I would get rid of these links.

I'll bet that very few people use them.

They "should not" be the problem since they are "nofollowed".  However, you never know what google is doing.

This massive number of links are going to evaporate a lot of pagerank.  No need to do that.

If it was really important to link to these other sites I would make one internal page such as "about us" or "our international sites" and link to them on that one page with a followed link.  That link will be anchored with the domain name or the official company name.

It sounds like you are prepared to spend a ton of money on engineering and promotion.  I would instead put all of that money into making the prize a spectacular amount.  Not $500, not $5000... but spectacular.  The promotion will take care of itself.   Consider opening the contest to a very broad geographic area - line the northern hemisphere.  Get permission to publish the essays.   Now you have what you need.

Substance will defeat SEO.

I was originally using on-page anchor links like this...

Brass Widgets

In my opinion these are some of the most powerful on-page optimization elements that you can use, ranking just below the title tag in their effectiveness.   Perhaps removing them caused the rankings drop and putting them back caused the rankings recovery?

I stopped using these types of pages.

In my opinion my long tail traffic suffered and visitor engagement dropped.

Same content before and after...

This was not an extensive text.  Just a few pages.  But I am not using them anymore.  Instead I am going to build bigass pages, which any noob will know how to use... and will appear more content rich to people who visit.