2021 AIME I #13

Circles \omega_1 and \omega_2 with radii 961 and 625, respectively, intersect at distinct points A and B. A third circle \omega is externally tangent to both \omega_1 and \omega_2. Suppose line AB intersections \omega at two points P and Q such that the measure of minor arc \widehat{PQ} is 120^{\circ}. Find the distance between the centers of \omega_1 and \omega_2.


2021 AIME I #12

Let A_1A_2A_3...A_{12} be a dodecagon (12-gon). Three frogs sit at A_4, A_8, and A_{12}. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is \frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.

2021 AIME #5

Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences.

2016 AIME I #14

Centered at each lattice point in the coordinate plane are a circle with radius 1/10 and a square with sides of length 1/5 whose sides are parallel to the coordinate axes. The line segment from (0,0) and (1001, 429) intersects m of the squares and n of the circles. Find m+n.