Problem A. 668. (April 2016)

A. 668. There is given a positive integer \(\displaystyle k\), some distinct points \(\displaystyle A_1,A_2,\ldots,A_{2k+1}\) and \(\displaystyle O\) in the plane, and a line \(\displaystyle \ell\) passing through \(\displaystyle O\). For every \(\displaystyle i=1,\ldots,2k+1\), let \(\displaystyle B_i\) be the reflection of \(\displaystyle A_i\) about \(\displaystyle \ell\), and let the lines \(\displaystyle OB_i\) and \(\displaystyle A_{i+k}A_{i+k+1}\) meet \(\displaystyle C_i\). (The indices are considered modulo \(\displaystyle 2k+1\): \(\displaystyle A_{2k+2}=A_1\), \(\displaystyle A_{2k+3}=A_2\), ..., and it is assumed that these intersections occur.) Show that if the points \(\displaystyle C_1,C_2,\ldots,C_{2k}\) lie on a line then that line passes through \(\displaystyle C_{2k+1}\) also.

(5 pont)

Deadline expired on May 10, 2016.

Statistics:

1 student sent a solution.
5 points:	Williams Kada.

Problems in Mathematics of KöMaL, April 2016