1. Little bees sell two types of honey : clover and acacia. The price of acacia honey (without cup) is twice the price of clover honey (also without cup). The bees can exchange empty cups for cups full of honey with the same value. Teddybear Pu has 20 empty cups what is enough for him to get an integral number of full cups. How many cups of honey can he get, when the price of twelve cups of clover honey equals the price of seven cups of acacia honey?

2. A dwarf lantern enlights the 90 degree angle. Prove that for any positions of four dwarves in the wood these dwarves can turn their lanterns so that thay enlight the whole wood. Each dwarf with lantern and each other object in wood is considered a point that has no shadow.

3. Honeycomb consists of regular hexagons with unit sides and the bees can move only on the edges of these hexagons. Little bee Maja sitting in the vertex of one hexagon wants to reach Vilko by the shortest path. After finishing this task she found out that she passed the distance 2000. Prove that she went at least one half of this distance in the same direction.

4. At the competition Miss Maruska each of 9 little moons judged 20 adepts by giving them the possitions from 1 to 20. The winner was the one that had the lowest sum of these marks. It came out that for each adept the difference between the best and worst mark did not exceed 3. What is the highest possible number of winners?

5 In Apache village there is n warriors and n squaws and each squaw knows each warrior. In Shoshon village there live n squaws s₁, s₂, ..., s_n and 2n-1 warriors b₁, b₂, ..., b_2n-1 and squaw s_i knows warriors b₁, b₂, ..., b_2i-1 and no other. For r=1, 2, ..., n let A(r) (S(r)) be the number of different ways in which r Apache (Shoshon) squaws can dance each with some warrior of her tribe she knows. Prove that A(r)=S(r) for each r=1, 2, ..., n.

Juraj Foldes (2, 3, 4) a Eugen Kovac (1, 5).