Problem A. 662. (February 2016)

A. 662. The points \(\displaystyle A_1\), \(\displaystyle A_2\), \(\displaystyle A_3\), \(\displaystyle A_4\), \(\displaystyle B_1\), \(\displaystyle B_2\), \(\displaystyle B_3\), \(\displaystyle B_4\) lie on a parabola in this order. For every pair \(\displaystyle (i,j)\) with \(\displaystyle 1\le i,j\le4\) and \(\displaystyle i\ne j\), let \(\displaystyle r_{ij}\) denote the ratio in which the line \(\displaystyle A_jB_j\) divides the segment \(\displaystyle A_iB_i\). (That is, if \(\displaystyle A_iB_i\) and \(\displaystyle A_jB_j\) meet at \(\displaystyle X\) then \(\displaystyle r_{ij}=\frac{A_iX}{XB_i}\).) Show that if two of the numbers \(\displaystyle r_{12} \cdot r_{21} \cdot r_{34} \cdot r_{43}\), \(\displaystyle r_{13} \cdot r_{31} \cdot r_{24} \cdot r_{42}\) and \(\displaystyle r_{14} \cdot r_{41} \cdot r_{23} \cdot r_{32}\) coincide then the third one is also equal to them.

(5 pont)

Deadline expired on March 10, 2016.

Statistics:

5 students sent a solution.
5 points:	Bodnár Levente, Bukva Balázs, Glasznova Maja, Imolay András, Williams Kada.

Problems in Mathematics of KöMaL, February 2016