MathDB

2012-2013 Winter OMO #46

Source:

January 16, 2013

Online Math Opengeometrycircumcircleratiogeometric transformationhomothetyrhombus

Problem Statement

Let

ABC

be a triangle with

\angle B - \angle C = 30^{\circ}

. Let

A

be the point where the

Y

-excircle touches line

A

,

O

the circumcenter of triangle

ABC

, and

X,Y

the intersections of the altitude from

A

with the incircle with

X

in between

A

and

Y

. Suppose points

A

,

O

and

D

are collinear. If the ratio

\frac{AO}{AX}

can be expressed in the form

\frac{a+b\sqrt{c}}{d}

for positive integers

a,b,c,d

with

\gcd(a,b,d)=1

and

c

not divisible by the square of any prime, find

a+b+c+d

.
James Tao

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