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1995 China mathematical olympiad problem 1

Source: 1995 China mathematical olympiad problem 1

September 14, 2013
inequalities unsolvedinequalities

Problem Statement

Let a1,a2,⋯ ,an;b1,b2,⋯ ,bn(n≥3)a_1,a_2,\cdots ,a_n; b_1,b_2,\cdots ,b_n (n\ge 3) be real numbers satisfying the following conditions: (1) a1+a2+⋯+an=b1+b2+⋯+bna_1+a_2+\cdots +a_n= b_1+b_2+\cdots +b_n ; (2) 0<a1=a2,ai+ai+1=ai+20<a_1=a_2, a_i+a_{i+1}=a_{i+2} (i=1,2,⋯ ,n−2i=1,2,\cdots ,n-2); (3) 0<b1≤b2,bi+bi+1≤bi+20<b_1\le b_2, b_i+b_{i+1}\le b_{i+2} (i=1,2,⋯ ,n−2i=1,2,\cdots ,n-2). Prove that an−1+an≤bn−1+bna_{n-1}+a_n\le b_{n-1}+b_n.
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