Pieces of the Puzzle

Algebra Level 4

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

\(N\) 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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