**A. 631.** Let \(\displaystyle k\ge1\) and let \(\displaystyle I_1,\ldots,I_k\) be non-degenerate subintervals of the interval \(\displaystyle [0, 1]\). Prove \(\displaystyle \sum \frac1{|I_i\cup I_j|} \ge k^2\) where the summation is over all pairs \(\displaystyle (i,j)\) of indices such that \(\displaystyle I_i\) and \(\displaystyle I_j\) are not disjoint.

Miklós Schweitzer competition, 2014

(5 points)

**B. 4670.** Let \(\displaystyle A_1\), \(\displaystyle B_1\) and \(\displaystyle C_1\) be the midpoints of the sides of a triangle \(\displaystyle ABC\). Drop a perpendicular from \(\displaystyle A_1\) to the angle bisector drawn from vertex \(\displaystyle A\), from \(\displaystyle B_1\) to the angle bisector drawn from vertex \(\displaystyle B\), and from \(\displaystyle C_1\) to the angle bisector drawn from vertex \(\displaystyle C\). Let \(\displaystyle A_2\) denote the intersection of the perpendiculars from \(\displaystyle B_1\) and from \(\displaystyle C_1\). The points \(\displaystyle B_2\) and \(\displaystyle C_2\) are obtained in a similar way. Show that the lines \(\displaystyle A_1A_2\), \(\displaystyle B_1B_2\) and \(\displaystyle C_1C_2\) are concurrent.

Suggested by *Zs. Sárosdi,* Veresegyház

(3 points)

**B. 4671.** Let \(\displaystyle AB_1B_2\dots B_6\) and \(\displaystyle AC_1C_2\dots C_6\) be regular heptagons with their vertices labelled in the same direction around the clock. Show that the lines \(\displaystyle B_1C_1, B_2C_2, \dots, B_6C_6\) are concurrent.

Suggested by *G. Holló,* Budapest

(5 points)

**B. 4672.** Determine all real functions \(\displaystyle f\) defined on the positive integers, such that for all positive integer \(\displaystyle n\), \(\displaystyle \frac{p}{f(1)+f(2)+\dots +f(n)} =\frac{p+1}{f(n)}
-\frac{p+1}{f(n+1)}\), where \(\displaystyle p\) is a fixed positive number.

Suggested by *B. Kovács,* Szatmárnémeti

(5 points)

**B. 4673.** \(\displaystyle E\) is the intersection of the diagonals of a cyclic quadrilateral \(\displaystyle ABCD\), and \(\displaystyle K\) is the centre of the circumscribed circle. The intersection of the lines of sides \(\displaystyle AB\) and \(\displaystyle CD\) is \(\displaystyle F\), and the intersection of the lines of sides \(\displaystyle BC\) and \(\displaystyle DA\) is \(\displaystyle G\). The second intersection of the circumscribed circles of triangles \(\displaystyle BFC\) and \(\displaystyle CGD\) is \(\displaystyle H\). Prove that the points \(\displaystyle K\), \(\displaystyle E\) and \(\displaystyle H\) are collinear.

Suggested by *Sz. Miklós,* Herceghalom

(4 points)

**B. 4674.** On the circumscribed circle of triangle \(\displaystyle ABC\), a point \(\displaystyle X\) is moving along the arc \(\displaystyle AC\) not containing vertex \(\displaystyle B\). Let \(\displaystyle Y\) and \(\displaystyle Z\) denote the points on the extensions of side \(\displaystyle BA\) beyond \(\displaystyle A\) and side \(\displaystyle BC\) beyond \(\displaystyle C\), respectively, for which \(\displaystyle AY=AX\) and \(\displaystyle CZ=CX\). What is the locus of the midpoint of line segment \(\displaystyle YZ\)?

Suggested by *E. Pozsonyi,* Budapest

(5 points)

**C. 1260.** The midpoints of sides \(\displaystyle AB\), \(\displaystyle BC\), \(\displaystyle CD\), \(\displaystyle DA\) of a unit square \(\displaystyle ABCD\) are \(\displaystyle E\), \(\displaystyle F\), \(\displaystyle I\), \(\displaystyle H\), respectively. Let \(\displaystyle M\) denote the intersection of the lines \(\displaystyle ED\) and \(\displaystyle HI\), and let \(\displaystyle G\) denote the intersection of the lines \(\displaystyle EC\) and \(\displaystyle FI\). Find the area of the quadrilateral \(\displaystyle MEGI\).

(5 points)

This problem is for grade 1 - 10 students only.

**C. 1264.** The interior angle bisector drawn from vertex \(\displaystyle A\) of triangle \(\displaystyle ABC\) intersects the opposite side at point \(\displaystyle P\), and the perpendicular bisector of line segment \(\displaystyle AP\) intersects side \(\displaystyle AC\) at point \(\displaystyle Q\). Given the line segments \(\displaystyle AB\), \(\displaystyle AQ\), and the angle \(\displaystyle BAQ\), express the area of quadrilateral \(\displaystyle ABPQ\).

(5 points)

This problem is for grade 11 - 12 students only.

**K. 439.** Each side of the figure in the diagram is 10 cm long, and each interior angle is \(\displaystyle 30^\circ\), \(\displaystyle 60^\circ\), \(\displaystyle 150^\circ\), or \(\displaystyle 300^\circ\). Find the area of the *figure. *

(6 points)

This problem is for grade 9 students only.