Impossible. You can check in a sentence like this by substituting "is" for "appears".

The correct word is "impossible". Impossible > Imposible. Impossibly > Imposiblemente. In Spanish it would sound very weird if you use "imposiblemente" instead of "imposible". "El problema parece imposiblemente de resolver." See? Sometimes all we need is to translate the words into our mother tongue. We don't need to think too much about adverbs and adjectives. Just a little bit of guesswork will sulfice. Hope it helps. =]

The correct word is impossible. Impossibly is the adverbial form.

Why an adjective afte a verb ? because /to appear/ is in the semntic category of /to be/, which usually requires adjectives as (in one of its several usages) it joins a noun to a state for it, i.e. an adjective. Or, just examine to the longer expression: The problem appears to be impossible to solve. I think you will not have any doubt that in THIS expression you need the adjective.

The correct word would be Impossible. Ed's substitution suggestion is a great way to check things like this. Another good way is to remove the extra information like "to solve" and see if the sentence works "The problem appears/is impossible" rather than "The problem appears/is impossibly" Hope this helps :)