SEEMOUS 2021

Let $f: [0, 1] \to \mathbb{R}$ be a continuous strictly increasing function such that $$\lim_{x \to 0^+} \frac{f(x)}{x}=1.$$ (a) Prove that the sequence $(x_n)_{n \ge 1}$ defined by $$x_n=f \left(\frac{1}{1} \right)+f \left(\frac{1}{2} \right)+\cdots+f \left(\frac{1}{n} \right)-\int_1^n f \left(\frac{1}{x} \right) \mathrm dx$$ is convergent. (b) Find the limit of the sequence $(y_n)_{n \ge 1}$ defined by $$y_n=f \left(\frac{1}{n+1} \right)+f \left(\frac{1}{n+2} \right)+\cdots+f \left(\frac{1}{2021n} \right).$$

Let $n \ge 2$ be a positive integer and let $A \in \mathcal{M}_n(\mathbb{R})$ be a matrix such that $A^2=-I_n$ . If $B \in \mathcal{M}_n(\mathbb{R})$ and $AB = BA$ , prove that $\det B \ge 0$ .

Let $A \in \mathcal{M}_n(\mathbb{C})$ be a matrix such that $(AA^*)^2=A^*A$ , where $A^*=(\bar{A})^t$ denotes the Hermitian transpose (i.e., the conjugate transpose) of $A$ . (a) Prove that $AA^*=A^*A$ . (b) Show that the non-zero eigenvalues of $A$ have modulus one.

For $p \in \mathbb{R}$ , let $(a_n)_{n \ge 1}$ be the sequence defined by $$a_n=\frac{1}{n^p} \int_0^n |\sin( \pi x)|^x \mathrm dx.$$ Determine all possible values of $p$ for which the series $\sum_{n=1}^\infty a_n$ converges.