Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem A. 650. (October 2015)

A. 650. There is given an acute-angled triangle \(\displaystyle ABC\) with a point \(\displaystyle X\) marked on its altitude starting from \(\displaystyle C\). Let \(\displaystyle D\) and \(\displaystyle E\) be the points on the line \(\displaystyle AB\) that satisfy \(\displaystyle \sphericalangle DCB=\sphericalangle ACE=90^\circ\). Let \(\displaystyle K\) and \(\displaystyle L\) be the points on line segments \(\displaystyle DX\) and \(\displaystyle EX\), respectively, such that \(\displaystyle BK=BC\) and \(\displaystyle AL=AC\). Let the line \(\displaystyle AL\) meet \(\displaystyle BK\) and \(\displaystyle BC\) at \(\displaystyle Q\) and \(\displaystyle R\), respectively; finally let the line \(\displaystyle BK\) meet \(\displaystyle AC\) at \(\displaystyle P\). Show that the quadrilateral \(\displaystyle CPQR\) has an inscribed circle.

(5 pont)

Deadline expired on November 10, 2015.


Statistics:

7 students sent a solution.
5 points:Bodnár Levente, Bukva Balázs, Lajkó Kálmán, Szabó 789 Barnabás, Williams Kada.
2 points:2 students.

Problems in Mathematics of KöMaL, October 2015