Problem

Source: Argentina IMO TST 2005 Problem 4

Tags: algebra unsolved, algebra



We have $ 150$ numbers $ x_1,x_2, \cdots , x_{150}$ each of which is either $ \sqrt 2 +1$ or $ \sqrt 2 -1$ We calculate the following sum: $ S=x_1x_2 +x_3x_4+ x_5x_6+ \cdots + x_{149}x_{150}$ Can we choose the $ 150$ numbers such that $ S=121$? And what about $ S=111$?