From personal experience I have found that once there are 50+ answers, there is just about no chance for an answer to reach one of the top 25 spots.
I've seen the contrary happen a lot too, especially to quality answers. In last year's contest, I posted my answer as the 34th entry. I ended up well in the top 10. This answer was posted around the 50 answer mark you mention, and ended up second. I know the knitting post from this year isn't a 'contest', but it's a fun post too. I uploaded my ugly sweater(s) as the 64th answer. They're now 'in fourth place'.
The current nr. 1 entry for the 'Time for more swag' contest was posted a day after it had started, as the 60-something answer. The current 4th answer was posted another day later (I'm going to spare myself the effort of counting through the timeline on that post again), and the answer now in 11th place was also posted after the 50 answer mark you mention.
The worst part is that this encourages low-effort speedy entries, ruining the chances for entries which take time to create.
First of all, low-effort doesn't mean bad. These contests are supposed to be fun, and saying someone shouldn't win because their answer was 'made' and posted within a few hours is not really nice towards the creativity and thoughts these people put into their entries. What does 'low-effort' mean? Sometimes, these entries can be just as fun, and fun is what the contest is about, not the amount of hours you spent creating something.
While there are quite a few speedy entries on the latest contest, I feel most quick entries are just as creative or fun as some later posted entries. I like watching plushies time-travel in a cardboard box, even if that didn't take days to create. In fact, when it came the 'Time for more swag' contest, I felt most of the good entries were made quickly, and there was a flood of one-liners, meme's, similar ideas, etc. only after those posts were already made.
I agree that those do clutter up the contest and that it's probably frustrating for that one person that does put time and effort in their answer, that after a few days still manages to upload an original, well-executed idea, only to have it drowned out by 4 pages of those kinds of answers. So there might be something to gain from e.g. random sorting, or showing the newest answers first (not active, newest. Edits don't count for more visibility).
Actively discouraging low-effort entries while actively encouraging great entries; the biggest gain. This will happen because everyone will know that there has been time for others to create great entries. To compete, their entry has to rely on its worth alone, not on speed.
Given the above, and my general faith in humanity, I don't really see how announcing the contest beforehand will stop answers from pouring in as soon as it becomes known there will be a contest. You're always going to have people in other time zones, are they going to have to stay up late to upload their answer ASAP?
An announcement isn't going to solve this FGITW problem you see, it only gives people a few days to prepare stuff, to prepare the same post they would've posted anyways. This includes meme's or one-liner jokes, or whatever else people consider 'low-effort, low quality'.
An announcement won't stop answers from cluttering things up, and I don't think it will lead to higher quality entries. A fun post that can be made in a few hours, is a fun post that can be made in a few hours. And someone that wants to upload a meme or one-liner joke or photoshopped whatever is going to do it anyways.
Giving answers an equal chance. Having lots of entries at a similar score as once will encourage users to look through the answers before just voting on high-scoring ones.
If you ask me, an announcement won't solve this enough. Having 80 answers in 2 hours is just as bad as having 80 answers over the course of two days: People that aren't interested aren't going to sift through it to see if there's something they like there. They'll see a few answers on the front-page, drop a few votes, and drop out again.
Even e.g. delaying the voting on answers until a contest is 'opened' (a bit like elections, only voting after having all nominations) and then displaying posts in random order isn't going to guarantee each post equal attention, in my opinion. 150 answers to a single swag contest is just too much, not many people will have the time and dedication to spend that much time reviewing each answer and judging it for how creative and fun it is.