$\#$AIME

Advanced Counting

AIME counting rarely yields to a formula: you build a recurrence, set up states, or find a symmetry. Count strings and tilings by their last move, compute expected waiting times with states, and let symmetry do the heavy lifting.

Polynomials

Newton's sums for power sums of roots you never compute, the roots-of-unity filter for every-third-coefficient sums, and the trick of building a helper polynomial that vanishes at the given points.

Geometry with Coordinates and Complex Numbers

When synthetic insight runs dry, compute. Place a figure on coordinates cleverly, use complex numbers for rotations, chase ratios with Ceva and Menelaus, and find lengths with Stewart, the law of cosines and the power of a point.

Cevians and Mass Points

Ratio-chasing recipes for the AIME: balance a triangle with mass points, read off every ratio along every cevian, and compute areas from ratios, including the famous triangle that is exactly one seventh of the whole.

Trig Formulas That Win

Double and triple angles, products that collapse with one clever multiplication, and sums of cosines that vanish because roots of unity balance. The trig moves that turn AIME problems into one line.

Recurrences and Characteristic Equations

Solve a linear recurrence the way you solve a differential equation: guess r^n, get a polynomial, combine the roots. Then spot recurrences hiding in counting problems, and catch sequences that secretly repeat.

Telescoping

Long sums and products that collapse to their first and last pieces. Learn to manufacture the difference that telescopes, using partial fractions, conjugates, logs and factorials, and turn 100-term computations into one line.

Heavy Machinery, Proved

The big AIME theorems, each with a proof you can rebuild in the exam: Stewart's theorem and the angle bisector theorem from the law of cosines, Ptolemy from similar triangles, and the roots-of-unity facts from one polynomial.