2013/14 AHSMC Part 1 Problem 11

Problem 11 of 16

The following statements are made about the integers $a$ , $b$ , $c$ , $d$ and $e$ : (i) $ab$ is even and $c$ is odd; (ii) $bc$ is even and $d$ is odd; (iii) $cd$ is even and $e$ is odd; (iv) $de$ is even and $a$ is odd; (v) $ea$ is even and $b$ is odd. The maximum number of these statements which may be correct is

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If $a$ , $c$ and $d$ are odd while $b$ and $e$ are even, then (i), (ii) and (iv) are all correct. Suppose at least four statements are correct. By symmetry, we may assume that they are (i), (ii), (iii) and (iv). However, by (i) and (ii), $c$ and $d$ are both odd, and yet by (iii), $cd$ is even. This is a contradiction. The answer is (C).

University of Alberta

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