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

2012 AMC 12 B #20

November 10, 2022 mathproblemsolvingskillsLeave a comment

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

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 $m+n$ .

2013 AMC 12 B #18

September 29, 2022 mathproblemsolvingskillsLeave a comment

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 #10

September 18, 2022 mathproblemsolvingskillsLeave a comment

I can tell when students are ramping up their preparation for AMC because my solution videos get a lot more views. This one has been receiving a lot of views lately.

Consider the sequence $(a_k)_{k \ge 1}$ of positive rational numbers defined by $a_1 = \frac{2020}{2021}$ and for $k \ge 1$ , if $a_k = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$ , then $a_{k+1} = \frac{m+18}{n+19}$ . Determine the sum of all positive integers $j$ such that the rational number $a_j$ can be written in the form $\frac{t}{t+1}$ for some positive integer $t$ .