I personally consider a problem fun to solve if the idea to solve involves some observations that you needed to stitch together to get the solution. Basically not some standard algorithm or do as the question says.

For many people, the main aspect is unfortunately the shortness of its statement.

I love some of those problems in which a very small concept or a fine detail sometimes makes everything supereasy

Positive factors:

• describes a situation from real life or game
• simple statement (NOT the same as short statement)
• solution doesn't require complicated techniques

Negative factors:

• Alice got an array of length N as a birthday present and want to answer M "xor of f(g(h(x)))" queries on range [L,R]
• calculate some useless math shit modulo 998353 nobody ever cares about

I like when in a task you need to make a few observations, understand / feel which of them is reasonable to develop (maybe several recursive observations), these observations mustn't immediately lead to the solution ofc. If the task is hard, then the solution itself can represent some serious technique, and ideally, the balance between the technique and the complexity of the observations should be kept.

That is, I like when there are a lot of possibilities where to go, what observations to use or even something to take somewhere on faith. I'm not very fan of problems where there is a one-step strange (magic) solution, but I think such problems will appear in future contests a lot, cause people like it.

Don't think that one can build good standard to rate problems it's more like opinion.

In my opinion

(+) Clear statement, easy to read in English, easy to understand the statement

(+) The problem doesnt required complicated combined tricks-algorithms-methods to solve

(+) Strong pretests & good example, maybe adding simple explanation will be good

(+) The hard of two near problems is not too different (like: easyA vs too hardB vs normal C)

(+) Can gain some experience or learning new things from solving the problems

(-) Complicated things hidden in the statement and hard to understand

(-) Long & Very long statement make coder hard to find the point of the problem

(-) Weird modulo and restriction can make some coder confuse when they dont read the statement carefully

Well, These are my views and I don't expect everyone agrees to them.

1) The solution of the problem should be based on a certain idea or observation and the implementation of the problem should not be super lengthy. See I have recently seen a problem Divisor Paths which I found very interesting because it was based on a certain idea and once you reach that idea you can easily implement it.

2) The statement should not be too lengthy or in other words, nobody wants to know why Nastya is throwing doors at the mountains to break it. I am not totally against the problem having a story but we can keep it separate like this contest

3) Like many times people tell that they had seen a similar question somewhere else and it really hurts when you were not able to solve that problem. So before adding a problem, setters should ensure that the problem or such a similar problem is not available anywhere else.

4) It's better when the test cases are explained properly by using diagrams and so.

Of course, beauty is in the eye of the beholder. Also, it's more about a problem in a problemset, not invidually.

• Most problems should be very original. No problem should be completely unoriginal except when it's specifically some training round for noobs.
• Difficulties should fit.
• Most problems shouldn't be too difficult to implement (compared to how difficult it is to figure out the idea) or just with too much casework.
• The topics should be sufficiently different between problems.
• "Decider" problems, i.e. last in division that you expect more than one person to solve, should be held to a higher standard.
• Things like deciphering the statement or printing in a specific format should be minimised.
• It's ok to have a "bad" problem if the rest of the problemset is "good".

Then, I don't like problems that involve too well-known or too unknown theory. Not in the sense that it's hard to figure out, but that if you have some specific background, it's very easy, and if you don't, it's very hard. FFT used to fit that many years ago before it became more common.

The number of times the word "statement" is found in this comment section is scary.

If I enjoyed solving the problem, it is good. If I didn't — it is bad.

I find that it is a lot more interesting to describe what makes a problem bad? and what doesn't make a problem bad? :P

There are some negative comments here about "complicated techniques". To me the use of advanced algorithms and data structures by itself is not a bad thing. I think that there is something wrong (and I hope that's what the others mean too) if knowing the complicated technique is the hardest part of the problem.

For example, here is a really bad problem (I think there was something like that in a real contest):

You are given an array of DAGs, there is a token on each DAG. Two players play a game, on every turn the player moves one of the tokens. The loser is the one who cannot move. You're given $$$Q$$$ queries of the form "who wins if we only consider the subarray $$$[l, r]$$$".

To me this is a really bad problem because you will solve it if and only if you understand prefix sums and the theory of Grundy numbers. It may be considered as an example problem in a tutorial or lecture, but IMO it has no place in a serious contest.

But here is something nice: ARC 061 F, coincidentally also about games. Probably some people commenting here wouldn't like it because it contains FFT which is some evil hard technique. But the problem isn't about FFT. You have to make some observations about the game. Then the problem becomes efficiently evaluating a certain combinatorial formula. Then you will notice that it makes sense to calculate together things that have $$$i + j = k$$$, for every $$$k$$$. And then you see the application of FFT. Although knowing that FFT exists is a part of the problem, it's not the problem.

PS. Grundy numbers and FFT may not be "advanced enough" for some readers, but the same idea holds with even harder concepts as well. I just wanted relatively accessible examples.

Uniqueness of the problem's idea.

Good: math. Bad: data structures shits.

For me the most interesting problems are simple and neat yet hard. I love problems where I ask myself how and why this isn't already well-known.

Also, sometimes complicated problems have really neat solutions so I would consider those also really nice.

In summary, the best problems for me have simple statements, are something which one could naturally ask and their solutions are original and creative.