2021 AIME I #13

October 26, 2022 mathproblemsolvingskillsLeave a comment

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

October 15, 2022 mathproblemsolvingskillsLeave a comment

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 I #8

July 17, 2022 mathproblemsolvingskillsLeave a comment

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

Find the number of integers $c$ such that the equation $||20|x|-x^2|-c|=21$ has $12$ distinct solutions.

2021 AIME I #7

July 5, 2022 mathproblemsolvingskillsLeave a comment

Find the number of pairs $(m,n)$ of positive integers with $1 \le m < n \le 30$ such that there exists a real number $x$ satisfying $\sin(mx)+\sin(nx)=2$ .

2021 AIME #5

June 18, 2022 mathproblemsolvingskillsLeave a comment

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

June 11, 2022 mathproblemsolvingskillsLeave a comment

Find the number of ways $66$ 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

June 3, 2022 mathproblemsolvingskillsLeave a comment

Circles $\omega_1$ and $\omega_2$ intersect at points $X$ and $Y$ . Line $l$ is tangent to $\omega_1$ and $\omega_2$ at $A$ and $B$ , respectively, with line $AB$ closer to point $X$ than to $Y$ . Circle $\omega$ passes through $A$ and $B$ intersecting $\omega_1$ again at $D \neq A$ and intersecting $\omega_2$ again at $C \neq B$ . The three points $C, Y,$ and $D$ are collinear, $XC = 67, XY = 47,$ and $XD = 37$ . Find $AB^2$ .

2016 AIME I #14

May 26, 2022 mathproblemsolvingskillsLeave a comment

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.