## Childhood Memories

I was browsing my analog archives, my high-school notes, and stumbled upon a couple of problems from a math battle we had.

Puzzle.A square is divided into 100 pieces of the same area, in two ways. Prove that you can find 100 points such that each piece in the first division has a point inside, and each piece in the second division also has a point inside.

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Puzzle.There is a finite number of points on a plane. All distances between any pairs of points are distinct. Each point is connected to its closest neighboring point. Prove that each point is connected to no more than 5 points.

## Gennardo:

Assume a point X is connected to 6 points A, B, C, D, E, F. Connect the 2 adjacent points respectively getting 6 triangles. One of this triangles (lets say XDE) must have an angle of at most 60° at X. This means the side DE can’t be larger than both sides XD and XE. Since there are no sides equal in length it must be shorter than one of the sides XD or XE. Contradiction.

19 August 2023, 2:51 am## James:

I like the first problem! Knowing a bit of graph theory there is an obvious way to tackle it. Might have been tougher at high-school level though….

24 August 2023, 7:21 am