**A. 579.** The circle *k*_{1} is internally tangent to the circle *k* which is externally tangent to *k*_{2}. The common external tangents of *k*_{1} and *k*_{2} are *u* and *v*. The line *u* is tangent to *k*_{1} and *k*_{2} at *P* and *Q*, respectively, and meets *k* at *A* and *B* in such a way that *B* lies between *P* and *Q*. Analogously, the line *v* is tangent to *k*_{1} and *k*_{2} at *R* and *S*, respectively, and meets *k *at *C* and *D* in such a way that *D* lies between *R* and *S *and *k*_{1} is tangent to that arc *BD* of *k* which does not contain *A* and *C*.

Show that

(5 points)

**B. 4505.** We have two containers with volumes of *p* litres and of *q* litres. It is allowed to fill up completely or empty either container at a tap, to fill up either container completely from the other one, and to pour the entire content of one into the other. Prove that if *p*>*q* are relatively prime positive integers, and *s *is a positive integer such that *s**p*, then it is possible to achieve that one container have exactly *s* litres of water in it.

Suggested by *G. Holló, *Budapest

(5 points)

**B. 4511.** The circle *k* touches the circle internally at point *P*. A line *p* passing through *P* intersects the circles again at the points *K* and *L*, respectively. *u* is the tangent drawn at a point *U* of circle *k*. One intersection of *u* with the circle is *V*, and the intersection of lines *KU* and *LV* is *T*. Determine the locus of the point *T* as *U* traverses the circle *k*. (Consider both intersections of *u* and ; if *U*=*K*, the line *KU* is the tangent to *k *at *K*. Analogously, if *V*=*L* then *LV* is the tangent drawn to at *L*.)

Suggested by *A. Hraskó,* Budapest

(6 points)

**C. 1151.** An arbitrary interior point *P* of a convex quadrilateral *ABCD* is connected to the midpoints *E*, *F*, *G*, *H* of the sides *AB*, *BC*, *CD*, *DA*, respectively. Prove that the sum of the areas of the quadrilaterals *AEPH* and *CGPF* is equal to the sum of the areas of the quadrilaterals *BFPE* and *DHPG*. (Any of them may be a degenerate quadrilateral.)

Suggested by *R. Gyimesi,* Budapest

(5 points)

**K. 362.** Peter takes 5 hours to do a certain job alone, and Paul takes 6 hours to do the same job alone. When they work together, they are both behindered by their continual quarrelling about everything, and their productivity decreases: each of them will do the work at a rate lowered by the same percentage. By what percentage is it lowered if the two of them take 3.5 hours to finish the job together?

(6 points)

This problem is for grade 9 students only.

**K. 363.** Big Ben announces every hour by as many strikes as the number of hours. When the clock has struck, the sound is heard a little time longer, and there are pauses between the sounds heard. Given that Big Ben takes 9 seconds to strike 3 and that it takes 10.5 seconds longer to strike 5 than to strike 2, calculate the total time, in seconds, of the sound of the strikes being heard during a day.

(6 points)

This problem is for grade 9 students only.

**K. 364.** One edge of a rectangular block of wood measures 10 cm. Three pairwise perpendicular bores, each of 2 cm by 2 cm square cross section, parallel to edges, are made through the block. (The three bores do not meet.) With the first bore made, the surface area of the block grew by 32 cm^{2}. The second bore further increased the surface area by 15.92%. The third bore was made parallel to the 10-cm edge. What was the remaining volume of the block after all three bores being made?

(6 points)

This problem is for grade 9 students only.