2008/09 AHSMC Part 1 Problem 16

We can guarantee that at least one ticket wins by buying the tickets $(1,2)$, $(1,3)$, $(2,3)$ and $(4,5)$. If neither $4$ nor $5$ is drawn, the last ticket wins. If at least one of them is drawn, then at most one of $1$, $2$ and $3$ is drawn. Then one of the first three tickets wins. If we only buy three tickets, we have to choose $6$ numbers. By the Pigeonhole Principle, one of the numbers $1$, $2$, $3$, $4$ and $5$ will appear on at least two of our tickets. If it appears on all three, none of them wins if this number is drawn. Suppose it appears only on the first two tickets. If this number is drawn along with one of the numbers on our third ticket, we will not have a winning ticket. It follows that the minimum number of tickets to guarantee that at least one of them wins is $4$. The answer is (B).