thuvientoan.net xin gửi đến các bạn tài liệu Tổng hợp đề thi đề nghị cho kỳ thi HSG Hình học IGO năm 2018.

Let ABCD be cyclic quadrilateral with circumcenter O. Let E be the point such that OE || AC and DE ⊥ OA. Let F be the point such that OF|| BD and AF ⊥ OD. Assume that AB meest CD at S. Prove that EF ⊥ OS.

Let ABC be a triangle inscribed in (O), P be an arbitrary point on A-median of triangle ABC. Circles (AP B), (AP C) meet AC, AB again at E, F, respectively. EF meets BC at G. Circle (AEF) meets AP again at T. Prove that the reflection of T wrt P G lies on (O).

Let ABC be an acute-angled triangle. The altitudes BE, CF meet at orthocenter H. O is circumcenter of triangle ABC. M is the midpoint of BC. P is the point on EF such that HP ⊥ HO. Q is the point on HA such that P Q ⊥ HM. Prove that QA = 3QH.

Let ABC be an acute-angled triangle with AB < AC. Let M, and E, F be the midpoint of BC, and the feet of the altitudes from B and C, respectively. Let I and J be the incenters of the triangles ABC and AEF, respectively. The lines respectively passing I perpendicular to AI and J perpendicular to MJ meet at a point K. Prove that if L is the point of intersection of KJ and EF then KJ = 2JL.

Tải tại đây.