MathDB

2

Part of 2003 IberoAmerican

Problems(2)

18th ibmo - argentina 2003/q2

Source: Spanish Communities

4/8/2006
Let CC and DD be two points on the semicricle with diameter ABAB such that BB and CC are on distinct sides of the line ADAD. Denote by MM, NN and PP the midpoints of ACAC, BDBD and CDCD respectively. Let OAO_A and OBO_B the circumcentres of the triangles ACPACP and BDPBDP. Show that the lines OAOBO_AO_B and MNMN are parallel.
trigonometrygeometrytrapezoidcircumcircleradical axiscyclic quadrilateralpower of a point
18th ibmo - argentina 2003/q5

Source: Spanish Communities

4/9/2006
In a square ABCDABCD, let PP and QQ be points on the sides BCBC and CDCD respectively, different from its endpoints, such that BP=CQBP=CQ. Consider points XX and YY such that X≠YX\neq Y, in the segments APAP and AQAQ respectively. Show that, for every XX and YY chosen, there exists a triangle whose sides have lengths BXBX, XYXY and DYDY.
trigonometrygeometrygeometric transformationreflectioncircumcirclepower of a pointradical axis