MathDB

2

Part of 2005 Iran MO (2nd round)

Problems(2)

PB is perpendicular to MK - Iran NMO 2005 - Problem5

Source:

9/21/2010
BCBC is a diameter of a circle and the points X,YX,Y are on the circle such that XY⊥BCXY\perp BC. The points P,MP,M are on XY,CYXY,CY (or their stretches), respectively, such that CY∣∣PBCY||PB and CX∣∣PMCX||PM. Let KK be the meet point of the lines XC,BPXC,BP. Prove that PB⊥MKPB\perp MK.
geometryparallelogramtrapezoidgeometry proposed
MN passes through a constant point- Iran NMO 2005 - Problem2

Source:

9/21/2010
In triangle ABCABC, ∠A=60∘\angle A=60^{\circ}. The point DD changes on the segment BCBC. Let O1,O2O_1,O_2 be the circumcenters of the triangles ΔABD,ΔACD\Delta ABD,\Delta ACD, respectively. Let MM be the meet point of BO1,CO2BO_1,CO_2 and let NN be the circumcenter of ΔDO1O2\Delta DO_1O_2. Prove that, by changing DD on BCBC, the line MNMN passes through a constant point.
geometrycircumcirclegeometry proposed