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complex numbers

Source: Ireland 1993

June 29, 2009
trigonometrycomplex numbersgeometry unsolvedgeometry

Problem Statement

For a complex number z\equal{}x\plus{}iy we denote by P(z) P(z) the corresponding point (x,y) (x,y) in the plane. Suppose z1,z2,z3,z4,z5,α z_1,z_2,z_3,z_4,z_5,\alpha are nonzero complex numbers such that: (i) (i) P(z1),...,P(z5) P(z_1),...,P(z_5) are vertices of a complex pentagon Q Q containing the origin O O in its interior, and (ii) (ii) P(αz1),...,P(αz5) P(\alpha z_1),...,P(\alpha z_5) are all inside Q Q. If \alpha\equal{}p\plus{}iq (p,q∈R) (p,q \in \mathbb{R}), prove that p^2\plus{}q^2 \le 1 and p\plus{}q \tan \frac{\pi}{5} \le 1.
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