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Problem 5 vietnamese tst 2006

Source: Vietnamese TST 2006

April 18, 2006

geometrycircumcircleparallelogramratioquadraticspower of a pointradical axis

Problem Statement

Given a non-isoceles triangle

ABC

inscribes a circle

(O,R)

(center

O

, radius

R

). Consider a varying line

l

such that

l\perp OA

and

l

always intersects the rays

AB,AC

and these intersectional points are called

M,N

. Suppose that the lines

BN

and

CM

intersect, and if the intersectional point is called

K

then the lines

AK

and

BC

intersect.

1

, Assume that

P

is the intersectional point of

AK

and

BC

. Show that the circumcircle of the triangle

MNP

is always through a fixed point.

2

, Assume that

H

is the orthocentre of the triangle

AMN

. Denote

BC=a

, and

d

is the distance between

A

and the line

HK

. Prove that

d\leq\sqrt{4R^2-a^2}

and the equality occurs iff the line

l

is through the intersectional point of two lines

AO

and

BC

.

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