MathDB

convex 1994-gon, closed polygonal line consists of 997 of its diagonals

Source: Czech And Slovak Mathematical Olympiad, Round III, Category A 1994 p3

February 20, 2020

combinatorial geometrycombinatoricsdiagonals

Problem Statement

A convex

1994

-gon

M

is given in the plane. A closed polygonal line consists of

997

of its diagonals. Every vertex is adjacent to exactly one diagonal. Each diagonal divides

M

into two sides, and the smaller of the numbers of edges on the two sides of

M

is defined to be the length of the diagonal. Is it posible to have (a)

991

diagonals of length

3

and

6

of length

2

? (b)

985

diagonals of length

6, 4

of length

8

, and

8

of length

3