MathDB
One-by-One Tuples Matching

Source: KJMO 2021 P1

November 13, 2021
combinatoricsmatchings

Problem Statement

For positive integers n,k,rn, k, r, denote by A(n,k,r)A(n, k, r) the number of integer tuples (x1,x2,…,xk)(x_1, x_2, \ldots, x_k) satisfying the following conditions.
[*] x1≥x2≥⋯≥xk≥0x_1 \ge x_2 \ge \cdots \ge x_k \ge 0 [*] x1+x2+⋯+xk=nx_1+x_2+ \cdots +x_k = n [*] x1−xk≤rx_1-x_k \le r
For all positive integers s,t≥2s, t \ge 2, prove that A(st,s,t)=A(s(t−1),s,t)=A((s−1)t,s,t).A(st, s, t) = A(s(t-1), s, t) = A((s-1)t, s, t).
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