BMO Round 1, 2018

Combinatorics Difficulty 6.0 National Olympiad Find the answer United Kingdom

If we take a $2 \times 100$ (or $100 \times 2$ ) grid of unit squares, and remove alternate squares from a long side, the remaining $150$ squares form a $100$ -comb. Henry takes a $200 \times 200$ grid of unit squares, and chooses $k$ of these squares and colours them so that James is unable to choose $150$ uncoloured squares which form a $100$ -comb. What is the smallest possible value of $k$ ?

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