AIME 2015 Problem 11

Geometry Level 5

The circumcircle of acute $\Delta ABC$ has center $O$ . The line passing through point $O$ perpendicular to $OB$ intersects lines $AB$ and $BC$ and $P$ and $Q$ , respectively. Also $AB=5$ , $BC=4$ , $BQ=4.5$ , and $BP=\dfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Try more problems of AIME from this set AIME 2015

The answer is 23.

1 solution

Niranjan Khanderia
Dec 15, 2015

$\text{Let OD and OE be the perpendicular bisectors of BC and AB resp. }~ \therefore ~ BD=2, ~and ~EB=\dfrac 52.\\ \because ~ \text{DOQ and OEP are right angles, draw } \bigodot ~ODQ ~ on ~ diameter ~ QO ~ and ~ \bigodot ~PEO ~ on ~ diameter ~ OP ~ resp.\\ \angle ~ s~QOB ~ and ~ BOP ~ are ~ rt. ~ \angle ~s.\\ \therefore ~ OB ~ is ~ a ~ TANGENT ~ to ~ both ~\bigodot ~s.\\ \therefore ~ BO^2=EB*BD=\dfrac 9 2 *2=9, ~ in ~ \bigodot ~ ODQ.\\ and ~ BO^2=PB*BE=PB*\dfrac 5 2, ~ in ~ \bigodot ~PEO.\\ \implies ~ PB=\dfrac 9 {\frac 5 2 }=\dfrac{18} 5.=\dfrac m n.\\ m+n=\Large \color{#D61F06}{23}\\$

AIME 2015 Problem 11

Try more problems of AIME from this set AIME 2015

The answer is 23.

1 solution

0 pending reports