Problem

Source: China TST 2004 Quiz

Tags: geometry, inequalities, circumcircle, geometry unsolved



Find the largest value of the real number $ \lambda$, such that as long as point $ P$ lies in the acute triangle $ ABC$ satisfying $ \angle{PAB}=\angle{PBC}=\angle{PCA}$, and rays $ AP$, $ BP$, $ CP$ intersect the circumcircle of triangles $ PBC$, $ PCA$, $ PAB$ at points $ A_1$, $ B_1$, $ C_1$ respectively, then $ S_{A_1BC}+ S_{B_1CA}+ S_{C_1AB} \geq \lambda S_{ABC}$.