"seem" or "seem to be"

Solution 1:

Grammatically, they are both correct and semantically, either will work, to be being elided in the second case. But yes, something is indeed missing or amiss here.

You didn't say why "Both sound weird to me." I would say the choice of words is inappropriate, which must be why.

seem syn. appear - look - sound
missing syn. lost - absent - wanting - lacking - gone
Naturally, directly juxtaposing seem with missing makes for a weird situation. It is correct, but it tends interfere with comprehension.

Using direct speech is one way out:

It seems the solution is missing.

Parenthesizing the contra word could also help:

The solution, apparently, is missing.

Solution 2:

They both work.

I personally prefer the sound of:

The solution seems to be missing.

If you are just worried about how the sentence sounds though, consider rearranging the words to give you things like:

It seems as though the solution is missing.

Depending on your preference, you might find you dispense with "seems" entirely and go with something like:

The solution appears to be missing.

Solution 3:

Either way is fine. They read a little weird being without apprpriate surrounding contexts that smooth them out, assumed to be reasoning prior to the conclusion that it looks like the actual solution is missing. By the way things can look like they're missing, either as a visual phenomena in surrounding context or metaphorically. In the latter case "seems" would be read as "apparently". 1, 2, 3, 4, 6, 7, 8, 9, 10. In this list 5 seems missing or seems to be missing. Either way, using "seems" is more succinct than most constructs to express that idea.