3

Part of 2019 VJIMC

Problems(2)

Source: VJIMC 2019

For an invertible

n\times n

M

with integer entries we define a sequence

\mathcal{S}_M=\{M_i\}_{i=0}^{\infty}

by the recurrence

M_0=M

M_{i+1}=(M_i^T)^{-1}M_i

i\geq 0

.
Find the smallest integer

n\geq 2

for wich there exists a normal

n\times n

matrix with integer entries such that its sequence

\mathcal{S}_M

is not constant and has period

P=7

M_{i+7}=M_i

M^T

means the transpose of a matrix

M

. A square matrix is called normal if

M^T M=M M^T

holds).
Proposed by Martin Niepel (Comenius University, Bratislava)..

linear algebramatrixVJIMC2019VJIMCVojtech JarnikAnnual Vojtech Jarnic

VJIMC 2019 P3 Category II

Source: VJIMC 2019

p

be an even non-negative continous function with

\int _{\mathbb{R}} p(x) dx =1

n

be a positive integer. Let

\xi_1,\xi_2,\xi_3 \dots ,\xi_n

be independent identically distributed random variables with density function

p

. Define
\begin{align*} X_{0} & = 0 \\ X_{1} & = X_0+ \xi_1 \\ & \vdotswithin{ = }\notag \\ X_{n} & = X_{n-1} + \xi_n \end{align*}
Prove that the probability that all random variables

X_1,X_2 \dots X_{n-1}

X_0

X_n

\frac{1}{n}

.
Proposed by Fedor Petrov (St.Petersburg State University).

real analysisVJIMC2019VJIMCVojtech JarnikAnnual Vojtech Jarnic