**A. 581.** In the plane, there are given two circles *k*_{1} and *k*_{2} with different radii, and a point *O* lying outside the circles. The end-points of the tangents drawn from *O *to *k*_{1} are *P* and *Q*, the end-points of the tangents drawn from *O* to *k*_{2} are *R* and *S*. The points *P*, *Q*, *R*, *S* are distinct. Let *H* be the external homothety center between *k*_{1} and *k*_{2}. Prove that if *PR* is not a external common tangent to the circles but passes through *H* then *QS* also passes through *H*.

(5 points)

**B. 4515.** If a token is inserted in a slot machine, it will roll a regular die and show the result. Then the player may choose to either accept the prize, which is 100 times the number rolled, and thereby end the game, or to insert a second token. In the latter case, the machine rolls the die again, and then the prize is 100 times the product of the two numbers rolled. The game then ends, there is no more choice offered. How much should a token cost so that, in the long run, the slot machine make a profit for its owner?

(4 points)

**B. 4516.** In a triangle *ABC*, tan =2, tan =1, and *b*=12. The midpoints of the sides opposite to *A* and *B* are *F*_{a} and *F*_{b}, and the feet of the corresponding altitudes are *T*_{a} and *T*_{b}, respectively. Prove that the centroid and the orthocentre of triangle *ABC* are collinear with the intersection of *T*_{a}*F*_{b} and *F*_{a}*T*_{b}.

(4 points)

**B. 4517.** *A**B* are interior points of the quadrant *XY* centred at *O*. The parallels drawn to the line *OX* through the points *A*, *B* intersect the radius *OY* at the points *A*_{Y} and *B*_{Y}. The lines drawn parallel to line *OY* intersect the radius *OX* at *A*_{X} and *B*_{X}. Determine the total area of the quadrilaterals *AA*_{X}*B*_{X}*B *and *AA*_{Y}*B*_{Y}*B* as a function of the length *AB*.

Suggested by *Gy. Károlyi,* Budapest, Brisbane

(4 points)

**B. 4520.** Given the positive numbers *a*, *b* and *c*, determine the values of the non-negative variables *x*, *y*, *z* such that the expression is a minimum.

Based on a problem by *Gy. Szöllősy,* Máramarossziget

(6 points)

**B. 4521.** *C* is an interior point of a segment *AB* of line *e*. *k*_{1}, *k*_{2} and *k*_{3} are semicircles drawn over the line segments *AB*, *AC* and *CB* on the same side of line *e*. The midpoints of the arcs of the semicircles are *F*_{1}, *F*_{2} and *F*_{3}. A circle touches semicircle *k*_{1} on the inside at *E*, and also touches the semicircles *k*_{2} and *k*_{3} on the outside. Show that the midpoint *M* of the line segment *AB* and the points *C*, *F*_{1}, *F*_{2}, *F*_{3} and *E* are all concyclic.

Suggested by *Sz. Miklós,* Herceghalom

(6 points)

**K. 367.** A school organized an ice cream building competition for the students. The participants built their ice cream towers on 10-cm tall cones, by placing the scoops one by one on top of each other. Initially, each scoop was a spherical ball of 4-cm diameter, but they were compressed under the weight of the overlying balls. The height of a ball decreased by 1 mm owing to each ball on top of it. The prize winning ice cream tower was 47.5 cm tall, measured from the bottom of the cone to the top of the uppermost ball, and one third of the height of the lowermost ball was inside the cone. How many balls were used to build this tower?

(6 points)

This problem is for grade 9 students only.

**K. 369.** At the beginning of a business meeting, everyone exchanged business cards with everyone. Mr. George joined the company later on. Since he knew some of the participants, he only gave his card to those he did not know, but he did not receive a card from anyone. Thus the total number of cards given out increased by 12.5% relative to the situation before his arrival. How many participants were there after the arrival of Mr. George?

(6 points)

This problem is for grade 9 students only.