MOPSS, 25th October 2025

(Brazil National Olympiad 2020 P1, AoPS) Prove that there are positive integers \(a_1, a_2, \dots, a_{2020}\) such that \[ \frac{1}{a_1} + \frac{1}{2a_2} + \frac{1}{3a_3} + \dots + \frac{1}{2020 a_{2020}} = 1 . \]

(Mexico National Olympiad 2013 P1, AoPS) All prime numbers are written in order, \(p_1 = 2, p_2 = 3, p_3 = 5, \dots\). Find all pairs of positive integers \(a\) and \(b\) with \(a - b \geq 2\), such that \(p_a - p_b\) divides \(2(a - b)\).

(Brazil National Olympiad 2020 P2, AoPS) The following statement is written on a board: \[ \text{The equation } x^2 - 824x + \blacksquare 143 = 0 \text{ has two integer solutions.} \] In the above, \(\blacksquare\) represents the first few digits of a number that is blurred. What are the possible equations originally written on the board?

(British Mathematical Olympiad 2008/9 P4) Find all positive integers \( n \) such that both \( n + 2008 \) divides \( n^2 + 2008 \) and \( n + 2009 \) divides \( n^2 + 2009 \).