IMO 1979 LL ISR45

For any positive integer n we denote by F(n) the number of

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IMO 1979 LL ISR45

Origin: ISR

Problem

For any positive integer n we denote by F(n) the number of ways in which n can be expressed as the sum of three different positive integers, without regard to order. Thus, since 10 = 7+2+1 = 6+3+1 = 5 + 4 + 1 = 5 + 3 + 2, we have F(10) = 4. Show that F(n) is even if n \equiv2 or 4 (mod 6), but odd if n is divisible by 6.