2008 USAMO Problems/Problem 4

Problem

(Gregory Galparin) Let $\mathcal{P}$ 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 $n$ .

Solution

We label the vertices of $\mathcal{P}$ as $P_0, P_1, P_2, \ldots, P_n$ . Consider a diagonal $d = \overline{P_a,P_{a+k}},\,k \le n/2$ in the triangulation. This diagonal partitions $\mathcal{P}$ into two regions $\mathcal{Q},\, \mathcal{R}$ , and is the side of an isosceles triangle in both regions. Without loss of generality suppose the area of $Q$ is less than the area of $R$ (so the center of $P$ does not lie in the interior of $Q$ ); it follows that the lengths of the edges and diagonals in $Q$ are smaller than $d$ . Thus $d$ must the be the base of the isosceles triangle in $Q$ , from which it follows that the isosceles triangle is $\triangle P_aP_{a+k/2}P_{a+k}$ , and so $2|k$ . Repeating this process on the legs of isosceles triangle ( $\overline{P_aP_{a+k/2}},\,\overline{P_{a+k}P_{a+k/2}}$ ), it follows that $k = 2^m$ for some positive integer $m$ .

Now take the isosceles triangle $P_xP_yP_z$ in the triangulation that contains the center of $\mathcal{P}$ in its interior; if a diagonal passes through the center, selecting either of the isosceles triangles with that diagonal is an edge will suffice. Without loss of generality, suppose $P_xP_y = P_yP_z$ . From our previous result, it follows that (sum of powers of $2$ ), eg $n = 2^{a+1} + 2^{b}$

We now claim that this condition is sufficient. Suppose WLOG that $a+1 \ge b$ ; then we rewrite this as $n = 2^{b}(2^{a-b+1}+1).$

Lemma 1: All $n = 2^k + 1$ and $n=4$ have triangulations that work.
Lemma 2: If $n$ -sided polygon works, then $2n$ -sided polygon works.

By induction, it follows that we can cover all the $n$ .

Proof 1: For $n = 3,4$ , this is trivial. For $k>1$ , construct $P_0P_{2^k}$ and $P_{2^k+1}P_O$ , then make midpoint isosceles triangles, repeat.

Proof 2: Construct every other edge, then reduce etc.

Template:Incomplete

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

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2008 USAMO (Problems • Resources)
Preceded by Problem 3	Followed by Problem 5
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2008 USAMO Problems/Problem 4

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