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roots of a quadratic

Source: 3rd National Women's Contest of Mexican Mathematics Olympiad 2024 , level 1+2 p7

June 14, 2024
Mexicoquadratics

Problem Statement

Consider the quadratic equation x2+a0x+b0x^2 + a_0 x + b_0 for some real numbers (a0,b0)(a_0, b_0). Repeat the following procedure as many times as possible:
Let ci=min⁡{ri,si}c_i = \min \{r_i, s_i\}, with ri,sir_i, s_i being the roots of the equation x2+aix+bix^2 + a_i x + b_i. The new equation is written as x2+bix+cix^2 + b_i x + c_i. That is, for the next iteration of the procedure, ai+1=bia_{i+1} = b_i and bi+1=cib_{i+1} = c_i.
We say that (a0,b0)(a_0, b_0) is an <spanclass=′latex−italic′>interesting</span><span class='latex-italic'>interesting</span> pair if, after a finite number of steps, the equation we obtain after one step is the same, so that (ai,bi)=(ai+1,bi+1)(a_i, b_i) = (a_{i+1}, b_{i+1}). Find all <spanclass=′latex−italic′>interesting</span><span class='latex-italic'>interesting</span> pairs.
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