53ème Olympiade Internationale de Maths

Ça tombe bien qu'on parle de géométrie,
le premier exercice est :
Given triangle $ABC$ the point $J$ is the centre of the excircle opposite the vertex $A$. This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$, respectively. The lines $ LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet at $G$. Let $S$ be the point of intersection of the lines $AF$ and $BC$, and let $T$ be the point of intersection of the lines $AG$ and $BC$. Prove that $M$ is the midpoint of $ST$.

Le second exercice :
If $a_2,a_3,\cdots, a_n$ are positive real numbers such that $a_2\times\cdots\times a_n=1$, then $$
\left(a_{2}+1\right)^{2}\left(a_{3}+1\right)^{3}\dots\left(a_{n}+1\right)^{n}>n^{n}
$$

Le troisième est :
The liar's guessing game is a game played between two players $A$ and $B$. The rules of the game depend on two positive integers $k$ and $n$ which are known to both players.

At the start of the game $A$ chooses integers $x$ and $N$ with $1 \le x \le N$. Player $A$ keeps $x$ secret, and truthfully tells $N$ to player $B$. Player $B$ now tries to obtain information about $x$ by asking player $ A$ questions as follows: each question consists of $B$ specifying an arbitrary set $S$ of positive integers (possibly one specified in some previuos question), and asking $A$ whether $x$ belongs to $S$. Player $B$ may ask as many questions as he wishes. After each question, player $A$ must immediately answer it with yes or no, but is allowed to lie as many times as she wants; the only restriction is that, among any $k+1$ consecutive answers, at least one answer must be truthful.

After $B$ has asked as many questions as he wants, he must specify a set $X$ of at most $n$ positive integers. If $x$ belongs to $X$, then $B$ wins; otherwise, he loses. Prove that:

1. If $n \ge 2^k$, then $B$ can guarantee a win.
2. For all sufficiently large $k$, there exists an integer $n \ge (1.99)^k$ such that $B$ cannot guarantee a win.

Source des problèmes : site mathlinks.ro

suite et fin des épreuves aujourd'hui :

Exercice 4 :
Find all functions $f\,:\, {\mathbb Z}\longrightarrow {\mathbb Z}$, such that for all $a+b+c=0$ holds:
$$
f(a)^2+f(b)^2+f(c)^2\quad =\quad 2f(a)f(b)+2f(b)f(c)+2f(c)f(a).
$$

Exercice 5 : (encore de la géométrie)
Let $\Delta ABC$ be a triangle with $\angle BCA=90^0$ and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $ K$ be the point on the segment $AX$ such that $BK=BC$. Similarly, let
$ L$ be the point on the segment $BX$ such that $AL=AC$. Let $M$ be the point of intersection of $AL$ and $BK$.

Show that $MK=ML$.

Exercice 6 :
Determine all positive integers $n$ for which there exist non-negative integers $a_1, a_2, \cdots, a_n$ such that:
$$
\frac{1}{2^{a_1}} + \frac{1}{2^{a_2}} + \cdots + \frac{1}{2^{a_n}} =
\frac{1}{3^{a_1}} + \frac{2}{3^{a_2}} + \cdots + \frac{n}{3^{a_n}}=1.
$$

Bonjour bs,

(1) le sujet est proposé dans la langue de choix pour chaque équipe, le responsable de l'équipe se charge de traduire l'énoncé pour les 6 élèves de sa délégation à partir de l'anglais, français, russe,...etc, mais disons que l'anglais c'est la langue "officielle" de communication entre les différentes équipes. Les élèves répondent dans la langue de leur pays.

(2) non, il n'est pas de coutume de chanter l'hymne national, mais les élèves sont beaucoup plus disciplinés que les footballeurs!!!!

Amicalement,
yan2