1

Part of 2005 Argentina Team Selection Test

Problems(2)

Is it possible to reach value of the expression?

Source: Argentina IMO TST 2005 Problem 4

150

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

numbers such that S\equal{}121? And what about S\equal{}111?

algebra unsolvedalgebra

covering a board...

Source: Argentina IMO 2005 TST, problem 1

Find all pairs of integers

(m,n)

m\times n

board can be totally covered with

1\times 3

2 \times 5

combinatorics proposedcombinatorics