In a circle with a radius $R$ a convex hexagon is inscribed. The diagonals $AD$ and $BE$,$BE$ and $CF$,$CF$ and $AD$ of the hexagon intersect at the points $M$,$N$ and$K$, respectively. Let $r_1,r_2,r_3,r_4,r_5,r_6$ be the radii of circles inscribed in triangles $ ABM,BCN,CDK,DEM,EFN,AFK$ respectively. Prove that.$$r_1+r_2+r_3+r_4+r_5+r_6\leq R\sqrt{3}$$.
Problem
Source: izho 2018
Tags: geometry, inequalities, geometric inequality
qweDota
14.02.2018 08:46
find inequality in convex trigonometric functions and use jensen
sqing
14.02.2018 11:34
qweDota wrote: In a circle with a radius $R$ a convex hexagon is inscribed. The diagonals $AD$ and $BE$,$BE$ and $CF$,$CF$ and $AD$ of the hexagon intersect at the points $M$,$N$ and$K$, respectively. Let $r_1,r_2,r_3,r_4,r_5,r_6$ be the radii of circles inscribed in triangles $ ABM,BCN,CDK,DEM,EFN,AFK$ respectively. Prove that.$$r_1+r_2+r_3+r_4+r_5+r_6\leq R\sqrt{3}$$. See also here https://artofproblemsolving.com/community/c6h1574314p9683141
qweDota
17.02.2018 05:19
Amazing problem.Convexity,concavity,Jensen,inequality Solution here:
Attachments:
solution.pdf (112kb)