MathDB

2

Part of 2022 Czech-Polish-Slovak Junior Match

Problems(2)

diophantine system x^2 = yz + 1 , y^2 = zx + 1, z^2 = xy + 1

Source: 2022 Czech-Polish-Slovak Match Junior, individual p2 CPSJ

9/3/2022
Solve the following system of equations in integer numbers: {x2=yz+1y2=zx+1z2=xy+1\begin{cases} x^2 = yz + 1 \\ y^2 = zx + 1 \\ z^2 = xy + 1 \end{cases}
number theorydiophantineDiophantine Equations
1st number divisible by 22, 2022 => 2020222 => 2020220222 => ...,

Source: 2022 Czech-Polish-Slovak Match Junior, team p2 CPSJ

9/3/2022
The number 20222022 is written on the board. In each step, we replace one of the 22 digits with the number 20222022. For example 2022⇒2020222⇒2020220222⇒...2022 \Rightarrow 2020222 \Rightarrow 2020220222 \Rightarrow ... After how many steps can a number divisible by 2222 be written on the board? Specify all options.
number theorydivisible