The following question comes from 2024 CSMC (Canadian Senior Math Contest) – Part B, Question 2. (There are three sub-questions, a, b, and c. We present only a. and c.)
a. The quadratic equation x2-2x-1= 0 has solutions x= r and x = s. Determine integers b and c for which quadratic equation x2 + bx + c = 0 has solutions x = 2r + s and x = r + 2s.
c. Suppose that A1=-6, B1 = 10 and C1 = -5. For each positive integer n \geq 2 , let \\ A_n = 2 B_{n-1} – (A_{n-1})^2, \\ B_n = (B_{n-1})^2 – 2 A_{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.
Now we look into how to solve the question above.
Analysis for a.
It shall hold x2 + px + q = (x-r) (x-s) if r, s are two roots of the equation. By expanding the right hand side, we obtain x2 – (r+s)x + rs. Now comparing it to x2 + px + q to find [this is indeed Viete’s Theorem]
r+s = -p, (rs) = q ~---~ -- (*)
Using (*) for the question in 2a), it holds that r+s = -p =2, and (rs) =q = -1; meanwhile we have -b = (2r+s) + (r+2s) = 3(r+s) etc. from which the value of b can be calculated.
Analysis for c.
We intend to show that for any n, equation f_n (x) = x^3 +A_n x^2 +B_n x + C_n has three distinct roots. (This is actually stronger than what the question is asked; yet if this is indeed true, then a clear path can be paved leading all the way to f100 (x). )
If one of the three roots (call it x1) is known (preferably an integer), then we have a linear factor (x – x_1) (by Factor Theorem), and the co-factor (quadratic) can be worked out with polynomial division. For the said quadratic co-factor, we argue it has real distinct roots by checking that the discriminant is positive, and be ensured neither root of the quadratic factor equals x_1 .
Note we find x=1 is a root for fn(x) (so let x1 =1.) And f1(x), f2(x), … each has three distinct roots. [Details to be worked out]
Next step: The readers are trusted to work out the solution to a. (analysis for a. was already given). For question c., in addition to the analysis given above, there are some gaps need to be filled. Please attempt to work them out by yourself, yet if you get stuck or want to check whether you have done it right, check out the next post. Using Factor Theorem (cont.) – selected details for a CSMC question