**A. 394.** Let *a*_{1},*a*_{2},...,*a*_{N} be nonnegative reals, not all 0. Prove that there exists a sequence 1=*n*_{0}<*n*_{1}<...<*n*_{k}=*N*+1 of integers such that

*n*_{1}*a*_{n0}+*n*_{2}*a*_{n1}+...+*n*_{k}*a*_{nk-1}<3(*a*_{1}+*a*_{2}+...+*a*_{N}).

(5 points)

**B. 3888.** For what integers *m*>1 is there an ordering *a*_{1},*a*_{2},...,*a*_{m} of the numbers 1,2,...,*m*, such that the sums *a*_{1},*a*_{1}+*a*_{2},...,*a*_{1}+*a*_{2}+...+*a*_{m} leave different remainders when divided by *m*?

(5 points)

**B. 3889.** Here is a magician's trick that I have seen: He asked one of the spectators to choose 5 cards at random from a deck of 52 French cards and hand them to his assistant. (The assistant was a permanent partner of the magician, so they could have talked before.) The assistant looked at the 5 cards, and handed four of them over to the magician one by one. Then the magician, without receiving any further information, named the fifth card. After the trick, the magician tried to convince mathematically minded spectators as follows:

``My assistant showed me four cards, and those four cards may be any four of the deck, therefore the only way he could convey information to me was through the order of the four cards he showed to me. Since, as you know, there are 4!=24 possible orders of four cards, and the fifth card could be any of 52-4=48, exactly 1 bit of information was missing. It is the guessing of this bit that needs a miracle.''

Does the trick really need supernatural powers?

Suggested by *M. Gáspár Előd,* physicist

(5 points)

**C. 840.** The values of 10 banknotes add up to 5000 forints (HUF). What banknotes may this sum consist of? (The Hungarian bank notes are: 200 HUF, 500 HUF, 1000 HUF, 2000 HUF, 5000 HUF, 10 000 HUF, 20 000 HUF.)

(5 points)

**K. 74.** The *Figure* shows two paint boxes. Ann, Bob, Clare and Diana are mixing paints for decorating eggs as follows: Wit eyes closed, they dip their paintbrush twice in a row in one compartment of one box, and then apply the colour obtained in this way to the egg. Ann dips her brush in the first box both times, Bob uses the second box both times, Clare dips her brush first in the first box and then in the second box, and Diana dips it in the first one and then in the second one. Which of them has the lowest chance of getting a purple colour (red + blue)?

(6 points)

This problem is for grade 9 students only.

**K. 75.** The point *K* lies on each of the circles of radius *r* centred at the points *K*_{1}, *K*_{2}, *K*_{3}. The other common points of two of the three circles are *P*, *Q*, and *R*. Show that the triangles *K*_{1}*K*_{2}*K*_{3} and *PQR* are congruent.

(6 points)

This problem is for grade 9 students only.

**K. 76.** 2006 small circles are placed in a large circle, so that each touches the next one in a row on the outside, the centres of the small circles all lie on the same diameter of the large circle, and the sum of the diameters of the small circles equals the diameter of the large circle. Compare the total perimeter of the small circles and the perimeter of the large circle.

(6 points)

This problem is for grade 9 students only.

**K. 77.** The Silly family bought a tape measure that can be hung on the wall for measuring heights of people. One end of the tape reads ``80 cm'' and the other end reads ``180 cm'' (If he person to be measured stands on the floor, the tape can measure heights between 80 and 180 cm if properly used.) The Sillies hung the tape in a vertical position, but its lower end was not fixed at the appropriate height.To make it even worse, it also turned out that even the centimetre scale on the tape was wrong: the divisions of were not actually 1 cm apart. Thus the head of the 130-cm-tall Pete reached up to the 120-cm mark while the tape read 145 cm for his 150-cm-tall sister. What is the maximum height of a person that can be measured with this measuring tape with improper scale hung improperly on the wall?

(6 points)

This problem is for grade 9 students only.