Tournament of the Towns (Selections from the A-Level, Fall 2010)

摘要

Two dueling wizards are at an altitude of 100 above the sea. They cast spells in turn, and each spell is of the form decrease the altitude by a for me and by b for my rival" where a and b are real numbers such that 0 < a < b. Different spells have different values for a and b. The set of spells is the same for both wizards, the spells may be cast in any order, and the same spell may be cast many times. A wizard wins if after some spell, he is still above water but his rival is not. Does there exist a set of spells.