Occult Research for azathoth on Eldritch Horror
Yes, resolving the encounter doesn't mean you have to resolve it successfully. Your clue gained during the encounter still entitles you to place it, as the Mystery states.
A few references to illustrate that the term resolve doesn't imply success or passing a test:
On page 9 of the rulebook, when describing Phase 2, the Encounter Phase:

During this phase, each investigator resolves one encounter.... is phase starts with the Lead Investigator. He resolves an encounter, then the investigator to his left resolves an encounter, and so on, until each investigator has resolved one encounter.

Under Token Encounters on the same page:

Clue tokens allow investigators to draw and resolve Research Encounter cards.
Gate tokens allow investigators to draw and resolve Other World Encounter cards.
Active Expedition token allows investigators to draw and resolve Expedition Encounter cards.

On Page 12 under Complex Encounters:

Expedition Encounters, Other World Encounters, and Special Encounters are rewarding but dangerous adventures called complex encounters. Each of these cards has three effects. To resolve these encounters, an investigator first resolves the initial effect (the top box of the card). Depending upon the result of this effect, he immediately resolves one of the other two effects.

These examples illustrate that resolving doesn't imply success, as the last case even talks of resolving and then branching based on the result of the effect (implying that 'resolve' could be a success or failure).