I'm collecting advanced exercises in geometry. Ideally, each exercise should be solved by one trick and this trick should be useful elsewhere (say it gives an essential idea in some theory).


If you have a problem like this please post it here.


**Remarks:**


* I have been collecting such problems for many years. The current collection is at [arXiv](https://arxiv.org/abs/0906.0290); the paper version is available at [amazon](https://rads.stackoverflow.com/amzn/click/com/B08QS38Z23).
* At the moment, I have just a few problems in topology and in geometric group theory and only one in algebraic geometry.
* Thank you all for nice problems --- I decided to add bounty once in a while and choose the best problem (among new or old).



One-step problems in geometry
Let $A$ be a set of intially labelled points in $\mathbb{R}^d$. We may take any line containing at least $k$ labelled points and label any point on this line. For which minimal size $|A|$ (as a function of $d, k$) it may occur that we can label (by performing finitely many such operations) every point of $\mathbb{R}^d$?


