3

Part of 1998 Bulgaria National Olympiad

Problems(2)

Regular polygons!

Source: Bulgaria 1998 Problem 3

On the sides of a non-obtuse triangle

ABC

a square, a regular

n

-gon and a regular

m

m

n > 5

) are constructed externally, so that their centers are vertices of a regular triangle. Prove that

m = n = 6

and find the angles of

\triangle ABC

trigonometrycomplex numbersgeometry proposedgeometry

Regular polygon coloring!

Source: Bulgaria 1998 Problem 6

The sides and diagonals of a regular

n

R

k

colors so that: (i) For each color

a

and any two vertices

A

B

R

AB

a

or there is a vertex

C

AC

BC

a

. (ii) The sides of any triangle with vertices at vertices of

R

are colored in at most two colors. Prove that

k\leq 2

combinatorics proposedcombinatorics