Question 3
Twelve students visit Nam Cát Tiên forest on six tandem bicycles. If two students $A,B$ do not know each other, the number of acquaintances of $A$ plus the number of acquaintances of $B$ is at least $11$.
(a) Prove that at least one pair of students know each other. (b) Prove that the students can be paired so that the two riders on every bicycle know each other.
Clarification: acquaintances are counted among these twelve students, knowing someone is mutual, and a common acquaintance is counted once for each of $A$ and $B$.
(a) Prove that at least one pair of students know each other. (b) Prove that the students can be paired so that the two riders on every bicycle know each other.
Clarification: acquaintances are counted among these twelve students, knowing someone is mutual, and a common acquaintance is counted once for each of $A$ and $B$.