Question 178
There are $m$ girls and $n$ boys participating in a duet singing contest ($m, n \ge 2$). At the contest, there will be one show in each section. Each show includes some boy-girl duets where each boy-girl couple will sing no more than one song and each participant will sing at least one song. Two shows are considered different if there exists a boy-girl couple sings in exactly one of these two shows. The contest will end if and only if every possible shows are performed, and each show is performed exactly once.
a) A show is called depend on a participant $X$ if when we cancel all duets that $X$ performs, then there will be at least one participant is not allowed to sing any song in that show. Prove that among every show that depends on $X$, the number of shows with odd number of songs equal to the number of shows with even number of songs.
b) Prove that the organizers can arrange the shows such that the number of songs in two consecutive shows have different parities.
a) A show is called depend on a participant $X$ if when we cancel all duets that $X$ performs, then there will be at least one participant is not allowed to sing any song in that show. Prove that among every show that depends on $X$, the number of shows with odd number of songs equal to the number of shows with even number of songs.
b) Prove that the organizers can arrange the shows such that the number of songs in two consecutive shows have different parities.