MathDB

some guy literally cry over this problem

Source: 2021 Thailand October Camp 4.2

May 6, 2023

geometry

Problem Statement

An acute triangle

ABC

has

AB

as one of its longest sides. The incircle of

ABC

has center

I

and radius

r

. Line

CI

meets the circumcircle of

ABC

at

D

. Let

E

be a point on the minor arc

BC

of the circumcircle of

ABC

with

\angle ABE > \angle BAD

and

E\notin \{B,C\}

. Line

AB

meets

DE

at

F

and line

AD

meets

BE

at

G

. Let

P

be a point inside triangle

AGE

with

\angle APE=\angle AFE

and

P\neq F

. Let

X

be a point on side

AE

with

XP\parallel EG

and let

S

be a point on side

EG

with

PS\parallel AE

. Suppose

XS

and

GP

meet on the circumcircle of

AGE

. Determine the possible positions of

E

as well as the minimum value of

\frac{BE}{r}

.

Back to Problems View on AoPS