Find That Lattice Point. Topic: Number Theory. Level: AIME/Olympiad.

Problem: (Minkowski's Thoerem) Any bounded plane convex region $ R $ symmetrical about the origin with area $ > 4 $ contains at least two other lattice points other than the origin itself.

Solution: WLOG, assume the lattice is a regular square lattice (with side length $ 1 $, of course). The result can easily be generalized to different types of lattices.

Consider partitioning the lattice into $ 2 \times 2 $ squares (with the origin being the vertex of four of them). Since the area of $ R $ is greater than $ 4 $, we can find two points in $ R $ such that their coordinates relative to the $ 2 \times 2 $ squares they are in are equivalent (consider stacking all the $ 2 \times 2 $ squares up; there must exist a point directly above another one or the area is $ \le 4 $).

Call these points

$ P(2a+\alpha, 2b+\beta) $ and $ Q(2c+\alpha, 2d+\beta) $

with $ a,b,c,d $ integers and $ 0 \le \alpha, \beta \le 2 $ reals. But since $ R $ is symmetrical about the origin, we know

$ P^{\prime}(-2a-\alpha, -2b-\beta) $ and $ Q^{\prime}(-2c-\alpha, -2d-\beta) $

also exist in $ R $. If $ R $ is convex, then any point along a line segment connecting two points in $ R $ is also in $ R $. In particular, the midpoint of any line segment between two points in $ R $ is in $ R $. Hence the midpoints of $ PQ^{\prime} $ and $ P^{\prime}Q $ are in $ R $. They are

$ M_1(a-c, b-d) $ and $ M_2(c-a, d-b) $,

which are lattice points not on the origin because $ a,b,c,d $ are integers and $ P \neq Q $. Therefore $ R $ contains at least two other lattice points $ M_1 $ and $ M_2 $. QED.

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Comment: We could have invoked symmetry at the end to find $ M_2 $ from $ M_1 $, too. From the proof, it shouldn't be too hard to tell what I assumed at the beginning for other lattices; basically you just need to define a coordinate system with the same properties . I first encountered this problem at PROMYS and it appears in most Number Theory texts, as far as I know.

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Practice Problem: Can you generalize this result to a $ 3 $-space? An $ n $-space?