1

Part of 2019 VJIMC

Problems(2)

Source: VJIMC 2019

\{a_n \}_{n=0}^{\infty}

be a sequence given recrusively such that

a_0=1

a_{n+1}=\frac{7a_n+\sqrt{45a_n^2-36}}{2}

n\geq 0

a_n

is a positive integer. b)

a_n a_{n+1}-1

is a square of an integer.
Proposed by Stefan Gyurki (Matej Bel University, Banska Bystrica).

number theoryalgebraDiscrete MathVJIMC2019VJIMCVojtech JarnikAnnual Vojtech Jarnic

VJIMC 2019 P1 Category II

Source: VJIMC 2019

a)Is it true that for every non-empty set

A

and every associative operation

*:A \times A \to A

x*x*y=y \;\;\; \text{and}\; \;\; y*x*x=y

x,y\in A

imply commutativity of

*

?
b)a)Is it true that for every non-empty set

A

and every associative operation

*:A \times A \to A

x*x*y=y

x,y\in A

implies commutativity of

*

?
Proposed by Paulius Drungilas, Arturas Dubickas (Vilnius University).

Discrete MathematicsVJIMC2019VJIMCVojtech JarnikAnnual Vojtech Jarnic