MathDB
IMO Long List 1986 - Find S(r, v, n)

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August 29, 2010
number theory proposednumber theory

Problem Statement

For given positive integers r,v,nr, v, n let S(r,v,n)S(r, v, n) denote the number of nn-tuples of non-negative integers (x1,⋯ ,xn)(x_1, \cdots, x_n) satisfying the equation x1+⋯+xn=rx_1 +\cdots+ x_n = r and such that xi≤vx_i \leq v for i=1,⋯ ,ni = 1, \cdots , n. Prove that S(r,v,n)=∑k=0m(−1)k(nk)(r−(v+1)k+n−1n−1)S(r, v, n)=\sum_{k=0}^{m} (-1)^k \binom nk \binom{r - (v + 1)k + n - 1}{n-1} Where m={n,[rv+1]}.m=\left\{n,\left[\frac{r}{v+1}\right]\right\}.
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