4

Part of 1993 Taiwan National Olympiad

Problems(1)

Source: 2-nd Taiwanese Mathematical Olympiad 1993

In the Cartesian plane, let

C

be a unit circle with center at origin

O

. For any point

Q

in the plane distinct from

O

Q'

to be the intersection of the ray

OQ

C

. Prove that for any

P\in C

k\in\mathbb{N}

there exists a lattice point

Q(x,y)

|x|=k

|y|=k

PQ'<\frac{1}{2k}

trigonometryanalytic geometrycombinatorics unsolvedcombinatorics