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Is it possible to reach value of the expression?

Source: Argentina IMO TST 2005 Problem 4

September 19, 2009
algebra unsolvedalgebra

Problem Statement

We have 150 150 numbers x1,x2,⋯ ,x150 x_1,x_2, \cdots , x_{150} each of which is either \sqrt 2 \plus{}1 or \sqrt 2 \minus{}1 We calculate the following sum: S\equal{}x_1x_2 \plus{}x_3x_4\plus{} x_5x_6\plus{} \cdots \plus{} x_{149}x_{150} Can we choose the 150 150 numbers such that S\equal{}121? And what about S\equal{}111?
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