IT-CESENATICO 1995 Problem 2

No two of $20$ students in a class have the same scores on both written and oral examinations in mathematics. We say that student $A$ is better than $B$ if his two scores are greater than or equal to the corresponding scores of $B$ . The scores are integers between $1$ and $10$ . (a) Show that there exist three students $A,B,C$ such that $A$ is better than $B$ and $B$ is better than $C$ . (b) Would the same be true for a class of $19$ students?