**A. 409.** For a positive integer *m*, let *s*(*m*) be the sum of the digits of *m*. For *n*2, let *f*(*n*) be the minimal *k* for which there exists a set *S* of *n* positive integers such that for any nonempty subset *X**S*. Prove that there are constants 0<*C*_{1}<*C*_{2} with *C*_{1}log_{10}*n**f*(*n*)*C*_{2}log_{10}*n*.

*U.S.A. Mathematical Olympiad, 2005*

(5 points)

**B. 3935.** Some characters in a Hamlet play are paired, for example, both members of a pair can play the role of Gertrude *and* the role of Player Queen. Before every performance it is decided by a lot who will play Gertrude and who Player Queen. Other pairs are also chosen by a similar lot. Sarah has already seen Hamlet, but she wants to see it again with a different actor playing the other member of the pair Gertrude/Player Queen, Claudius/Player King and Ophelia/Fortinbras, although not necessarily in one performance. How many tickets should she still buy to see all three roles in the other casting with 90% probability?

(4 points)

**B. 3940.** Let *a*, *o* and *c* be three lines in the plane. Consider all squares *ABCD* such that vertex *A* lies on line *a*, the opposite vertex *C* on line *c*, and the centre of the square *O *lies on line *o*. Find the locus of vertices *B* and *D*.

Suggested by *M. Danka* and *B. Kalló,* 9th-grade students of Fazekas Mihály Gimnázium, Budapest

(5 points)

**C. 867.** Starting from the origin in the usual Cartesian coordinate system, a broken line segment is drawn. We arrive back at the *y*-axis in every fourth step, see the figure.

Using a certain ball-pen and a coordinate system with unit length of 0.5 cm, we draw broken line segments of length 8000 meters as in the figure. Count how many times we arrive back at the *y*-axis.

(5 points)

**K. 92.** Pilots use the face of a clock to express bearings. The *actual* flight direction is identified with 12 o'clock, so, for example, instead of ``90 degrees to the right'' they say ``3 o'clock'', or ``behind me'' equals to ``6 o'clock''. Follow the flight path of a reconnaissance plane that takes off from the base and flies 3 minutes in a given direction, then turns to 2 o'clock and flies 4 minutes, then turns again to 2 o'clock and flies 3 minutes, finally the plane turns to 4 o'clock and flies 9 minutes. In what clock direction should the plane be turned to in order to fly straight back to the base? How many minutes does this take?

(6 points)

This problem is for grade 9 students only.

**K. 93.** Sherlock Holmes was investigating a crime in the Musgrave property famous for a beautiful elm tree. Looking for the hidden treasure, he was to ``walk ten and ten steps to the north, five and five steps to the east, two and two steps to the south, and one and one step to the west.'' Holmes found out, however, that one of the directions was wrong, one should walk in the opposite direction instead (for example, south instead of north). Since he did not know which was wrong, he tried all possibilities. Each walk ended within the park of the Musgraves. Given that Holmes' steps are 80 cm long and the park is a rectangle with sides in the north-south and east-west directions, find the minimum possible area of the park.

(6 points)

This problem is for grade 9 students only.

**K. 94.** When the five-digit number is multiplied by 4, the five-digit number is obtained. Find the value of . (*A*, *B*, *C*, *D*, *E *denote different digits.)

(6 points)

This problem is for grade 9 students only.