**A. 614.** In the triangle \(\displaystyle A_1A_2A_3\), denote by \(\displaystyle k_i\) the excircle opposite to \(\displaystyle A_i\), and let \(\displaystyle P_i\) be the point of \(\displaystyle k_i\) for which the circle \(\displaystyle A_{i+1}A_{i+2}P_i\) is tangent to \(\displaystyle k_i\). (\(\displaystyle i=1,2,3\); the indices are considered modulo \(\displaystyle 3\).) Show that the line segments \(\displaystyle A_1P_1\), \(\displaystyle A_2P_2\) and \(\displaystyle A_3P_3\) are concurrent.

(5 points)

**B. 4624.** In a trapezium \(\displaystyle ABCD\), let \(\displaystyle E\) and \(\displaystyle F\) denote the midpoints of the bases \(\displaystyle AB\) and \(\displaystyle CD\), respectively, and let \(\displaystyle O\) denote the intersection of the diagonals. A line parallel to the bases intersects the line segments \(\displaystyle OA\), \(\displaystyle OE\) and \(\displaystyle OB\) at points \(\displaystyle M\), \(\displaystyle N\) and \(\displaystyle P\), respectively. Show that the quadrilaterals \(\displaystyle APCN\) and \(\displaystyle BNDM\) have equal areas.

Suggested by *L. Longáver,* Nagybánya

(3 points)

**B. 4627.** The angle bisector drawn from right-angled vertex \(\displaystyle C\) of triangle \(\displaystyle ABC\) intersects the circumscribed circle at point \(\displaystyle P\), and the angle bisector from vertex \(\displaystyle A\) intersects the circumscribed circle at point \(\displaystyle Q\). \(\displaystyle K\) is the intersection of line segments \(\displaystyle PQ\) and \(\displaystyle AB\). The centre of the inscribed circle is \(\displaystyle O\), and its point of tangency on side \(\displaystyle AC\) is \(\displaystyle E\). Prove that points \(\displaystyle E\), \(\displaystyle O\) and \(\displaystyle K\) are collinear.

Suggested by *Zs. Sárosdi,* Veresegyház

(4 points)