MGGA! Geometry Mock Competition
Notice:
1.Problem releasing time: Part A: 27th July, 2022; Part B: 28th July, 2022; Ending of submitting
your answers: 6th August, 2022;
2.There are 6 geometry problems in this competition. The competition is divided into two parts,
with 3 problems each part and 7 points for each problem (full mark: 42 points).
Part A.(Release time: 27th July, 0:00 UTC)
1.In △ABC, ∠A is the largest and ∠B, ∠C are both greater than 45◦ . Point D, E are on ray
CA, BA respectively such that ∠ABC = ∠ABD, ∠ACB = ∠ACE. F, G ∈ BC(F is on the left of B
and G is on the right of C) satisfy ∠ABC + ∠AF C = ∠ACB + ∠AGB = 90◦ . Line DG, EF
intersect at H. Prove that: HF = HG.
(Proposed by LoloChen)
2.In triangle ABC, D is the midpoint of BC, AD′ is the symmedian midline(passing the symmedian
point K = X6 ). E, F, E ′ , F ′ are similarly defined. Prove that the radical center of
(ADD ′ ), (BEE ′ ), (CF F ′ ) is on the Brocard axis (line OK, where O denotes the circumcenter).
(Proposed by LoloChen)
3.O, I are circumcenter and incenter of △ABC, resp. A-mixtilinear incircle touches (O) at T . IO
and BC intersect at K. A′ is the symmetric point of A WRT line BC. Extend A′ I to L such that
∠ALO = ∠IT K. L∗ is the isogonal conjugate point of L WRT △ABC. Let (BOC) intersects line
AB, AC at point U, V , resp. U V ∩ BC = D. Line LL∗ intersect line AO, BC at X, Y , resp. Prove
that: X, A, D, Y are concyclic.
(Proposed by Inequality.)