MEMO 2007 pojedinačno problem 3
Dodao/la:
arhiva28. travnja 2012. Let
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
be a circle and
![k_{1},k_{2},k_{3},k_{4}](/media/m/3/0/f/30f35a29586933da889f4e1f931789f2.png)
four smaller circles with their centres
![O_{1},O_{2},O_{3},O_{4}](/media/m/4/0/b/40b3d1b45fe90c5eb4205143e8556e23.png)
respectively, on
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
. For
![i = 1,2,3,4](/media/m/9/8/1/981d796117d7c617dcc9685358aa32f7.png)
and
![k_5=k_1](/media/m/7/c/6/7c6b5c4f95a07afb8babfe8a9b4c0aed.png)
the circles
![k_i](/media/m/a/0/6/a062f4d1ad84207038d6f4a6b7f02553.png)
and
![k_{i+1}](/media/m/9/6/4/964009b839557aab591dfd4313a91dba.png)
meet at
![A_i](/media/m/5/f/0/5f0935569a883b13bb70b83ea33eee14.png)
and
![B_i](/media/m/1/4/5/14587d3e0ae49b15b1042914a7f802f4.png)
such that
![A_i](/media/m/5/f/0/5f0935569a883b13bb70b83ea33eee14.png)
lies on
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
. The points
![O_{1},A_{1},O_{2},A_{2},O_{3},A_{3},O_{4},A_{4}](/media/m/d/6/9/d69c5f6f354a843e1aa36161a5dde88d.png)
lie in that order on
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
and are pairwise different.
Prove that
![B_{1}B_{2}B_{3}B_{4}](/media/m/f/2/9/f29986cbeaa8304e9eb85b787335e16b.png)
is a rectangle.
%V0
Let $k$ be a circle and $k_{1},k_{2},k_{3},k_{4}$ four smaller circles with their centres $O_{1},O_{2},O_{3},O_{4}$ respectively, on $k$. For $i = 1,2,3,4$ and $k_5=k_1$ the circles $k_i$ and $k_{i+1}$ meet at $A_i$ and $B_i$ such that $A_i$ lies on $k$. The points $O_{1},A_{1},O_{2},A_{2},O_{3},A_{3},O_{4},A_{4}$ lie in that order on $k$ and are pairwise different.
Prove that $B_{1}B_{2}B_{3}B_{4}$ is a rectangle.
Izvor: Srednjoeuropska matematička olimpijada 2007, pojedinačno natjecanje, problem 3