2016 AIME I #10

A strictly increasing sequence of positive integers a_1, a_2, a_3, \dots has the property that for every positive integer k, the subsequence a_{2k-1}, a_{2k}, a_{2k+1} is geometric and the subsequence a_{2k}, a_{2k+1}, a_{2k+2} is arithmetic. Suppose that a_{13} = 2016. Find a_1.


2020 AIME II #9 Video Solution

While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next to each other after the break. Find the number of possible seating orders they could have chosen after the break.