RO-NMO 2010 Problem 73

Each of the small squares of a $50\times 50$ table is coloured in red or blue. Initially all squares are red. A step means changing the colour of all squares on a row or on a column. a) Prove that there exists no sequence of steps, such that at the end there are exactly $2011$ blue squares. b) Describe a sequence of steps, such that at the end exactly $2010$ squares are blue.

Adriana & Lucian Dragomir