IMO Shortlist 1986 problem 17
Dodao/la:
arhiva2. travnja 2012. Given a point
![P_0](/media/m/c/a/f/caf7613f56f456dd36b5bcda1f1ec1c0.png)
in the plane of the triangle
![A_1A_2A_3](/media/m/b/b/c/bbcede562021e40de971618cb504b791.png)
. Define
![A_s=A_{s-3}](/media/m/c/a/5/ca51e2347671f15dd4cef028a5096e4c.png)
for all
![s\ge4](/media/m/d/3/f/d3ff7b88ae4e3617f22234116ba5d1eb.png)
. Construct a set of points
![P_1,P_2,P_3,\ldots](/media/m/d/a/3/da32bc6fbc25212448ba0ec3cda68f76.png)
such that
![P_{k+1}](/media/m/c/6/2/c6288e3bab8473ba4524e899ff6235fb.png)
is the image of
![P_k](/media/m/3/8/9/389a080ec4c6cb1cdfcf3fa1a01f3e6f.png)
under a rotation center
![A_{k+1}](/media/m/e/3/8/e384117b5404dac67ebb27ff51a162f5.png)
through an angle
![120^o](/media/m/a/b/e/abe8ae3362a821968fe779881aad2ad5.png)
clockwise for
![k=0,1,2,\ldots](/media/m/f/1/8/f18fe87eb60a626ad7ad4813840529c2.png)
. Prove that if
![P_{1986}=P_0](/media/m/d/4/1/d41f8fec3685f288733229d54be18615.png)
, then the triangle
![A_1A_2A_3](/media/m/b/b/c/bbcede562021e40de971618cb504b791.png)
is equilateral.
%V0
Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.
Izvor: Međunarodna matematička olimpijada, shortlist 1986