MathDB

A7 / B8

Part of 2011 Princeton University Math Competition

Problems(2)

2011 PUMaC Algebra A7 / B8

Source:

9/24/2019
Let α1,α2,…,α6\alpha_1,\alpha_2,\dots,\alpha_6 be a fixed labeling of the complex roots of x6−1x^6-1. Find the number of permutations {αi1,αi2,…,αi6}\{\alpha_{i_1},\alpha_{i_2},\dots,\alpha_{i_6}\} of these roots such that if P(α1,…,α6)=0P(\alpha_1, \dots, \alpha_6) = 0, then P(αi1,…,αi6)=0P(\alpha_{i_1},\dots,\alpha_{i_6}) = 0, where PP is any polynomial with rational coefficients.
algebra
2011 PUMaC Combinatorics A7 / B8

Source:

9/24/2019
At the start of the PUMaC opening ceremony in McCosh auditorium, the speaker counts 9090 people in the audience. Every minute afterwards, either one person enters the auditorium (due to waking up late) or leaves (in order to take a dreadful math contest). The speaker observes that in this time, exactly 100100 people enter the auditorium, 100100 leave, and 100100 was the largest audience size he saw. Find the largest integer mm such that 2m2^m divides the number of different possible sequences of entries and exits given the above information.
combinatorics