IMO Shortlist 2003 problem C2
Dodao/la:
arhiva2. travnja 2012. Let
![D_1](/media/m/f/e/6/fe67388584f844e56a8db45e4e8768ca.png)
,
![D_2](/media/m/e/0/e/e0efa1bccf5ba2b6dd748dce78cc1cd6.png)
, ...,
![D_n](/media/m/c/3/7/c374cd937c9fb1d42806c1b7dee8e528.png)
be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most
![2003](/media/m/b/1/2/b12e42bc0cb8847464f1eabd933daa3a.png)
discs
![D_i](/media/m/e/5/a/e5abe4fbd94087a725334f4cb2766eb7.png)
. Prove that there exists a disc
![D_k](/media/m/e/6/6/e663dff03f5148f3c661d628fec1f2a7.png)
which intersects at most
![7\cdot 2003 - 1 = 14020](/media/m/6/e/d/6ed29f8843fc615637ece2044bef54b7.png)
other discs
![D_i](/media/m/e/5/a/e5abe4fbd94087a725334f4cb2766eb7.png)
.
%V0
Let $D_1$, $D_2$, ..., $D_n$ be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most $2003$ discs $D_i$. Prove that there exists a disc $D_k$ which intersects at most $7\cdot 2003 - 1 = 14020$ other discs $D_i$.
Izvor: Međunarodna matematička olimpijada, shortlist 2003