MathDB

3

Part of 2016 Korea National Olympiad

Problems(1)

Weird Ineq

Source: 2016 KMO Senior #3

11/12/2016
Acute triangle △ABC\triangle ABC has area SS and perimeter LL. A point PP inside △ABC\triangle ABC has dist(P,BC)=1,dist(P,CA)=1.5,dist(P,AB)=2dist(P,BC)=1, dist(P,CA)=1.5, dist(P,AB)=2. Let BC∩AP=DBC \cap AP = D, CA∩BP=ECA \cap BP = E, AB∩CP=FAB \cap CP= F. Let TT be the area of △DEF\triangle DEF. Prove the following inequality.
(AD⋅BE⋅CFT)2>4L2+(AB⋅BC⋅CA24S)2 \left( \frac{AD \cdot BE \cdot CF}{T} \right)^2 > 4L^2 + \left( \frac{AB \cdot BC \cdot CA}{24S} \right)^2
geometrygeometric inequalityKorea