# Divisible by this year??? (Part 11: The Last Stand)

Calculus Level 5

Let's make the end of 2014 very memorable.

I have a dartboard which has the shape identical to the intersection of $y\quad =\quad x^x +{ x }^{ \left\lfloor x \right\rfloor }+{ \left\lfloor x \right\rfloor }^{ x }+{\left\lfloor x \right\rfloor }^{ \left\lfloor x \right\rfloor }$, $x - 1 = 0$, $x - 5 = 0$, and $y = 0$

Its bull's-eye, located at the center of the dartboard, has the shape identical to the intersection of $y\quad=\quad x +\left| x \right| + x\left| x \right|$, $x + 7 = 0$, $x - 7 = 0$, and $y = 0$

Me and my friend, Devonna, made a bet. If I hit the bull's-eye 2 OR 5 times, I will win. Otherwise, she will win. We will do this while wearing blindfolds.

Given that we have 5 darts each, and also assuming that all of the darts hit the dartboard, let $$p$$ be the probability that I will win or in other words, the probability that I hit the bull's-eye 2 OR 5 times.

Also let $$A$$ be the value of $\left\lfloor 10000p \right\rfloor$.

Then find the smallest positive integer $$n$$ such that ${ A }^{ n } - B \equiv 0(mod \quad 2014)$ where $$B$$ is a perfect square less than $$2014$$.

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