“Deliberately” vs. “intentionally” vs. “on purpose”

Solution 1:

The levels of formality are, in descending order: intentionally, deliberately, and on purpose. If you look at published books and journals that have blank pages, you'll find some with printed statements "This page intentionally left blank". That's because it's the most formal and the most neutral. Yes, deliberately is a synonym that means intentionally, but it more often has a negative connotation than intentionally does (especially among high school students, at least that was true when I was in high school), although both words are used both positively and negatively, e.g., That was {a deliberate / an intentional snub} (negative use, and Google Ngrams shows more frequent use of the former than the latter). Maybe that's merely because deliberately is more frequently used than intentionally is.

Wikipedia has an article on deliberate practice, however, which shows that word's very positive connotation. There are also Web pages for intentional practice that show its equally positive connotation. Google Ngrams shows that deliberate practice is more often used than intentional practice. On purpose is what children say for intentionally and deliberately, which is why it's the at the lowest level of formality of those three terms. They are not always interchangeable: register and context must be taken into account.

Solution 2:

They are synonyms, which as with all synonyms means there is a slight difference in nuance (no two words are exactly identical).

Intentionally and on purpose differ mainly in register. Deliberately also carries another meaning of doing something with great care (compare deliberated), which while not necessarily implied in the sense you are talking about, does colour the word.

So, you can use any of them to convey the same meaning, but you might consider those subtle differences when aiming for the perfect choice in more considered writing.