Competition problem — Geometry — Maths Olympiad Prep

Library / /15 of 53

Geometry Difficulty 5.7 AIME, harder Prove it China

Let $S = \{(a, b) \mid 1 \le a, b \le 5,\ a, b \in \mathbb{Z}\}$ . Let $T$ be the set of integer points in the plane such that for any point $P$ in $S$ , there exists a different point $Q$ in $T$ such that $PQ$ does not contain integer points except $P$ and $Q$ . Find the minimum value of $|T|$ , where $|T|$ denotes the number of elements of the finite set $T$ .

Solution

We first prove that $|T| \ne 1$ .
If $|T| = 1$ , let $T = \{Q(x_0, y_0)\}$ . We may take point $P(x_1, y_1)$ in $S$ satisfying the conditions: (1) $(x_1, y_1) \ne (x_0, y_0)$ , (2) $x_1$ and $x_0$ have the same parity, $y_1$ and $y_0$ have the same parity. Then, the midpoint of $PQ$ is an integer, which is a contradiction.

If $|T| = 2$ , see the following figure satisfying the conditions of the problem:
• • • • •
• • ◐ • •
• • • • • ◐
• • • • • •
• • • • • •

as desired.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.