MathDB

49

Part of 1992 IMO Longlists

Problems(1)

4k+2 real numbers

Source:

9/2/2010
Given real numbers xi (i=1,2,⋯ ,4k+2)x_i \ (i = 1, 2, \cdots, 4k + 2) such that ∑i=14k+2(−1)i+1xixi+1=4m( x1=x4k+3 )\sum_{i=1}^{4k +2} (-1)^{i+1} x_ix_{i+1} = 4m \qquad ( \ x_1=x_{4k+3} \ ) prove that it is possible to choose numbers xk1,⋯ ,xk6x_{k_{1}}, \cdots, x_{k_{6}} such that ∑i=16(−1)ikiki+1>m( xk1=xk7 )\sum_{i=1}^{6} (-1)^{i} k_i k_{i+1} > m \qquad ( \ x_{k_{1}} = x_{k_{7}} \ )
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