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First rectangle can be placed within the second one

Source: IMO Shortlist 1996, G6

August 9, 2008

geometryrectanglerotationIMO ShortlistInequalityfour variables

Problem Statement

Let the sides of two rectangles be

\{a,b\}

and

\{c,d\},

respectively, with

a < c \leq d < b

and

ab < cd.

Prove that the first rectangle can be placed within the second one if and only if \left(b^2 \minus{} a^2\right)^2 \leq \left(bc \minus{} ad \right)^2 \plus{} \left(bd \minus{} ac \right)^2.

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