MathDB

5

Part of 2021 239 Open Mathematical Olympiad

Problems(2)

Hard inequality involving complex numbers from Saint Petersburg

Source: Open math olympiad of lyceum 239

4/8/2021
Let a,b,ca,b,c be some complex numbers. Prove that ∣a2ab+ac−bc∣+∣b2ba+bc−ac∣+∣c2ca+cb−ab∣≥32|\dfrac{a^2}{ab+ac-bc}| + |\dfrac{b^2}{ba+bc-ac}| + |\dfrac{c^2}{ca+cb-ab}| \ge \dfrac{3}{2} if the denominators are not 0
complex numbersinequalities
exists triangle of sides BD, DF,AF with area >1/2 ABC if<ABC =<AED, AF // BC

Source: 239 MO 2021 8-9 p5

5/2/2021
The median ADAD is drawn in triangle ABCABC. Point EE is selected on segment ACAC, and on the ray DEDE there is a point FF, and ∠ABC=∠AED\angle ABC = \angle AED and AF//BCAF // BC. Prove that from segments BD,DFBD, DF and AFAF, you can make a triangle, the area of ​​which is not less half the area of ​​triangle ABCABC.
geometrytriangle inequalitygeometric inequality