9/15/2023

Today I did some more probability stuff, and in the time that I had left, I did a couple problems from the final five of the AMCs.

They were fine, but one of them specifically stood out to me because I found it pretty cool.

Problem:

For how many positive integers $n \le 1000$ is$$\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor$$not divisible by $3$? (Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.)

I'll do something different this time -- instead of posting the solution, I'll give everyone 3 days to try their attempts on the problem. Feel free to post your progress in the comments! If no one solves the problem in 3 days, I'll post my own solution.