Prove that for every $n \geq 3$ there exist $n$ points on the plane , not all colinear, such that the distance between any of them is actually an integer number.
Solution
Let us consider points of the form $(\sin 2\theta, \cos 2\theta)$ where $\theta$ is chosen in such way that both $\cos \theta$ , $\sin \theta$ are both rational numbers. For instance pick
$$\sin \theta = \frac{2t}{1+t^2} \quad , \quad \cos \theta = \frac{1-t^2}{1+t^2} \quad , \;t \in \mathbb{Q}$$
The chord ${\rm AB}$ on the circle of center the origin and radius $1$ connecting two such points ${\rm A} \left ( \sin 2 \theta , \cos 2\theta \right )$ and ${\rm B} \left ( \sin 2 \varphi , \cos 2 \varphi \right )$ has length $\ell = 2 \left |\sin \left ( \theta-\varphi \right ) \right |$.
However $\sin \left ( \theta - \varphi \right )= \sin \theta \cos \varphi -\cos \theta \sin \varphi$ which is clearly rational due to the choice of $\varphi, \theta$ we made at the very beginning. This way we have indeed found $n$ points , whose in between distance is rational. Multiplying by an appropriate constant (e.g the product of all denominators that show up) we can achieve the wanted result.
The exercise can also be found at mathematica.gr
Solution
Let us consider points of the form $(\sin 2\theta, \cos 2\theta)$ where $\theta$ is chosen in such way that both $\cos \theta$ , $\sin \theta$ are both rational numbers. For instance pick
$$\sin \theta = \frac{2t}{1+t^2} \quad , \quad \cos \theta = \frac{1-t^2}{1+t^2} \quad , \;t \in \mathbb{Q}$$
The chord ${\rm AB}$ on the circle of center the origin and radius $1$ connecting two such points ${\rm A} \left ( \sin 2 \theta , \cos 2\theta \right )$ and ${\rm B} \left ( \sin 2 \varphi , \cos 2 \varphi \right )$ has length $\ell = 2 \left |\sin \left ( \theta-\varphi \right ) \right |$.
However $\sin \left ( \theta - \varphi \right )= \sin \theta \cos \varphi -\cos \theta \sin \varphi$ which is clearly rational due to the choice of $\varphi, \theta$ we made at the very beginning. This way we have indeed found $n$ points , whose in between distance is rational. Multiplying by an appropriate constant (e.g the product of all denominators that show up) we can achieve the wanted result.
The exercise can also be found at mathematica.gr
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