There is a dartboard with labeled numbers 1 through 20. The score obtained by throwing a dart is the number that the dart lands on, so if it lands on 10, the score is 10. If 11 darts are thrown, and all of them hit, prove that there are two darts who either landed on the same number, or whose sum is divisible by 10.
My work: If we divide all 20 integers into 10 pairs, we have $(10,20),(11,9),(12,8),(13,7),(14,6),(15,5),(16,4),(17,3),(18,2),(19,1)$. There are two possibilities, either there was some dart that landed on the same number, or there was no repetition. Considering the second possibility, since each dart landed on a unique number, there must be at least 2 darts that belong in the same pair. Since each pair adds up to 20, we conclude that there must be two darts whose sum is divisible by 10. Is this line of reasoning correct? Can someone verify if this is a valid proof?