MathDB

5

Part of 2017 China Northern MO

Problems(2)

China Northern Mathematical Olympiad 2017, Problem 5

Source: China Northern Mathematical Olympiad 2017

7/29/2017
Triangle ABCABC has AB>ACAB > AC and ∠A=60∘\angle A = 60^\circ . Let MM be the midpoint of BCBC, NN be the point on segment ABAB such that ∠BNM=30∘\angle BNM = 30^\circ. Let D,ED,E be points on AB,ACAB, AC respectively. Let F,G,HF, G, H be the midpoints of BE,CD,DEBE, CD, DE respectively. Let OO be the circumcenter of triangle FGHFGH. Prove that OO lies on line MNMN.
geometrySpiral Similaritycircumcircle
2017 CNMO Grade 11 P5

Source: 2017 China Northern MO, Grade 11, Problem 5

2/24/2020
Length of sides of regular hexagon ABCDEFABCDEF is aa. Two moving points M,NM,N moves on sides BC,DEBC,DE, satisfy that ∠MAN=π3\angle MAN=\frac{\pi}{3}. Prove that AM⋅AN−BM⋅DNAM\cdot AN-BM\cdot DN is a definite value.
geometry