MathDB

2

Part of 1996 Iran MO (3rd Round)

Problems(3)

In a parallelogram show that PA+PB+AD≥PE

Source: Iran Third Round 1997, E1, P2

3/27/2011
Let ABCDABCD be a parallelogram. Construct the equilateral triangle DCEDCE on the side DCDC and outside of parallelogram. Let PP be an arbitrary point in plane of ABCDABCD. Show that PA+PB+AD≥PE.PA+PB+AD \geq PE.
geometryparallelogramgeometric transformationrotationgeometry unsolved
PQRS is a square ==> ABCD is a square

Source: Iran Third Round 1996, E2, P2

3/25/2011
Let ABCDABCD be a convex quadrilateral. Construct the points P,Q,R,P,Q,R, and SS on continue of AB,BC,CD,AB,BC,CD, and DADA, respectively, such that BP=CQ=DR=AS.BP=CQ=DR=AS. Show that if PQRSPQRS is a square, then ABCDABCD is also a square.
inequalitiesgeometrysimilar trianglesgeometry proposed
Radical axis perpendicular to MK

Source: Iran 1996 Third Round

6/7/2010
Consider a semicircle of center OO and diameter ABAB. A line intersects ABAB at MM and the semicircle at CC and DD s.t. MC>MDMC>MD and MB<MAMB<MA. The circumcircles od the AOCAOC and BODBOD intersect again at KK. Prove that MK⊥KOMK\perp KO.
geometrycircumcircleEulercyclic quadrilateralpower of a pointradical axis