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Two perpendicular chords of a circle with centre O

Source: IMO LongList 1988, Greece 1, Problem 21 of ILL

October 22, 2005

geometrygeometry unsolved

Problem Statement

Let "AB" and

CD

be two perpendicular chords of a circle with centre

O

and radius

r

and let

X,Y,Z,W

denote the cyclical order of the four parts into which the disc is thus divided. Find the maximum and minimum of the quantity

\frac{A(X) + A(Z)}{A(Y) + A(W)},

where

A(U)

denotes the area of

U.