# 2020 AMC 12 A #9

Watch me draw the worst approximations of trig graphs. Yikes.

# 2012 AMC 12 B #20

Good luck to everyone on the AMC 10 A and 12 A tomorrow! Here’s an old problem from 2012 about the side lengths that result in a valid trapezoid. (Spoiler: it’s an application of triangle inequality.)

# 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 $\omega_2$.

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# 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 $m+n$.

# 2013 AMC 12 B #18

Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbara’s turn, she must remove 2 or 4 coins, unless only one coin remains, in which case she loses her turn. When it is Jenna’s turn, she must remove 1 or 3 coins. A coin flip determines who goes first. Whoever removes the last coin wins the game. Assume both players use their best strategy. Who will win when the game starts with 2013 coins and when the game starts with 2014 coins?

# 2021 AIME I #8

(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

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

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.