Circles and with radii and , respectively, intersect at distinct points and . A third circle is externally tangent to both and . Suppose line intersections at two points and such that the measure of minor arc is . Find the distance between the centers of and .

# Tag: AIME

# 2021 AIME I #12

Let be a dodecagon (-gon). Three frogs sit at and . 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 , where and are relatively prime positive integers. Find .

# 2021 AIME I #8

(Note: I have the wrong problem number written in the video whiteboard.)

Find the number of integers such that the equation has distinct solutions.

# 2021 AIME I #7

Find the number of pairs of positive integers with such that there exists a real number satisfying .

# 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.

# 2021 AIME I #4

Find the number of ways identical coins can be separated into three nonempty piles so that there are fewer coins in the first pile than in the second pile and fewer coins in the second pile than the third pile.

# 2016 AIME I #15

Circles and intersect at points and . Line is tangent to and at and , respectively, with line closer to point than to . Circle passes through and intersecting again at and intersecting again at . The three points and are collinear, and . Find .

# 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.

# 2016 AIME I #13

This one is the classic jumping frog with state diagrams, this time in the coordinate plane between a fence and river.

# 2016 AIME I #12

Find the least positive integer such that is a product of at least four not necessarily distinct primes.