**A. 662.** The points \(\displaystyle A_1\), \(\displaystyle A_2\), \(\displaystyle A_3\), \(\displaystyle A_4\), \(\displaystyle B_1\), \(\displaystyle B_2\), \(\displaystyle B_3\), \(\displaystyle B_4\) lie on a parabola in this order. For every pair \(\displaystyle (i,j)\) with \(\displaystyle 1\le i,j\le4\) and \(\displaystyle i\ne j\), let \(\displaystyle r_{ij}\) denote the ratio in which the line \(\displaystyle A_jB_j\) divides the segment \(\displaystyle A_iB_i\). (That is, if \(\displaystyle A_iB_i\) and \(\displaystyle A_jB_j\) meet at \(\displaystyle X\) then \(\displaystyle r_{ij}=\frac{A_iX}{XB_i}\).) Show that if two of the numbers \(\displaystyle r_{12} \cdot r_{21} \cdot r_{34} \cdot r_{43}\), \(\displaystyle r_{13} \cdot
r_{31} \cdot r_{24} \cdot r_{42}\) and \(\displaystyle r_{14} \cdot r_{41} \cdot r_{23} \cdot
r_{32}\) coincide then the third one is also equal to them.

(5 points)

**A. 664.** Let \(\displaystyle a_1<a_2<\ldots<a_n\) be an arithmetic progression of positive integers. Prove that \(\displaystyle [a_1,a_2,\ldots,a_n] \ge [1,2,\ldots,n]\). (The symbol \(\displaystyle [\ldots]\) stands for the least common multiple.)

(5 points)

**B. 4771.** In an aeroplane, there are one hundred seats, booked by one hundred passengers, each having their assigned seat. However, the first passenger does not care, and sits down on a random seat. When the other passengers enter one by one, each of them tries to take his or her own seat, or, if that seat is already taken, selects another one at random. What is the probability that the hundredth passenger is able to take his own seat?

Proposed by *N. Nagy,* Budapest

(5 points)

**B. 4775.** Find those pairs \(\displaystyle (n,k)\) of positive integers for which

\(\displaystyle
\sum_{i=1}^{2k+1} {(-1)}^{i-1} a_{i}^{n}\ge \bigg(\sum_{i=1}^{2k+1} {(-1)}^{i-1} a_{i}\bigg)^{\!\!n}
\)

for all real numbers \(\displaystyle a_1\ge a_2\ge \dots \ge a_{2k+1}\ge 0\).

Proposed by *Á. Somogyi,* Budapest

(6 points)

**C. 1337.** Csongor's wife sewed a leather sheath decorated with 77 beads for his husband's mouth harp, and gave it to him as a birthday present. Csongor liked it so much that he decided to surprise every member of his heritage preservation mouth harp band with a sheath like his own. He presented the sheaths in the main yurt erected for the celebration of the winter solstice. The yurt had a maximum capacity of 50 persons. Since the beads were bought in packets of 100, 7 beads remained, so Csongor's wife decorated her traditional headdress with them. How many members are there in Csongor's mouth harp band?

(5 points)

This problem is for grade 1 - 10 students only.

**C. 1340.** Points \(\displaystyle P\), \(\displaystyle Q\), \(\displaystyle R\), \(\displaystyle S\) lie on sides \(\displaystyle AB\), \(\displaystyle BC\), \(\displaystyle CD\), \(\displaystyle DA\) of a rectangle \(\displaystyle ABCD\), respectively. Line segments \(\displaystyle PR\) and \(\displaystyle QS\) are perpendicular. Prove that the midpoints of line segments \(\displaystyle SP\), \(\displaystyle PQ\), \(\displaystyle QR\) and \(\displaystyle RS\) form a rectangle, which is similar to \(\displaystyle ABCD\).

(5 points)

**K. 493.** Is it possible to write the numbers 1, 2, 3, 4, 5, 6, 7 and 8 on the vertices of a cube so that the sum of the numbers on the vertices of each face is a prime number?

(6 points)

This problem is for grade 9 students only.