2016 AIME I #10

May 7, 2022 mathproblemsolvingskillsLeave a comment

A strictly increasing sequence of positive integers $a_1, a_2, a_3, \dots$ has the property that for every positive integer $k$ , the subsequence $a_{2k-1}, a_{2k}, a_{2k+1}$ is geometric and the subsequence $a_{2k}, a_{2k+1}, a_{2k+2}$ is arithmetic. Suppose that $a_{13} = 2016$ . Find $a_1$ .

2020 AIME II #8 Video Solution

October 25, 2021April 16, 2022 mathproblemsolvingskillsLeave a comment

Define a sequence recursively by $f_1(x)=|x-1|$ and $f_n(x)=f_{n-1}(|x-n|)$ for integers $n>1$ . Find the least value of $n$ such that the sum of the zeros of $f_n$ exceeds $500,000$ .

2020 AIME II #9 Video Solution

October 18, 2021April 16, 2022 mathproblemsolvingskillsLeave a comment

While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next to each other after the break. Find the number of possible seating orders they could have chosen after the break.

2020 AIME II #12 video solution

October 9, 2021April 16, 2022 mathproblemsolvingskillsLeave a comment

I could not find a great solution to the 2020 AIME II #12, so I created a video solution myself. It is a nice geometric problem with lots of inequalities.