MO in Albania 2004.

[rsoldo]

Let $O$ be a given point inside the rectangle $ABCD$. Determine the points $M, N, P$ on the sides $BC, CD, DA$, so that the length of the broken line $OMNPB$ is minimal.

[zykov]

Reflect $ABCD$ about line $BC$ and get $A_1BCD_1$.
Reflect $A_1BCD_1$ about line $CD_1$ and get $A_2B_2CD_1$.
Reflect $A_2B_2CD_1$ about line $A_2D_1$ and get $A_2B_3C_3D_1$.

Intersect line $OB_3$ with $BC$, $CD_1$, $D_1A_2$ getting points $M$, $N_1$, $P_2$.
Reflect $N_1$ about line $BC$ and get $N$.
Reflect $P_2$ about line $CD_1$ and get $P_1$. Reflect $P_1$ about line $BC$ and get $P$.

[zykov]

In this solution points $M$ and $N$ may happen to be outside the intervals $BC$ and $CD$.

In this case both $M$ and $N$ will coincide with the point $C$ (happens if the point $O$ was on or below the line $CB_3$). In this case point $P_2$ will be the intersection between lines $CB_3$ and $D_1A_2$. The rest is the same.