2. Let $A B C D$ be a square of side 4. Determine the largest positive integer $k$ with the following property: For an arbitrary arrangement of $k$ points strictly inside square $A B C D$, one can always find a square of side 1 , entirely contained in square $A B C D$ (with sides not necessarily parallel to the sides of square $A B C D$ ), whose strict interior contains none of the $k$ given points.

(Bojan Bašić)
2. The answer is $k=15$.

It is clear that $k=15$ fulfils the requirements. Indeed, divide square $A B C D$ into 16 unit squares; given any 15 points inside square $A B C D$, at least one of these unit square contains no given points.

We now show that $ka$, so it contains at least one marked point.