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ASU 497 All Soviet Union MO 1989 area XQYP/area ABCD < mn/(m^2 + mn + n^2)

Source:

August 13, 2019

equal ratiosgeometryareasgeometric inequality

Problem Statement

ABCD

is a convex quadrilateral.

X

lies on the segment

AB

with

\frac{AX}{XB} = \frac{m}{n}

Y

lies on the segment

CD

with

\frac{CY}{YD} = \frac{m}{n}

AY

and

DX

intersect at

P

, and

BY

and

CX

intersect at

Q

. Show that

\frac{S_{XQYP}}{S_{ABCD}} < \frac{mn}{m^2 + mn + n^2}