MathDB

The lateral surface of a cylinder of revolution is divided by

n-1

planes parallel to the base and

m

parallel generators into

mn

cases

( n\ge 1,m\ge 3)

. Two cases will be called neighbouring cases if they have a common side. Prove that it is possible to write a real number in each case such that each number is equal to the sum of the numbers of the neighbouring cases and not all the numbers are zero if and only if there exist integers

k,l

such that

n+1

does not divide

k

and

\cos \frac{2l\pi}{m}+\cos\frac{k\pi}{n+1}=\frac{1}{2}

Ciprian Manolescu

Problem Statement