When you do something resolutely, you are determined to make it happen. If you resolutely sit down and do your math homework, you'll work hard at it until it's done.
The adverb resolutely is perfect for describing something you do in a strong, purposeful way. You might resolutely decide to clean your room, and immediately make your bed and start filling up a garbage bag. Strangely, the original meaning of resolutely and its adjective form, resolute, was almost completely opposite to its meaning now: it meant "dissolved, or of loose structure," from a Latin root word, resolvere, "untie, unfasten, or loose."