Olympiad problems

2015 China Northern MO, 6

The figure obtained by removing one small unit square from the $2\times 2$ grid table is called an $L$ ''shape". .Put $k$ L-shapes in an $8\times 8$ grid table. Each $L$-shape can be rotated, but each $L$ shape is required to cover exactly three small unit squares in the grid table, and the common area covered by any two $L$ shapes is $0$, and except for these $k$ $L$ shapes, no other $L$ shapes can be placed. Find the minimum value of $k$.

1995 Poland - Second Round, 6

Tags: tiling , combinatorics , combinatorial geometry

Determine all positive integers $n$ for which the square $n \times n$ can be cut into squares $2\times 2$ and $3\times3$ (with the sides parallel to the sides of the big square).

2011 Bundeswettbewerb Mathematik, 1

Tags: square , hexagon , tiling , combinatorial geometry , tiles , geometry , angle

Prove that you can't split a square into finitely many hexagons, whose inner angles are all less than $180^o$.