2019 SEEMOUS

Part of SEEMOUS

Subcontests

(4)

1

SEEMOUS 2019, problem 4

n

is a positive integer. Calculate

\displaystyle \int_0^1 x^{n-1}\ln x\,dx

.\\
(b) Calculate

\displaystyle \sum_{n=0}^{\infty}(-1)^n\left(\frac{1}{(n+1)^2}-\frac{1}{(n+2)^2}+\frac{1}{(n+3)^2}-\dots \right).

1

Equidistributed sequence

\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1

is called "Devin" if for any

f\in C[0,1]

\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n f(x_i)=\int_0^1 f(x)\,dx

Prove that a sequence

\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1

is "Devin" if and only if for any non-negative integer

k

\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n x_i^k=\frac{1}{k+1}.

Remark. I left intact the text as it was proposed. Devin is a Bulgarian city and SPA resort, where this competition took place.

1

Traces of matrices

A_1, A_2,\dots,A_m\in \mathcal{M}_n(\mathbb{R})

. Prove that there exist

\varepsilon_1,\varepsilon_2,\dots,\varepsilon_m\in \{-1,1\}

\rm{tr}\left( (\varepsilon_1 A_1+\varepsilon_2A_2+\dots+\varepsilon_m A_m)^2\right)\geq \rm{tr}(A_1^2)+\rm{tr}(A_2^2)+\dots+\rm{tr}(A_m^2)

1

Ranks of matrices

A,B

n\times n

n\geq 2

B^2=B

\text{rank}\,(AB-BA)\leq \text{rank}\,(AB+BA)