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27 handshakes of m x n students (HOMC 2018 Ind. p13)

Source:

January 31, 2020
combinatorics

Problem Statement

A competition room of HOMC has m×nm \times n students where m,nm, n are integers larger than 22. Their seats are arranged in mm rows and nn columns. Before starting the test, every student takes a handshake with each of his/her adjacent students (in the same row or in the same column). It is known that there are totally 2727 handshakes. Find the number of students in the room.