Show that if x,y,z are positive integers, then (xy+1)(yz+1)(zx+1) is a perfect square if and only if xy+1,yz+1,zx+1 are all perfect squares.

Source: the probem posed and resolved by Euler.

Two magicians A and B perform the following trick:

A leaves the room and B chooses 4 members from the audience at random. Each member chooses a card numbered from 1 to 100 (each chooses a different card) and after B has seen their cards he chooses a card from the remaining deck of cards. The 5 chosen cards are shuffled by an audience member and handed to A who just returned to the room. Prove that A is able to figure out which cards each member picked. Consider that the chosen members form a row and e.g. the leftmost member picks the first card and the rightmost member (B) picks the last card.