**B. 4034.** The midpoint of one side of a triangle is *F*, and the points dividing another side into three equal parts are *H*_{1} and *H*_{2}. The third side is divided into *n* equal parts by the points . Consider all triangles *FH*_{i}*N*_{j} where *i*=1,2, . Show that for any triangle selected out of these triangles there is exactly one other triangle that has the same area.

Suggested by *T. Káspári,* Paks

(3 points)

**K. 142.** In the Mathematical Olympiad of 2100, gold and silver medals will be made of pure gold and silver (and bronze medals will be made of bronze). The diameter of the silver medal will be 3 cm, with a thickness of 5 mm. (The shape of the medals will be as usual.) What will be the diameters of the gold and bronze medals if all the three kinds of medal have the same mass and thickness? (Densities: gold , silver , bronze 8930 kg/m^{3}.)

(6 points)

This problem is for grade 9 students only.

**K. 143.** Tickets cost 1500 forints a piece in a cinema, but few people bought them for so much. When the managers decided to reduce the price, the average number of spectators per day increased by 50%, while the total daily income increased by 25%. What was the reduced price of a ticket?

(6 points)

This problem is for grade 9 students only.

**K. 144.** Consider the seven-digit numbers made out of the digits 1, 2, 3, 4, 5, 6, 7 by using each of them once. What is the sum of all such numbers?

(6 points)

This problem is for grade 9 students only.