MathDB

2012-2013 Winter OMO #40

Source:

January 16, 2013

Online Math Openratiotrigonometrygeometrysimilar trianglestrig identitiesLaw of Cosines

Problem Statement

Let

ABC

be a triangle with

AB=13

,

BC=14

, and

AC=15

. Let

M

be the midpoint of

BC

and let

\Gamma

be the circle passing through

A

and tangent to line

BC

at

M

. Let

\Gamma

intersect lines

AB

and

AC

at points

D

and

E

, respectively, and let

N

be the midpoint of

DE

. Suppose line

MN

intersects lines

AB

and

AC

at points

P

and

O

, respectively. If the ratio

MN:NO:OP

can be written in the form

a:b:c

with

a,b,c

positive integers satisfying

\gcd(a,b,c)=1

, find

a+b+c

.
James Tao

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