Learn Russian: Быстрый ум помогает в любой ситуации.

While both translate to "any," любой emphasizes absolute freedom of choice (whichever situation you pick, it doesn't matter). всякий tends to mean "every kind of" or "various." Because this sentence is about a universal tool that applies to literally any scenario imaginable, любой is the most natural choice.

Why is the Prepositional case used in в любой ситуации rather than the Accusative?

The phrase describes the circumstances where the help happens, answering the question "where?" or "in what state?". This requires the preposition в plus the Prepositional case. The Accusative (в любую ситуацию) would imply motion or direction into a situation, such as getting into trouble.

Could I say быстрый мозг instead of быстрый ум?

No, that would sound strange. The word мозг refers to the physical, biological organ (the brain). When talking about intellect, wit, or the abstract mind, Russian uses ум.

Why does this sentence use the imperfective verb помогает instead of the perfective поможет?

The imperfective помогает is used because this is a general, timeless truth (a quick mind always helps). If you used the perfective поможет, which means "will help," it would change the meaning to a prediction about a specific, single situation in the future.