2020 SEEMOUS

Part of SEEMOUS

Subcontests

(4)

1

A Matrix and Its Adjugate

A\in \mathcal{M}_{2020}(\mathbb{C})

(1)\begin{cases} A+A^{\times} =I_{2020},\\ A\cdot A^{\times} =I_{2020},\\ \end{cases}

A^{\times}

is the adjugate matrix of

A

, i.e., the matrix whose elements are

a_{ij}=(-1)^{i+j}d_{ji}

d_{ji}

is the determinant obtained from

A

, eliminating the line

j

i

. Find the maximum number of matrices verifying

(1)

such that any two of them are not similar.

1

SEEMOUS 2020 P2

k>1

be a real number. Calculate: (a)

L=\lim_{n\to \infty} \int_0^1\left( \frac{k}{\sqrt[n]{x}+k-1}\right)^n\text{d} x.

\lim_{n\to \infty} n\left\lbrack L- \int_0^1\left( \frac{k}{\sqrt[n]{x}+k-1}\right)^n\text{d} x\right\rbrack.

1

Rank of Matrix and Its Trance

n

be a positive integer,

k\in \mathbb{C}

A\in \mathcal{M}_n(\mathbb{C})

\text{Tr } A\neq 0

\text{rank } A +\text{rank } ((\text{Tr } A) \cdot I_n - kA) =n.

\text{rank } A

1

SEEMOUS 2020 P4

0<a<T

D=\mathbb{R}\backslash \{ kT+a\mid k\in \mathbb{Z}\}

f:D\to \mathbb{R}

T-

periodic and differentiable function which satisfies

f' > 1

(0, a)

f(0)=0,\lim_{\substack{x\to a\\x<a}}f(x)=+\infty \text{ and }\lim_{\substack{x\to a\\ x<a}}\frac{f'(x)}{f^2(x)}=1.

[*]Prove that for every

n\in \mathbb{N}^*

f(x)=x

has a unique solution in the interval

(nT, nT+a)

x_n

y_n=nT+a-x_n

z_n=\int_0^{y_n}f(x)\text{d}x

\lim_{n\to \infty}{y_n}=0

and study the convergence of the series

\sum_{n=1}^{\infty}{y_n}

\sum_{n=1}^{n}{z_n}