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(230) 4

Part of 1989 Tournament Of Towns

Problems(1)

TOT 230 1989 Autumn A J4 no of triples (a, b, c) , a + b + c = N

Source:

3/12/2021
Given the natural number N, consider triples of different positive integers (a,b,c)(a, b, c) such that a+b+c=Na + 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)K(N). Prove that: (a) K(N)>N6−1K(N) >\frac{N}{6}-1
(b) K(N)<2N9K(N) <\frac{2N}{9}
(L.D. Kurliandchik, Leningrad)
combinatoricsnumber theoryinequalities