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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