MathDB

3

Part of 2003 Iran MO (2nd round)

Problems(2)

Volleyball teams - Iran NMO 2003 (Second Round) - Problem3

Source:

10/4/2010
nn volleyball teams have competed to each other (each 22 teams have competed exactly 11 time.). For every 22 distinct teams like A,BA,B, there exist exactly tt teams which have lost their match with A,BA,B. Prove that n=4t+3n=4t+3. (Notabene that in volleyball, there doesn’t exist tie!)
combinatorics proposedcombinatorics
Chessboard & Robot - Iran NMO 2003 (Second Round) - Problem6

Source:

10/4/2010
We have a chessboard and we call a 1×11\times1 square a room. A robot is standing on one arbitrary vertex of the rooms. The robot starts to move and in every one movement, he moves one side of a room. This robot has 22 memories A,BA,B. At first, the values of A,BA,B are 00. In each movement, if he goes up, 11 unit is added to AA, and if he goes down, 11 unit is waned from AA, and if he goes right, the value of AA is added to BB, and if he goes left, the value of AA is waned from BB. Suppose that the robot has traversed a traverse (!) which hasn’t intersected itself and finally, he has come back to its initial vertex. If v(B)v(B) is the value of BB in the last of the traverse, prove that in this traverse, the interior surface of the shape that the robot has moved on its circumference is equal to ∣v(B)∣|v(B)|.
geometryrectanglecombinatorics proposedcombinatorics