Pigeons in Geometry

Problem 6. Fifty-one points are scattered inside a square with a side of 1 meter. Prove that some set of three of these points can be covered by a square with side 20 cm.

Solution. If we divide the square into 25 smaller squares with sides 20 centimeters, the General Pigeon Hole Principle assures us that one of these squares includes at least 3 of the 51 scattered points.

Problem 7. Show that an equilateral triangle cannot be completely covered by 2 smaller equilateral triangles

Hint. If that would be possible, at least one of these 2 smaller triangles would cover 2 of the vertices of the bigger triangle.