Author: mathproblemsolvingskills

I coach students preparing for MathCounts and the AMCs, and I teach using curricula published by Art of Problem Solving.

2021 AIME I #13

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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

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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

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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

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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.