2. There were 1009 gnomes with 2017 cards, numbered from 1 to 2017. Ori had one card, and each of the other gnomes had two. All the gnomes knew only the numbers on their own cards. Each gnome, except Ori, said: "I am sure that I cannot give Ori any of my cards so that the sum of the numbers on his two cards would be 2018." What number is on Ori's card?
Answer: 1009

Solution: If one of the gnomes has a card with the number 1, then he must be sure that Ori does not have a card with the number 2017. This can only be certain if the card with the number 2017 is also in the hands of this gnome. Similarly, the gnome with the card that has the number 2 has another card with the number 2016, the gnome with the card that has the number 3 has another card with the number 2015, and so on. All the cards in the gnomes' hands will be divided into pairs that sum to 2018. But all the cards except the card with the number 1009 can be divided into such pairs. It is this card that Ori has.

Comment: It is proven that some of the gnomes' cards can be divided into pairs that sum to 2018 - 3 points. It is proven that all the cards in the gnomes' hands can be divided into pairs that sum to 2018, but the answer is not found - 5 points.