## BdMO 2013 Higher Secondary Problem 9

Discussion on Bangladesh Mathematical Olympiad (BdMO) National
FahimFerdous
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Joined: Thu Dec 09, 2010 12:50 am

### BdMO 2013 Higher Secondary Problem 9

Six points \$A, B, C, D, E, F\$ are chosen on a circle anticlockwise. None of \$AD, BE, CF\$ is a diametre. Extended \$AB\$ and \$DC\$ meet at \$Z\$, \$CD\$ and \$FE\$ at \$X\$, \$EF\$ and \$BA\$ at \$Y\$. \$AC\$ and \$BF\$ meets at \$P\$, \$CE\$ and \$BD\$ at \$Q\$ and \$AE\$ and \$DF\$ at \$R\$. If \$O\$ is the point of intersection of \$YQ\$ and \$ZR\$, find \$\angle XOP\$.

FahimFerdous
Posts: 176
Joined: Thu Dec 09, 2010 12:50 am

### Re: BdMO 2013 Higher Secondary Problem 9

Let \$AD \cap EF=T\$, \$BE \cap CD=S\$, \$AB \cap CF=K\$, \$AE \cap BD=L\$, \$CR \cap FQ=M\$. Then use Pascal's theorem and Desaurge's theorem to prove that \$P, Q, T; P, R, S; R, Q, K; X, O, M; L, M, P; X, L, P\$ are collinear (I'm leaving this tedious job to you guys, I can't write it). And it implies that \$X, O, P\$ are collinear.

I still can't believe I missed this and problem 8.

sourav das
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### Re: BdMO 2013 Higher Secondary Problem 9

You spin my head right round right round,
When you go down, when you go down down......
(-\$from\$ "\$THE\$ \$UGLY\$ \$TRUTH\$" )

*Mahi*
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### Re: BdMO 2013 Higher Secondary Problem 9

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Use \$L^AT_EX\$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi