IMO 1986 LL MON53

For given positive integers r, v, n let S(r, v, n) denote the num-

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IMO 1986 LL MON53

Origin: MON

Problem

For given positive integers r, v, n let S(r, v, n) denote the num- ber of n-tuples of nonnegative integers (x1, . . . , xn) satisfying the equation x1 + \cdot \cdot \cdot + xn = r and such that xi \leqv for i = 1, . . . , n. Prove that S(r, v, n) = m  k=0 (−1)k n k r −(v + 1)k + n −1 n −1  , where m = min n,  r v+1 4 .