4

Part of 2015 Vietnam National Olympiad

Problems(1)

VMO 2015 problem 4

Source: VMO 2015

Given a circumcircle

(O)

and two fixed points

B,C

(O)

BC

is not the diameter of

(O)

A

(O)

ABC

is an acute triangle.

E,F

is the foot of the altitude from

B,C

respectively of

ABC

(I)

is a variable circumcircle going through

E

F

I

.
a) Assume that

(I)

BC

D

\frac{DB}{DC}=\sqrt{\frac{\cot B}{\cot C}}

(I)

BC

M

N

H

be the orthocenter and

P,Q

be the intersections of

(I)

(HBC)

. The circumcircle

(K)

P,Q

(O)

T

T

is on the same side with

A

PQ

). Prove that the interior angle bisector of

\angle{MTN}

passes through a fixed point.

geometrycircumcircleangle bisector