why does allocating "pi^2 for a gold, pi for a silver and 1 for a bronze" mean "there could never be any tie in the total scores except when two countries had exactly the same medal counts in all three categories"?
This is actually a slightly stronger condition than is needed, because no country can earn negative medals and there are a finite number of medals available, so you could actually have the weights for eg. gold and silver medals differ by a rational factor as long as the denominator was large enough.