MathLabs

Problem 5

Let nn be an integer greater than or equal to 33. Prove that there is a set of nn points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.
Step 1 of 5: Choose n lattice points on a parabola
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In plain words

A parabola gives a simple infinite supply of lattice points while automatically preventing three of them from lining up.

S={(x,x2):x∈Z, 1≤x≤n}S=\{(x,x^2):x\in\mathbb Z,\ 1\le x\le n\}
The selected points lie on the parabola y=x2y=x^2.
A graph of the parabola y equals x squared, illustrating the curve containing the chosen lattice points.
Detailed analysis

Take S={(x,x2):x∈Z,1≤x≤n}S=\{(x,x^2):x\in\mathbb Z,1\le x\le n\}. It contains exactly nn distinct points, and every coordinate is an integer. All points lie on the parabola y=x2y=x^2.