Let \(S\) be a finite set of \(n\) points in \(\mathbb{R}^2\). Find the maximal number of segments that can be drawn in the plane such that:
- the extremeties of the segments belong to \(S\),
- the segments do not intersect, except maybe at their extremities.
In particular, how does that number depend on \(S\) ?
A well-ordered set of dwarves \(E\) plays a game with a giant. They decide of a strategy before the game begins. Once it is established, the giant places a hat, black or white, on the head of each dwarf. The dwarves are aligned such that a dwarf \(a\) sees all the hats of dwarves \(b\) for \(b>a\). The dwarves then try to guess the color of their hats in order. They can only say "black" or "white" and all the other dwarves hear them, meaning that when a dwarf \(a\) takes his guess, he is aware of the guesses of all dwarves \(b < a\).
1. If \(E\) is finite, does there exist a strategy such that every dwarf but one always guesses right ?
2. In general, does there exist a strategy such that every dwarf but a finite number of them guesses right ?
Let \(n\in\mathbb{N}\). Find subsets \(E_1,\dots,E_{2n} \subset [0,1]\) such that:
- \(\sum_{i=1}^{2n} \mathbf{I}_{E_i} = n\mathbf{I}_{[0,1]}\),
- \(\bigcup_{i\in J}E_i \neq [0,1]\) for all sets of indices \(J\) such that \( \left| J\right| < n\).
It is well-known that every complete metric space has Baire's property. Is the converse true? More precisely, is every metrizable topological space with Baire's property metrizable with a complete metric?
Consider an object \(O\) which is modelised by a piecewise linear, compact, connexe curve in \(\mathbb{R}^3\). Suppose that the weight of the object is uniformly distributed on the curve. Let \(n\) be the number of linear pieces that constitute \(O\). How many ways is there to place \(O\) on a perfectly flat, horizontal and infinite table, such that \(O\) does not move? (two positions are considered to be the same if they can be obtained by moving the table in its plane.) More precisely, give an optimal upper bound depending on \(n\).
Four people, a biologist, a chemist, a physicist and a philosopher attempt to escape a horde of buffaloes running after them in the night. They have a 17 minute head start but arrive at a windy and shaky bridge. They are well aware that only two of them can be on the bridge at the same time or else it would surely break. Also, they only have one lantern and anyone who crosses the bridge needs to hold or stand very close to that lantern to avoid an otherwise inescapable fall. It also happens quite luckily that they have a precise idea of how much time each of them would take to cross the bridge. The biologist, who is afraid of heights, cannot complete the crossing in less than 10 minutes. The chemist, not very sure footed, takes at least 5 minutes. The physicist, who is in decent shape, can make it in 2 minutes. And finally, the philosopher, a frequent runner, would be able to cross in 1 minute only. Is there a way that they can be safely on the other side of the bridges in less than 17 minutes, when the buffaloes will inevitably destroy the bridge?