K. 163. A team of eight members is paid 224 000 forints (Hungarian currency, HUF) for doing a certain job. The leader of the team (one of the eight members) gets 7 000 forints more than the average pay per member. Given that all the other members get the same amount, calculate how the sum is divided by the members of the team.
K. 164. Eight points are marked on a circle, at equal distances between adjacent points. Consider all scalene (non-isosceles) triangles formed by the eight points. What may be the angles of the triangle?
K. 165. Billy and Sophie are making rings out of paper strips and link them together to form chains. Each of them uses a 21×30 cm sheet of paper for making the chain. Billy cuts up the sheet parallel to the 21-cm sides, and a Sophie cuts it parallel to the 30-cm sides. Both of them make strips 1 cm wide, curve them into rings and glue them with 0-cm overlaps. The rings link together to form a chain without any deformation. Which chain is longer? (In calculating the perimeter of a circle, use 3.14 for the value of , and disregard the thickness of the paper.)
K. 166. Alex is building a pyramid out of congruent cubes, following to the pattern. (He uses no glue, only places the cubes simply next to on top of each other.) Each vertical layer is two cubes taller than the previous one, and forms ascending steps on both sides. When Alex has reached the height of 10 cubes, he goes on decreasing the height of the layers by two cubes in the same fashion. How many cubes does he use altogether?
K. 167. In honour to the Scottish mathematician Dudley Langford, numbers of the following property are called DudLa numbers: Every digit of the number occurs at least twice, and between any pair of identical digits there is as many other digits as the value of the digit in question. For example, 723 121 327 is a DudLa number since there is 1 other digit between two 1's, 2 other digits between two 2's, 3 other digits between two 3's and 7 other digits between two 7's. Find as many 7-digit DudLa numbers as you can.
B. 4076. Lisa and Sam are playing the following game: Lisa marks a point L1 on side AB of a given triangle ABC. Then Sam chooses a point S on side BC. Finally, Lisa marks a point L2 on side CA. Lisa is trying to make the area of triangle L1SL2 as large as possible, and Sam is trying to make it as small as possible. What will be the area of triangle L1SL2 if both of them play as well as possible?
B. 4079. Let G be a simple graph in which the longest path consists of k3 edges and the degree of each vertex is at least k/2. Show that every edge of G belongs to a circuit. (A path is a sequence of linking edges that visits each vertex at most once.)
A. 449. In the convex quadrilateral ABCD, denote by rA, rB, rC and rD the inradii of triangles BCD, CDA, DAB and ABC, respectively. Prove that the quadrilateral ABCD is cyclic if and only if rA+rC=rB+rD.