National and Regional Contests USA Contests USA - College-Hosted Events Harvard-MIT Mathematics Tournament 2019 Harvard-MIT Mathematics Tournament 10 HMMT Algebra/NT 2019/10: How hard could it be? Problem Statement The sequence of integers { a i } i = 0 ∞ \{a_i\}_{i = 0}^{\infty} { a i } i = 0 ∞ a 0 = 3 a_0 = 3 a 0 = 3 a 1 = 4 a_1 = 4 a 1 = 4 a n + 2 = a n + 1 a n + ⌈ a n + 1 2 − 1 a n 2 − 1 ⌉ a_{n+2} = a_{n+1} a_n + \left\lceil \sqrt{a_{n+1}^2 - 1} \sqrt{a_n^2 - 1}\right\rceil a n + 2 = a n + 1 a n + ⌈ a n + 1 2 − 1 a n 2 − 1 ⌉ n ≥ 0 n \ge 0 n ≥ 0 ∑ n = 0 ∞ ( a n + 3 a n + 2 − a n + 2 a n + a n + 1 a n + 3 − a n a n + 1 ) . \sum_{n = 0}^{\infty} \left(\frac{a_{n+3}}{a_{n+2}} - \frac{a_{n+2}}{a_n} + \frac{a_{n+1}}{a_{n+3}} - \frac{a_n}{a_{n+1}}\right). n = 0 ∑ ∞ ( a n + 2 a n + 3 − a n a n + 2 + a n + 3 a n + 1 − a n + 1 a n ) .