(Canadian Senior Mathematics Contest 2024, Part B, Question 2, CEMC - UWaterloo)

(a) The quadratic equation $x^2 - 2x - 1 = 0$ has solutions $x = r$ and $x = s$. Determine integers $b$ and $c$ for which the quadratic equation $x^2 + bx + c = 0$ has solutions $x = 2r + s$ and $x = r + 2s$.
(b) This problem has been adapted by the MCR to fit the online experience.
Suppose that $m$ and $p$ are real numbers for which the polynomial $f(x) = x^2 + mx + p$ has two distinct positive real roots. Prove that the polynomial $g(x) = x^2 - (m^2 - 2p)x + p^2$ has two distinct positive real roots. (do not answer)
(c) This problem has been adapted by the MCR to fit the online experience.
Suppose that $A_1 = -6$, $B_1 = 10$, and $C_1 = -5$. For each positive integer $n \geq 2$, let:
$A_n = 2B_{n-1} - (A_{n-1})^2$
$B_n = (B_{n-1})^2 - 2A_{n-1}C_{n-1}$
$C_n = -(C_{n-1})^2$
Prove that the polynomial $f_{100}(x) = x^3 + A_{100}x^2 + B_{100}x + C_{100}$ has three distinct positive real roots. (do not answer)

Note: only part a needs to be answered, so this problem is worth only $7$ points. Your answer should be the pair $(b,c)$ with no spaces.