MathDB

3

Part of 2017 Vietnam National Olympiad

Problems(2)

Not so hard geometry problem

Source: 2017 VMO Problem 3

1/11/2017
Given an acute, non isoceles triangle ABCABC and (O)(O) be its circumcircle, HH its orthocenter and E,FE, F are the feet of the altitudes from BB and CC, respectively. AHAH intersects (O)(O) at DD (D≠AD\ne A).
a) Let II be the midpoint of AHAH, EIEI meets BDBD at MM and FIFI meets CDCD at NN. Prove that MN⊥OHMN\perp OH.
b) The lines DEDE, DFDF intersect (O)(O) at P,QP,Q respectively (P≠D,Q≠DP\ne D,Q\ne D). (AEF)(AEF) meets (O)(O) and AOAO at R,SR,S respectively (R≠A,S≠AR\ne A, S\ne A). Prove that BP,CQ,RSBP,CQ,RS are concurrent.
geometrycircumcircle
Projective geometry

Source: 2017 VMO Problem 7

1/11/2017
Given an acute triangle ABCABC and (O)(O) be its circumcircle. Let GG be the point on arc BCBC that doesn't contain OO of the circumcircle (I)(I) of triangle OBCOBC. The circumcircle of ABGABG intersects ACAC at EE and circumcircle of ACGACG intersects ABAB at FF (E≠A,F≠AE\ne A, F\ne A).
a) Let KK be the intersection of BEBE and CFCF. Prove that AK,BC,OGAK,BC,OG are concurrent.
b) Let DD be a point on arc BOCBOC (arc BCBC containing OO) of (I)(I). GBGB meets CDCD at MM , GCGC meets BDBD at NN. Assume that MNMN intersects (O)(O) at PP nad QQ. Prove that when GG moves on the arc BCBC that doesn't contain OO of (I)(I), the circumcircle (GPQ)(GPQ) always passes through two fixed points.
geometrycircumcircle