MathDB

3

Part of 2015 Azerbaijan JBMO TST

Problems(3)

AZE JBMO TST

Source: AZE JBMO TST

5/2/2015
Let ABCABC be a triangle such that ABAB is not equal to ACAC. Let MM be the midpoint of BCBC and HH be the orthocenter of triangle ABCABC. Let DD be the midpoint of AHAH and OO the circumcentre of triangle BCHBCH. Prove that DAMODAMO is a parallelogram.
geometrycircumcircleAZE JBMO TST
AZE JBMO TST

Source: AZE JBMO TST

5/2/2015
Acute-angled △ABC\triangle{ABC} triangle with condition AB<AC<BCAB<AC<BC has cimcumcircle C,C^, with center OO and radius RR.And BDBD and CECE diametrs drawn.Circle with center OO and radius RR intersects ACAC at KK.And circle with center AA and radius ADAD intersects BABA at LL.Prove that EKEK and DLDL lines intersects at circle C,C^,.
geometrycircumcircle
AZE JBMO TST

Source: AZE JBMO TST

5/2/2015
There is a triangle ABCABC that ABAB is not equal to ACAC.BDBD is interior bisector of ∠ABC\angle{ABC}(D∈ACD\in AC) MM is midpoint of CBACBA arc.Circumcircle of △BDM\triangle{BDM} cuts ABAB at KK and J,J, is symmetry of AA according KK.If DJ∩AM=(O)DJ\cap AM=(O), Prove that J,B,M,OJ,B,M,O are cyclic.
geometrycircumcirclesymmetry