Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# 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