USAMO $2008/4$ Triangulation of regular n-gonLet $ \mathcal{P}$ be a convex polygon with $ n$ sides, $ n\ge3$. Any set of $ n - 3$ diagonals of $ \mathcal{P}$ that do not intersect in the interior of the polygon determine a triangulation of $ \mathcal{P}$ into $ n - 2$ triangles. If $ \mathcal{P}$ is regular and there is a triangulation of $ \mathcal{P}$ consisting of only isosceles triangles, find all the possible values of $ n$.
Sol- In this [1] solution ,page number $9$, I have some doubts which i mention below,
a) In the last claim when we are repeating the argument with the triangles on each half why the number $n − 1$ must be a power of $2$ ?
$n=17$ (for reference)
[2]
thankyou very much
In the last claim when we are repeating the argument with the triangles on each half why the number $n − 1$ must be a power of $2$ ?
We shall take a look at small examples. For $n=7$:
For $n=9$:
As we can see from above, the inner lines that connect the outer points(which are paired) must be also paired. That is to say, the number of points $\dfrac{n-1}2$ should be always divisible by $2$ whenever doing $N\to N/2$. Thus, $\dfrac{n-1}{2}$ is a power of $2$ and so is $n-1$.
