MS Lesson Inspired by the IMO

[Sean Corey]

Hey Yall,

My legg is still broken, which gives me plenty of time to try and tackle some IMO problems. While struggling through my first IMO problem I realized that they are chock full of incredible mathematical properties that could be posed to middle schoolers. 

From the following problem taken fromt the 2011 IMO:

I came up with this list of MS level challenges:

Any thoughts?

-Sean

[Sean Corey]

New Problem  

From the 2012 Manhattan Math Olympiad grades 7-8:

A new problem for 9-12 might be:

The "five-torial" of a number is the normal factorial, except each factor of five has multiplicity 2.  For instance, 7 five-torial is 7*6*5*5*4*3*2*1.

In a "five five-torial" each factor of five has multiplicity of 3.  

What is the minimum number of fives in a five...five-torial such that the last non-zero digit of 2013 five....five-torial is odd?

-Sean