SIMON-MARAIS 2025 A3

Consider the matrix

$$M = \begin{bmatrix} t + 1 & t \\ 1 & 1 \end{bmatrix},$$

whose entries are polynomials in the variable $t$. Let $a(t)$ denote the polynomial in the first row and first column of $M^{2025}$ and let $b(t)$ denote the polynomial in the first row and second column of $M^{2025}$.

Prove that $a(t)$ has 2025 distinct real roots $\alpha_1, \alpha_2, \ldots, \alpha_{2025}$ and that $b(t)$ has 2025 distinct real roots $\beta_1, \beta_2, \ldots, \beta_{2025}$ satisfying

$$\alpha_1 < \beta_1 < \alpha_2 < \beta_2 < \cdots < \alpha_{2025} < \beta_{2025}.$$