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Is an inconvenience without a solution still a problem?
An example would be you are stuck in a war. You obviously cannot fix the war (safe to assume that), so would it be a problem to you, or simply an inconvenience?
To add some maths: x + y > 0 where x > y and x is a negative number. Would this be a problem?
I think you are just redefining things, the problem still exists
in your example, you seem to have a problem that you can't solve, but the problem itself is solvable ( just not by you )
its like having the problem 2+2 = ??
and you can't solve it, so you say, pffff, math is inconvenient. The problem still exists, and has a solution, but its just you have decided not to care about it / reclassify it.
Now this is different from a problem that provably has no solution. Like "the halting problem" and leads into Gödel's incompleteness theorems. You may regarded it as inconvenience, but in essence it defines a 'limitation' of what one can know.
We can make a broad and sweeping general description of what a problem is: There is some way you'd like the world to look. The world does not look like that currently, and you are not sure what to do to make the world more like you want it to be.
The thing is, if faced with an alternative to the current world, you can rate it better or worse.
Problems are all those things that you can imagine or create a world which is better because of the existence of a solution.
This makes even unsolvable things like the Halting "Problem" into problems, just look at computer science discussion forums that sport a "What would you do with an infinitely fast computer?" thread.
An example would be you are stuck in a war. You obviously cannot fix the war (safe to assume that), so would it be a problem to you, or simply an inconvenience?
I may be able to help by answering the analogy of your question.
From the movie, Letters from Iwo Jima, General Kuribayashi truly did have compassion for the issues that Japan caused with the United States [he has visited the U.S. a many of times] by the bombing of Pearl Harbor. As in the movie, as soon as he got on the Island, he walked it to see what he could do for an already lost battle which he sacrificed himself for with his men... there was no way for Japan to save Iwo Jima from the United States.
To add some maths: x + y > 0 where x > y and x is a negative number.
Also, with the math side, a negative number is still on the number line, so it may not work out on the positive side of "idealism" but, there is still an answer to it.
How does the situation change if, let's say, you are sky diving. You jump out of the plane and realize that neither one both of your chutes will open, do you have a problem? I think you do not have a problem because there is no solution: you are going to die, 99.999%. I don't know what to call it but "problem" just doesn't seem to fit.
