3

Part of 2020 New Zealand MO

Problems(2)

unlimited supply of square tiles and equilateral triangle tiles

Source: New Zealand MO 2020 Round 1 p3

You have an unlimited supply of square tiles with side length

1

and equilateral triangle tiles with side length

1

. For which n can you use these tiles to create a convex

n

-sided polygon? The tiles must fit together without gaps and may not overlap.

combinatorial geometrycombinatoricsTilingtiles

3 of 13 points on circle of r=1 form a triangle with area <13

Source: 2020 New Zealand MO Round 2 p3 NZMO

13

marked points on the circumference of a circle with radius

13

. Prove that we can choose three of the marked points which form a triangle with area less than

13

geometrycombinatorial geometrycombinatorics