2012/13 AHSMC Part 1 Problem 8

Note that $a > 2$ and $a < 3$ cannot both be true as there are no integers between $2$ and $3$. Similarly, $a > 4$ and $a < 5$ cannot both be true. Since exactly three of the statements are true, $a < 1$ must be true. Hence the largest possible value is $a = 0$, and for this value, the three statements $a < 1$, $a < 3$ and $a < 5$ are true and the two statements $a > 2$ and $a > 4$ are false. The answer is (A).