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Found problems: 1

2024 Brazil National Olympiad, 5
Tags: algebra , quadratic , board , sequence , sum and product
Esmeralda chooses two distinct positive integers \(a\) and \(b\), with \(b > a\), and writes the equation \[ x^2 - ax + b = 0 \] on the board. If the equation has distinct positive integer roots \(c\) and \(d\), with \(d > c\), she writes the equation \[ x^2 - cx + d = 0 \] on the board. She repeats the procedure as long as she obtains distinct positive integer roots. If she writes an equation for which this does not occur, she stops. a) Show that Esmeralda can choose \(a\) and \(b\) such that she will write exactly 2024 equations on the board. b) What is the maximum number of equations she can write knowing that one of the initially chosen numbers is 2024?