Let the tangent line passing through a point $A$ outside the circle with center $O$ touches the circle at $B$ and $C$. Let $[BD]$ be the diameter of the circle. Let the lines $CD$ and $AB$ meet at $E$. If the lines $AD$ and $OE$ meet at $F$, find $|AF|/|FD|$.
2009 Turkey Junior National Olympiad
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In the beginnig, each square of a strip formed by $n$ adjacent squares contains $0$ or $1$. At each step, we are writing $1$ to the squares containing $0$ and to the squares having exactly one neighbour containing $1$, and we are writing $0$s into the other squares. Determine all possible values of $n$ such that whatever the initial arrangement of $0$ and $1$ is, after finite number of steps, all squares can turn into $0$.
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The integer $n$ has exactly six positive divisors, and they are: $1<a<b<c<d<n$. Let $k=a-1$. If the $k$-th divisor (according to above ordering) of $n$ is equal to $(1+a+b)b$, find the highest possible value of $n$.