Saudi Arabia Mathematical Competitions

Question

Prove that for any positive integer n there is an equiangular hexagon whose side-lengths are n+1, n+2, , n+6 in some order. FIGURE_0

Accepted Answer

Assume that the equiangular hexagon has the side-lengths

a_{1}, a_{2}, \dots, a_{6}

. Since all angles of the hexagon are

12 0^{\circ}

, extending its sides we get an equilateral triangle. It is clear that

a_{1} + a_{2} + a_{6} = a_{2} + a_{3} + a_{4} = a_{4} + a_{5} + a_{6}

that is

a_{1} + a_{6} = a_{3} + a_{4} and a_{2} + a_{3} = a_{5} + a_{6} (1)

which are the necessary and sufficient conditions for a hexagon with side-lengths

a_{1}, a_{2}, \dots, a_{6}

to be equiangular.

If

a_{1} = n + 1

,

a_{2} = n + 6

,

a_{3} = n + 2

,

a_{4} = n + 4

,

a_{5} = n + 3

,

a_{6} = n + 5

, then (1) is verified and we are done.