## Monday, March 5, 2007

### Criss Cross Applesauce. Topic: Geometry. Level: AIME.

Problem: (2007 Mock AIME 6 - #2) Draw in the diagonals of a regular octagon. What is the sum of all distinct angle measures, in degrees, formed by the intersections of the diagonals in the interior of the octagon?

Solution: Consider the circumscribed circle of the octagon. Each diagonal is a chord of this circle, and we know that the angle between two chords that intercept arcs of measure $\alpha$ and $\beta$ is $\frac{\alpha+\beta}{2}$.

Now, any pair of diagonals can together intercept $2$, $3$, $4$, $5$, or $6$ little arcs (between vertices of the octagon). $1$ is not possible, because then on one side there is no little arc which means the intersection is not in the interior. This also means $7$ is not possible (they come in supplement pairs - except $90^{\circ}$ of course). Since each little arc is $45^{\circ}$ and we have to account for the division by $2$, the final sum is

$45 \cdot \frac{1}{2} \cdot (2+3+4+5+6) = 450$.

QED.

--------------------

Comment: This wasn't an extremely hard problem, but it needed some clever thinking. Most people overcounted, which is reasonable given that there are $20$ diagonals in an octagon. Pretty hard to draw accurately. Using arcs on a circle proved to be much easier and less prone to careless mistakes.

--------------------

Practice Problem: Let $R$ be a set of $13$ points in the plane, no three of which lie on the same line. At most how many ordered triples of points $(A,B,C)$ in $R$ exist such that $\angle ABC$ is obtuse?