Trang chủ Monday, December 11th, 2017

Đề thi chọn đội tuyển Việt Nam (tiếp theo)

[ Wednesday, 08 January, 2014 ]

Đầu năm đi đó đi đây

Đầu năm nước Việt thì đầy bài thi…

(Đúng! Năm nào cũng thi chọn đội tuyển Việt Nam vào ngày mồng hai DL. Đây là bài mà độc giả thế giới chưa có câu trả lời cho phần b).

This is the problem without solution for part b.

Problem 5 of Vietnam Mathematical Olympiad 2014

ABC is an acute triangle inscribed in circle Γ with center O and with fixed vertices B and C whereas vertex A moves on circle Γ.  On lines AB and AC pick points M and N, respectively such that MA = MC and NA = NB. The circumcircles of triangles AMN and ABC intersect at P (P ≠ A). The line MN intersects BC at Q.

a) Prove that the three points A, P and Q are collinear.

b) Let D be the midpoint of BC. The circles with centers at M, N and pass through point A intersect at K (K ≠ A). The line that passes through point A and perpendiculars to AK intersects BC at E. The circumcircle of triangle ADE cuts Γ at F (F ≠ A). Prove that the line AF passes through a fixed point.

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