MathDB

Italian MO Problem 6

Source: ITAMO 2018

May 5, 2018

geometry

Problem Statement

Let

ABC

be a triangle with

AB=AC

and let

I

be its incenter. Let

\Gamma

be the circumcircle of

ABC

. Lines

BI

and

CI

intersect

\Gamma

in two new points,

M

and

N

respectively. Let

D

be another point on

\Gamma

lying on arc

BC

not containing

A

, and let

E,F

be the intersections of

AD

with

BI

and

CI

, respectively. Let

P,Q

be the intersections of

DM

with

CI

and of

DN

with

BI

respectively.
(i) Prove that

D,I,P,Q

lie on the same circle

\Omega

(ii) Prove that lines

CE

and

BF

intersect on

\Omega

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