MathDB

3

Part of 2022 Vietnam National Olympiad

Problems(2)

VMO 2022 problem 3 day 1

Source: Vietnam Mathematical Olympiad 2022 problem 3 day 1

3/4/2022
Let ABCABC be a triangle. Point E,FE,F moves on the opposite ray of BA,CABA,CA such that BF=CEBF=CE. Let M,NM,N be the midpoint of BE,CFBE,CF. BFBF cuts CECE at DD a) Suppost that II is the circumcenter of (DBE)(DBE) and JJ is the circumcenter of (DCF)(DCF), Prove that MN∥IJMN \parallel IJ b) Let KK be the midpoint of MNMN and HH be the orthocenter of triangle AEFAEF. Prove that when EE varies on the opposite ray of BABA, HKHK go through a fixed point
vmogeometrycircumcircle
VMO 2022 problem 7 day 2

Source: Vietnam Mathematical Olympiad problem 7 day 2

3/5/2022
Let ABCABC be an acute triangle, B,CB,C fixed, AA moves on the big arc BCBC of (ABC)(ABC). Let OO be the circumcenter of (ABC)(ABC) (B,O,C(B,O,C are not collinear, AB≠AC)AB \ne AC), (I)(I) is the incircle of triangle ABCABC. (I)(I) tangents to BCBC at DD. Let IaI_a be the AA-excenter of triangle ABCABC. IaDI_aD cuts OIOI at LL. Let EE lies on (I)(I) such that DE∥AIDE \parallel AI. a) LELE cuts AIAI at FF. Prove that AF=AIAF=AI. b) Let MM lies on the circle (J)(J) go through Ia,B,CI_a,B,C such that IaM∥ADI_aM \parallel AD. MDMD cuts (J)(J) again at NN. Prove that the midpoint TT of MNMN lies on a fixed circle.
vmogeometrycircumcircle