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VMO 2022 problem 7 day 2

Source: Vietnam Mathematical Olympiad problem 7 day 2

March 5, 2022
vmogeometrycircumcircle

Problem Statement

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.
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