2013 Aime I New! -

A complex number geometry problem involving regular polygons on the complex plane. It required using roots of unity and distance formulas. The solution elegantly reduced to a trigonometric sum. Students who memorized (\sum \cos^2(k\theta)) formulas had a significant advantage.

This problem required a deep command of similar triangles and coordinate geometry. The configuration was complex, involving two circles tangent to the sides of a right triangle (since $3-4-5$ is right). The computation involved setting up equations based on the tangency conditions. Many students who attempted this problem spent the better part of an hour on it, only to fall victim to an algebraic slip. The solution relied on identifying the centers of the circles and utilizing the slope of the lines effectively, eventually yielding an answer that was not an integer (which is unique for AIME problems, as answers are always 2013 aime i

To appreciate the intricacy of the 2013 AIME I, let’s look at a few notable problems that challenged students. A complex number geometry problem involving regular polygons

The AIME is unique because it removes the safety net of guessing. On the AMC, a student can sometimes eliminate answers or plug in choices to find the solution. On the AIME, if a student makes a minor calculation error—misplacing a negative sign or miscalculating a modulo—the answer is simply wrong. The 2013 AIME I exemplified this rigorous standard. Students who memorized (\sum \cos^2(k\theta)) formulas had a