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