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Upgrade One-by-One Tuples Matching

Source: KMO 2021 P2

November 13, 2021
combinatorics

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 m,s,tm, s, t, prove that A(m,s,t)=A(m,t,s).A(m, s, t)=A(m, t, s).
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