Question 9
Andrija and Boris have 2014 cards labelled with numbers from 1 to 2014. Andrija has all the cards with even, and Boris all the cards with odd numbers. Andrija arranged his cards in a circle clockwise, from 2 to 2014 respectively, and put them upside down, so that the numbers on the cards can not be seen. Boris knows that the cards are arranged in that order and that direction, but does not know the position of the card with number 2. After that, Boris puts one of his cards on top of each Andrija's card thus forming 1007 pairs of cards.
For each pair they compare the numbers on the cards and give one point to the player whose card has the larger number.
Find the largest possible $N$ such that Boris can be certain he will gain at least $N$ points. (Ukraine 2010)
For each pair they compare the numbers on the cards and give one point to the player whose card has the larger number.
Find the largest possible $N$ such that Boris can be certain he will gain at least $N$ points. (Ukraine 2010)