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find all (p,q) that forms Z^2

Source: STEMS 2024, CAT B - P1, CAT A - P2

December 17, 2023
combinatorics

Problem Statement

Let S=Z×ZS = \mathbb Z \times \mathbb Z. A subset PP of SS is called nice if
[*] (a,b)∈P  ⟹  (b,a)∈P(a, b) \in P \implies (b, a) \in P [*] (a,b)(a, b), (c,d)∈P  ⟹  (a+c,b−d)∈P(c, d)\in P \implies (a + c, b - d) \in P
Find all (p,q)∈S(p, q) \in S so that if (p,q)∈P(p, q) \in P for some nice set PP then P=SP = S.
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