# Problem 6 : Strange Fractions

Geometry Level 5

In a non-isoceles triangle $$ABC$$ let $$O$$ and $$I$$ be the circumcenter and the incenter respectively. Let $$D, E, F$$ be the midpoints of the sides $$BC, AC, AB$$ respectively. Let $$T$$ be the foot of perpendicular from $$I$$ to $$AB$$, $$P$$ be the circumcenter of triangle $$DEF$$, and $$Q$$ be the midpoint of $$OI$$. If $$A, P, Q$$ are collinear, the maximum possible value of $$|\frac{AO}{OD} - \frac{BC}{AT}|$$ is $$\frac{a\sqrt{b}}{c}$$, where $$b$$ is not divisible by the square of any prime and $$a$$ and $$c$$ are relatively prime positive integers. Determine the value of $$a + b + c$$.

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