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Regional Olympiad - FBH 2016 Grade 11 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2016

September 22, 2018

geometrycircumcircle

Problem Statement

h_a

h_b

and

h_c

are altitudes,

t_a

t_b

and

t_c

are medians of acute triangle,

r

radius of incircle, and

R

radius of circumcircle of acute triangle

ABC

. Prove that

\frac{t_a}{h_a}+\frac{t_b}{h_b}+\frac{t_c}{h_c} \leq 1+ \frac{R}{r}