EMIC 2012
6. In a village, friendship among girl is mutual. Each girl has either exactly one friend or exactly two friends among themselves. One morning, all girl with two friends wear red hats and the other girls all wear blue hats. It turn out that any two friends wear hats of different colors. In the afternoon. 10 girls change their red hats into blue hats and 12 girls change their blue hats into red hats. Now it turn out that any two friends wear hats of the same color. How many girls are there in the village? ( A girl can only change her hat once.)
8. The sum of 1006 different positive integers is 1019057. If none of them is greater than 2012, what is the minimum number of these integers which must be odd?
9. The desks in the TAIMC contest room are arranged in a 6x6 configuration. Two constants are neighbours if they occupy adjacent along a row , a colunm, or a diagonal. Thus a contestant in a seat at a corner of the room has 3 neighbours, a contestant in a seat on an edge of the room has 5 neighbours, and a contestant in a seat in the interior of the room has 8 neighbours . After the contest, a contestant get a prize if at most one neighbour has a score greater than or equal to the score of the contestant. What is maximum number of prize-winners?