Person A passing through a wall? (AI)

Problem Statement

"Throwing a ball at a solid wall will not cause it to break through."

In our everyday lives (the macroscopic world), this is a fundamental rule. However, in the realm of quantum mechanics (the microscopic world), there exists a phenomenon where particles can pass through an energy barrier—which they otherwise should not be able to overcome—as if a tunnel had been bored through it. This is known as the "quantum tunneling effect."

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Person A, who is unhealthy and exceptionally prone to boredom, has shut themselves away in a room. Eventually, growing tired of staying put, they decide they want to escape.

However, a sturdy, high "wall" (an energy barrier) of height $V$V stands in the way of the exit. Given that Person A possesses an amount of "motivation" (energy) $E$E such that $E < V$E<V, it should be physically impossible for them to overcome the wall and get out.

(In simpler terms: "Person A only has 50 units of power, but the wall requires 100 units of power to overcome.")

Yet, in the world of quantum mechanics, it is possible for Person A to pass through the wall and escape.

The probability $T$T that Person A will successfully pass through the wall and escape (the tunneling probability) is given by the following relationship:

$T = (\text{Person A's personality constant}) \times \left(\dfrac{E}{V}\right) \times \left(1 - \dfrac{E}{V}\right)$T=(Person A’s personality constant)×(VE​)×(1−VE​)

(Note: "Person A's personality constant" is a positive constant less than or equal to 1; think of it as a scaling factor to ensure the probability does not exceed 1.)

Person A wishes to escape the room and go out to have some fun—ideally with the highest possible probability of success. Given a fixed wall height $V$V, determine the ratio $E/V$E/V—where $E$E represents "A's level of motivation"—that maximizes the probability $T$T that person A can pass through the wall and escape.

Constraints

• $V > 0$V>0
• $E > 0$E>0

Input Format

The ratio $E/V$E/V can be expressed as an irreducible fraction $p/q$p/q (where $p$p and $q$q are coprime natural numbers). Provide the value of $p + q$p+q.