IMO 1972 LL ROM32

If n1, n2, . . . , nk are natural numbers and n1+n2+\cdot \cdot \cdot+nk = n,

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IMO 1972 LL ROM32

Origin: ROM

Problem

If n1, n2, . . . , nk are natural numbers and n1+n2+\cdot \cdot \cdot+nk = n, show that max n1+\cdot\cdot\cdot+nk=n n1n2 \cdot \cdot \cdot nk = (t + 1)rtk−r, where t = [n/k] and r is the remainder of n upon division by k; i.e., n = tk + r, 0 \leqr \leqk −1.