MathDB

pmo problem 8

Source: PMO 2021

March 20, 2021

geometryPMO

Problem Statement

In right triangle

ABC

,

\angle ACB = 90^{\circ}

and

\tan A > \sqrt{2}

.

M

is the midpoint of

AB

,

P

is the foot of the altitude from

C

, and

N

is the midpoint of

CP

. Line

AB

meets the circumcircle of

CNB

again at

Q

.

R

lies on line

BC

such that

QR

and

CP

are parallel,

S

lies on ray

CA

past

A

such that

BR = RS

, and

V

lies on segment

SP

such that

AV = VP

. Line

SP

meets the circumcircle of

CPB

again at

T

.

W

lies on ray

VA

past

A

such that

2AW = ST

, and

O

is the circumcenter of

SPM

. Prove that lines

OM

and

BW

are perpendicular.

Back to Problems View on AoPS