# Pieces of the Puzzle

Algebra Level 5

$$f(x) = DHx^{5} - D^{2}EH^{2}x^{4} + FG^{F}x^{3} - C^{F}DEFHx^{2} + EHx - DE^{2}H^{2}.$$

$$N$$ is a real root of $$f(x)$$.

An analog clock is inscribed in an equilateral triangle of side length $$P$$. The clock reads 12:45, and the hands of the clock enclose a circular sector of area $$Q$$. $$P$$ can be expressed as $$A\sqrt{\frac{B\sqrt{C}}{Dπ}}$$, when simplified.

$$\overline{WXYZ}$$ is a $$4$$-digit integer, whose digits add to a power of $$2$$ and multiply to a power of $$3$$, with $$(H-D-B)$$ distinct prime factors. $$\overline{WX}=\overline{ZY}=E$$ and $$W≠X$$.

A regular hexagon of area $$Q$$ is inscribed in the graph of $$y=x^{2}$$, touching the graph at four vertices. The distance from the center of the hexagon to the origin can be expressed as $$\frac{F}{G}$$ when simplified.

$$J=A-B$$. There is a $$JxJ$$ grid and a $$DxD$$ grid. The $$JxJ$$ grid contains each integer from $$1$$ to $$J^{2}$$ inclusive. The $$DxD$$ grid contains each integer from $$1$$ to $$D^{2}$$ inclusive. A number is randomly chosen from the $$JxJ$$ grid; then, one is randomly chosen from the $$DxD$$ grid. $$H$$% is the probability that the first number is not a factor of $$234$$ and the second is a factor of $$2340$$.

Find $$N$$.

Details and Assumptions:

• $$A, B, C, D, E, F,$$ and $$G$$ are positive integers.

• $$B$$ and $$D$$ are coprime.

• $$F$$ and $$G$$ are coprime.

• $$B, C,$$ and $$D$$ are not divisible by any perfect square greater than $$1$$.

• A program may be used to calculate roots of 5th degree polynomials.

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