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set A contains at least (p - 1)(q + 1)/8 pairs whose entries are both even

Source: 2019 RMM Shortlist N1

June 19, 2020
number theoryEven

Problem Statement

Let pp and qq be relatively prime positive odd integers such that 1<p<q1 < p < q. Let AA be a set of pairs of integers (a,b)(a, b), where 0≤a≤p−1,0≤b≤q−10 \le a \le p - 1, 0 \le b \le q - 1, containing exactly one pair from each of the sets {(a,b),(a+1,b+1)},{(a,q−1),(a+1,0)},{(p−1,b),(0,b+1)}\{(a, b),(a + 1, b + 1)\}, \{(a, q - 1), (a + 1, 0)\}, \{(p - 1,b),(0, b + 1)\} whenever 0≤a≤p−20 \le a \le p - 2 and 0≤b≤q−20 \le b \le q - 2. Show that AA contains at least (p−1)(q+1)/8(p - 1)(q + 1)/8 pairs whose entries are both even.
Agnijo Banerjee and Joe Benton, United Kingdom
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