Deceptively Complex

Geometry Level 5

In an Argand diagram, \(O\) is the origin and \(P\) is the point \(2+0i\). The points \(Q, R\) and \(S\) are such that the length of \(OP, PQ, QR\) and \(RS\) are equal, and that \(\angle OPQ = \angle PQR = \angle QRS = \frac{5\pi}{6}\). Note that the points \(O,P,Q,R,S\) are \(5\) vertices of a regular \(12\)-sided polygon lying on the upper half of the Argand diagram.

The point \(C\) is the circumcenter of triangle \(OPQ\). If the polygon is rotated anticlockwise about \(O\) until \(C\) first lies on the real axis, the new position of \(S\) can be written as

\(( \large -\frac{A}{B}(\sqrt{C}+\sqrt{D}), \frac{E\sqrt{F}+\sqrt{G}}{H}i)\),

where \(A,B,C,D,E,F,G,H\) are not necessarily distinct positive integers with \(C,D,F,G\) squarefree.

Find \(A+B+C+D+E+F+G+H\).

This problem comes from STEP (Sixth Term Examinations Paper) year 2001.


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