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