2023 SEEMOUS

Part of SEEMOUS

Subcontests

(4)

1

Properties of a certain function

f:\mathbb{R}\to\mathbb{R}

be a continuous, strictly decreasing function such that

f([0,1])\subseteq[0,1]

.
[*]For all positive integers

n{}

prove that there exists a unique

a_n\in(0,1)

, solution of the equation

f(x)=x^n

(a_n){}

is the sequence defined as above, prove that

\lim_{n\to\infty}a_n=1

f

has a continuous derivative, with

f(1)=0

f'(1)<0

x\in\mathbb{R}

F(x)=\int_x^1f(t) \ dt.

\alpha{}

be a real number. Study the convergence of the series

\sum_{n=1}^\infty F(a_n)^\alpha.

1

Matrix equals its conjugate transpose

A{}

n\times n

matrix with complex entries such that

A+A^*=A^2A^*

A=A^*

. (Here, we denote by

M^*

the conjugate transpose

\overline{M}^t

M{}

1

Complicated looking limit

For the sequence

S_n=\frac{1}{\sqrt{n^2+1^2}}+\frac{1}{\sqrt{n^2+2^2}}+\cdots+\frac{1}{\sqrt{n^2+n^2}},

\lim_{n\to\infty}n\left(n\cdot\left(\log(1+\sqrt{2})-S_n\right)-\frac{1}{2\sqrt{2}(1+\sqrt{2})}\right).

1

Strange matrix identity

A{}

B{}

n\times n

matrices with complex entries which satisfy

A=AB-BA+A^2B-2ABA+BA^2+A^2BA-ABA^2,

\det(A)=0