MathDB

Luffy is playing with some magic boxes and a machine. Each box has a value(number) inside. Opening a box, Luffy sees the value, adds the value to his score(if the box value is negative, Luffy loses points) and destroys the box. Putting a box of value

X

in the machine, this box vanishes and it is replaced by two new boxes of values

X+1

and

X-1

(it's not known which one has the respective value, but he can identify the new boxes). At the beginning, Luffy has

0

points and has a box whose value is known(it is zero). a) Prove that Luffy can reach at least

1000

points b) Is it possible that Luffy reaches at least

1000000

points, without have less than

-42

points in any moment?

Problem Statement