Thread: Pidgeon hole principle problem

A large calculus class has n students, with n between
100 and 200. Each student is identified with one of the numbers 1, 2,
. . . , n. The teacher randomly chooses three students to report on their
performances. When he entered the first student number on the computer,
his assistant on seeing this on the screen, immediately says that
the probability that the other two numbers are both smaller than this first
number is exactly 50-50.
What is this first number, and how many students are there in the
class? Thanks for your help.

Sorry, I don't know the proper solution, but the answer found by brute force is 86 and 121. Then after the 86th student is chosen, there are 85 students on the one hand and 35 on the other side. So 2 * C(85, 2) = C(120, 2), if my calculations are right.