k=2p2qbk>0\sum_{k=2p}^{2q}b_k>0. Prove that ∑i=02n(−1)iaibi≥0\sum_{i=0}^{2n}(-1)^i a_i b_i\geq 0, and determine when the equality holds.
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