Lo Jac' le pirate owns a rectangular carpet of unknown dimensions. Unfortunately, a tick has kidnapped his ruler and he has no other device to measure lengths. However, he notices that he can place his carpet (flat) in two rectangular rooms of his Grésillon castle so that each corner of the carpet touches a different wall of the rooms. He also knows that the length and width of his carpet are an integer number of feet, that the widths of these two rooms are 38 feet and 50 feet respectively, and that the lengths of these two rooms are the same (the exact value is unknown). Go fetch an enigma from the cell corresponding to the length of his carpet.

## 3 Solutions

## Common Enigmas
Let's first name the variables we are looking for: $x$ and $y$ the length and width of the carpet, and $d$ the common length of the two rooms. Now, we place the carpet $E F G H$ in the rectangular room $50 \times d A B C D$, with $E \in[A B], F \in[B C]$, etc. We note all the lengths: $A B=50, A E=a, B C=d, A H=C F=b, E F=y, F G=x$. We also note $k=\frac{y}{x}$. We see that the triangles $A E H, B F E, C G F, D H G$ are similar with ratios of $1, k$ or $1 / k$. This allows us to see that $E B=k b$ and $D H=k a$, so we have the system of two equations $a+k b=50$ and $k a+b=q$, from which we deduce the values of $a$ and $b$:

$$
\begin{gathered}
a=\frac{q k-50}{k^{2}-1} \text { and } b=\frac{50 k-q}{k^{2}-1} \\
x^{2}=a^{2}+b^{2}=\frac{(q k-50)^{2}+(50 k-q)^{2}}{\left(k^{2}-1\right)^{2}}
\end{gathered}
$$

By placing the carpet in the other room, we get

$$
x^{2}=a^{2}+b^{2}=\frac{(q k-38)^{2}+(38 k-q)^{2}}{\left(k^{2}-1\right)^{2}}
$$

These two expressions are equal, so we have

$$
(q k-50)^{2}+(50 k-q)^{2}=(q k-38)^{2}+(38 k-q)^{2}
$$

This simplifies to

$$
k q=22\left(k^{2}+1\right)
$$

By writing $k$ in its irreducible form $\frac{c}{d}$ with $\operatorname{PGCD}(c, d)=1$, we expand everything,

$$
c d q=22\left(c^{2}+d^{2}\right)
$$

so $c$ and $d$ divide 22. This gives the possible conditions

$$
k=\frac{1}{22}, \frac{1}{11}, \frac{2}{11}, \frac{1}{2}, 1,2, \frac{11}{2}, 11,22
$$

Since $k$ and $1 / k$ give the same result, it suffices to check $1,2, \frac{11}{2}, 11,22$. I leave it to the reader to verify that only $k=2$ gives an integer solution, and then $x=25, y=50$.

## VIII. Memorable Quotes

- "I imagine that your parents don't have a total order, so they can only ask you for a partial order, which you already have" [Bodo]
- "Vincent's room is partially ordered" [Michel]
- "It's wonderful for you that these two theorems exist" [Bodo]
- "It helps to tidy up rooms" [Michel]
- "According to the theorem of Séginus, it takes more time to explain this to your parents than to tidy up the room" [Vincent]
- "A problem is in $N P$ (respectively $C o-N P$) if and only if we can quickly convince the parents that the answer is yes (respectively no)" [François]
- "Something tells me that this is not the real definition..." [Vincent]
- Regarding the marriage theorem:
- "We have a few men and a few women..." [Bodo]
- "A few?!? No, many" [Michel]
- "In February, countries propose a problem, send it (with the solution!), and the host country will then establish and reveal a pre-selection." [François]
- "And publish the problems on the internet right after" [Michel]
- Talking about a long proof: "While waiting for Bodo"
- "A 7 is always congruent to 0 modulo 3" [Mathieu while playing set]
- "It's true...twice true, since we proved it twice" [Pierre]
- "I won't do it, because there are sums everywhere, logs...beurk. Believe me, it's true" [Pierre]
- "Now the Cauchy-Schwartz inequality... [Pierre]
- "Cheers! Schwartz, it's written without a t!" [Michel]
- "For a not quite regular polyhedron, it might not be quite true..." [Ilia]
- "Imagine a hypercube hanging by a hyperstring from a hyperceiling" [Ilia]
- "Can you write a bit louder, please?" [Michel]
- "Thank you to Ilia for this hyperconference that closed this hyperday!" [François]
- "Hyperbolas are used to drink hypersoup" [Antoine]
- "Do you know Dirichlet's theorem? Because it allows you to wipe the $7 \gg$ [Emmanuel]
- "I can come and formalize this properly" [Vincent?!?]
- "I will modify the statement a bit, because otherwise the proof becomes a bit too complex with my method." [Emmanuel]
- "Are the windows of your rooms closed?" [Antoine]
- "Closed, that means..." [Vincent]
- "OK! Shut up!" [Antoine]
- "Is Kantian physics the physics of Kant?" [Anonymous]
- "And if we go to class?"
- "For what?" [A card player]
- "Is there life after math?" [Jérémy]
- "Is there math after life?" [Jérémy]
- "Do you borrow shoes, or put on socks, or walk on your hands..." [Igor to Vincent]
- "If you put the candle in the same plane, it's the problem that this branch could start burning" [Bodo]
- ...?... (really sounds funny, but impossible to understand) [Walid trying to speak Italian]
- "On your table, you see a map of the projective reality, but you are still missing the line at infinity that you can't see well" [Bodo]
- "You can do this during the day but only if your parents are absent" [Bodo]
- "And now you call on the sun so that its ray is in the direction of $\left(C C^{\prime}\right)$" [Bodo]
- "Between two consecutive integers, there is no integer" [François]
- "There are -39 minutes left!" [Vincent]
- "Ah, I see! That's more than enough" [Bodo]
- "And classically, how do we do this?" [Michel about the projective proof of Ménélaüs' theorem]
- "Today we are not doing stupid proofs!" [Bodo]
- "This is the extreme left... This is the extreme right... As you can see, it comes back, so extreme left = extreme right" [Bodo]
- "The game of Mao is a game of 3 cards that starts to the left of the dealer and starts... now!" [many people]
- "As you know, Nobel Prizes in physics, chemistry... are often awarded to people who are a bit old... who look a bit like nothing" [Igor]
- "When you are a mathematician, you have to be old to be rich" [Igor]
- "Everyone, uh... can't see anything, right?" [Igor]
- "So, what is percolation? Well, we are going to percolate" [Igor]
- "Does anyone have a red chalk in their pocket?" [Igor]
- "Since $f$ is bijective and sufficiently nice, as I can look sufficiently close to this point, everything happens quite nicely" [Igor]
- "A simply connected domain... or a potato without a hole" [Igor]
- "There are no probabilities at all in the preparatory class"
- "WHOOF!" [Michel]
- "Where are Piggy and Piggety? We haven't seen them much since just now!" [Antoine, at a late hour of the poly writing]

## IX. Appendix: Selection Test for the Grésillon 2010 Workshop