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.


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