MathDB

3

Part of 2024 India Regional Mathematical Olympiad

Problems(2)

RMO 2024 Q3

Source: RMO 2024 Q3

11/3/2024
Let ABCABC be an acute triangle with AB=ACAB = AC. Let DD be the point on BCBC such that ADAD is perpendicular to BCBC. Let O,H,GO,H,G be the circumcenter, orthocenter and centroid of triangle ABCABC respectively. Suppose that 2⋅OD=23⋅HD2 \cdot OD = 23 \cdot HD. Prove that GG lies on the incircle of triangle ABCABC.
geometrycircumcircle
RMO KV 2024 Q3

Source: RMO KV 2024 Q3

11/3/2024
Let ABCABC be an equilateral triangle. Suppose DD is the point on BCBC such that BD+DC=1:3BD+DC = 1:3. Let the perpendicular bisector of ADAD intersect AB,ACAB,AC at E,FE,F respectively. Prove that 49⋅[BDE]=25⋅[CDF]49 \cdot [BDE] = 25 \cdot [CDF], where [XYZ][XYZ] denotes the area of the triangle XYZXYZ.
geometryperpendicular bisector