Question 107
A computer program reads the numbers displayed, identifies the selected ones and does one of the following actions:
- If $A$ is pressed, every selected number is changed to its successor;
- If $B$ is pressed, every selected number is changed to its triple.
On Andrei and Bogdan's computer screens are displayed the numbers $1$, $3$, $3^2$, $3^3$, ..., $3^{19}$.
a) Andrei will select $5$ numbers and then presses $A$, this is a type $A$ step.
Determine if there is a succession of type $A$ steps in order to obtain the sum of all displayed numbers to be equal to $2019^{2020}$.
b) Bogdan will select $5$ numbers and then presses $B$, this is a type $B$ step.
What is the minimum number of type $B$ steps needed in order to obtain all the displayed numbers to be equal?
- If $A$ is pressed, every selected number is changed to its successor;
- If $B$ is pressed, every selected number is changed to its triple.
On Andrei and Bogdan's computer screens are displayed the numbers $1$, $3$, $3^2$, $3^3$, ..., $3^{19}$.
a) Andrei will select $5$ numbers and then presses $A$, this is a type $A$ step.
Determine if there is a succession of type $A$ steps in order to obtain the sum of all displayed numbers to be equal to $2019^{2020}$.
b) Bogdan will select $5$ numbers and then presses $B$, this is a type $B$ step.
What is the minimum number of type $B$ steps needed in order to obtain all the displayed numbers to be equal?