To submit an answer, you'd also have to do some "pointless" proof of work too for each submission. So you can't just spam an infinite amount of fake answers. You have to do real work that is easily verifiable also.
If you spam a million false answers, you also have to have the computing resources to solve a million proof of work problems too. And the prize for any one fraction of a problem would be minuscule. So the reward for lying through spam, and spending all that processing time and electricity, wouldn't be worthwhile.
Now you are just walking in circles.
So Bitcoin is bad because it "wasted" computing cycles doing "worthless work" (you still haven't realized that work is very valuable but we will save that for later).
So "solution" have Bitcoin do "useful work" however to prevent cheating we also require some "useless work" too.
You haven't prevented cheating.
If the value of useful work >0 then it is is always better to cheat. Right?
If I have x computing power I can operate legit and earn x reward OR I can cheat and for the % of total workload which is "useful work" I gain that as a bonus.
So if the network reward is based 70% useless (uncheatable) work and 30% useful (cheatable work) then I simply do the useless work and cheat on the rest and gain a 30% bonus in revenue.
Logically the only way to avoid that is make the useful work 0% and ... drumroll ...
you just invented Bitcoins' proof of work.Only a small percentage of the work miners would do would be "useless". Most of it is useful. The useless work would be a tiny token added on just to increase the cost of fraud.
The reason why you wouldn't want to cheat is that it wouldn't be profitable, even if you pull it off.
For example, let's say a researcher submits a problem which is divided into several million smaller problems. Each micro-problem is only worth a teensy fraction of a cent. Unless you have a supercomputer, even someone with a mining rig can only handle a small percentage of the total micro-problems, so everyone takes a random grab bag of micro-problems.
Let's say 10,000 miners are working on micro-problem #1,704,423 (along with many other micro-problems but this is one they all have in common). If you play fair and submit one answer for it, then your odds of winning are 1/10,000. This is profitable despite the low odds because you work on many micro-problems, and only expect to win a small percentage of them.
So by submitting one low cost "useless" proof of work answer you get a 1/10,000 chance of winning.
What if you want to cheat? Well, you have to overwhelm the network with false answers. So let's say you send in 10,000 false answers, to outnumber the other 9,999 people sending in answers for that micro-problem. So it's like buying $10,000 of tickets in a raffle to win $10,000. At best you break even.
And you can't predict in advance how many other people are going to submit answers to any micro-problem. So might end up spending too few entries and losing everything. Or you might spend more than you need to, and spend your entries for less than they're worth.
For example, you might only submit 4,000 false answers, and be outnumbered and have wasted all that proof of work. Or you might submit 15,000 answers, and have spent more than you needed to win a contest with 1/10,000 odds.