IMO Shortlist 2006 problem G8
Dodao/la:
arhiva2. travnja 2012. Let
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
be a convex quadrilateral. A circle passing through the points
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
and
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
and a circle passing through the points
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
and
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
are externally tangent at a point
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
inside the quadrilateral. Suppose that
![\angle{PAB}+\angle{PDC}\leq 90^\circ](/media/m/8/9/2/8928755a70a1f20cc53288d14e45e3a7.png)
and
![\angle{PBA}+\angle{PCD}\leq 90^\circ](/media/m/3/4/5/345a2f5cc1fcbc1d82dee9555cf92715.png)
.
Prove that
![AB+CD \geq BC+AD](/media/m/0/0/c/00c8f7f2cf251d87919bcb4d3b094f78.png)
.
%V0
Let $ABCD$ be a convex quadrilateral. A circle passing through the points $A$ and $D$ and a circle passing through the points $B$ and $C$ are externally tangent at a point $P$ inside the quadrilateral. Suppose that $\angle{PAB}+\angle{PDC}\leq 90^\circ$ and $\angle{PBA}+\angle{PCD}\leq 90^\circ$.
Prove that $AB+CD \geq BC+AD$.
Izvor: Međunarodna matematička olimpijada, shortlist 2006