MathDB

Given the natural number N, consider triples of different positive integers

(a, b, c)

such that

a + b + c = N

. Take the largest possible system of these triples such that no two triples of the system have any common elements. Denote the number of triples of this system by

K(N)

. Prove that: (a)

K(N) >\frac{N}{6}-1

(b)

K(N) <\frac{2N}{9}

(L.D. Kurliandchik, Leningrad)

Problem Statement