Chinese Girls' Mathematical Olympiad 2015,P1

[tanmoy]

Let $\triangle ABC$ be an acute-angled triangle with $AB>AC$, $O$ be its circumcenter and $D$ the midpoint of side $BC$. The circle with diameter $AD$ meets sides $AB,AC$ again at points $E,F$ respectively. The line passing through $D$ parallel to $AO$ meets $EF$ at $M$. Show that $EM=MF$

[asif e elahi]

Let $P$ be the reflection of $A$ across $D$. Then prove that $ \triangle DEF \cup M \sim \triangle CAP \cup D.$

[tanmoy]

$\text{My Solution}$:

Let $M$ be a midpoint of $EF$.I'll prove that $AO \parallel DM$.Let ${A}'$ be the antipode of $A$ and let $AD \cap \odot ABC=X$.
Now,$\Delta DEF\sim \Delta BXC\Rightarrow \Delta DFM\sim \Delta DCX$.$\therefore$ $\measuredangle FDM=\measuredangle ADF-\measuredangle ADM=\measuredangle ABC......(i)$

Again,$\measuredangle ABC=\frac{\Pi }{2}-\measuredangle {A}'BC=\frac{\Pi }{2}-\measuredangle OAC=\frac{\Pi }{2}-\measuredangle DAF-\measuredangle OAD=\measuredangle ADF-\measuredangle OAD...........(ii)$

From (i) and (ii),we get that $\measuredangle ADF-\measuredangle ADM=\measuredangle ADF-\measuredangle OAD \Rightarrow \measuredangle ADM=\measuredangle OAD$.S0,$AO\parallel DM$.

[rah4927]

Can you please give me a link to CGMO problems of 2015? Aops doesn't seem to have them yet.

[tanmoy]

Can you please give me a link to CGMO problems of 2015? Aops doesn't seem to have them yet.

The question paper has not yet been published.