MathDB

5

Part of 1993 Irish Math Olympiad

Problems(2)

complex numbers

Source: Ireland 1993

6/29/2009
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.
trigonometrycomplex numbersgeometry unsolvedgeometry
unit squares

Source: Ireland 1993

6/29/2009
(a) (a) The rectangle PQRS PQRS with PQ\equal{}l and QR\equal{}m (l,m∈N) (l,m \in \mathbb{N}) is divided into lm lm unit squares. Prove that the diagonal PR PR intersects exactly l\plus{}m\minus{}d of these squares, where d\equal{}(l,m). (b) (b) A box with edge lengths l,m,n∈N l,m,n \in \mathbb{N} is divided into lmn lmn unit cubes. How many of the cubes does a main diagonal of the box intersect?
geometryrectangle3D geometrygeometry proposed