Question 67
$n$ children take part in an arm wrestling tournament. Before the start of the tournament, each child received a serial number (first, second, ..., $n$-th contestant). The tournament will take place on two competition days under the following system:
on the first day, first the first and the second contestant compete; the winner competes with the third contestant; the winner competes with the fourth contestant; and so on...;
on the second day, first the $n$-th and the $(n-1)$-th contestant compete; the winner competes with the $(n-2)$-th contestant; the winner competes with the $(n-3)$-th contestant; and so on...
Show that some two contestants will compete against each other both on the first and on the second day.
on the first day, first the first and the second contestant compete; the winner competes with the third contestant; the winner competes with the fourth contestant; and so on...;
on the second day, first the $n$-th and the $(n-1)$-th contestant compete; the winner competes with the $(n-2)$-th contestant; the winner competes with the $(n-3)$-th contestant; and so on...
Show that some two contestants will compete against each other both on the first and on the second day.