A vizsga után a barátom olyan fáradt, hogy azonnal hazamegy.

Because Hungarian uses the definite article a / az much like English the.

• A vizsga = the exam
• a barátom = my friend

Even though barátom already means my friend, Hungarian still often uses the article before possessed nouns:

• a barátom = my friend
• a testvérem = my sibling
• a házam = my house

So a barátom is completely normal Hungarian.

Because után is a postposition. Hungarian often uses postpositions where English uses prepositions.

• vizsga után = after the exam
• literally: exam after

The noun before után usually stays in its basic form, often with an article if needed:

• az óra után = after the lesson
• ebéd után = after lunch

So a vizsga után is built as:

• a vizsga = the exam
• után = after

Barátom means my friend.

It is made from:

• barát = friend
• -om = my (a possessive ending)

So Hungarian often says possession with an ending on the noun instead of a separate word like my.

Examples:

• barátom = my friend
• könyvem = my book
• házam = my house

The exact form of the possessive ending changes depending on vowel harmony and the shape of the word.

In Hungarian, the verb van = is / are is usually left out in the present tense, third person singular and plural when the predicate is a noun or adjective.

So:

• A barátom fáradt. = My friend is tired.
• literally: My friend tired.

But in other persons, van is not omitted:

• Fáradt vagyok. = I am tired.
• Fáradt vagy. = You are tired.

So a barátom olyan fáradt is normal Hungarian.

This is a very common Hungarian structure meaning so ... that.

• olyan fáradt, hogy... = so tired that...

It links a degree with a result:

• Olyan magas, hogy mindenkit lát. = He is so tall that he can see everyone.
• Olyan hideg van, hogy nem megyünk ki. = It is so cold that we are not going out.

So in your sentence:

• olyan fáradt = so tired
• hogy azonnal hazamegy = that he goes home immediately

Here hogy introduces a clause, like English that in so tired that...

So:

• olyan fáradt, hogy... = so tired that...

Hungarian uses hogy very often to introduce subordinate clauses.

Examples:

• Tudom, hogy igaz. = I know that it is true.
• Azt mondta, hogy jön. = He said that he is coming.

In this sentence, hogy connects the cause/intensity to the consequence.

Because haza and megy form a single verb meaning go home.

• haza = homeward / to home
• megy = goes

Together:

• hazamegy = goes home

This is very common in Hungarian. Directional elements often combine with verbs:

• bemegy = goes in
• kimegy = goes out
• felmegy = goes up
• lejön = comes down
• hazajön = comes home

So hazamegy is not two separate words in normal spelling.

Yes, Hungarian word order is flexible, but it is not random. It often shows focus, emphasis, or what is being presented as most important.

In ...hogy azonnal hazamegy, the adverb azonnal naturally comes before the verb:

• azonnal hazamegy = he goes home immediately

This is a very neutral and natural order.

Other orders are possible, but they can sound more marked or emphasize something else. For example:

• hazamegy azonnal can sound less neutral in many contexts
• AZONNAL hazamegy strongly emphasizes immediately

So the sentence uses a normal, straightforward word order.

Because Hungarian verb forms usually show the subject clearly enough, so subject pronouns are often omitted.

• megyek = I go
• mész = you go
• megy = he/she goes

Here hazamegy already means he/she goes home, and the subject is understood from a barátom earlier in the sentence.

Hungarian uses pronouns like ő only when needed for emphasis, contrast, or clarity.

Yes, hazamegy is present tense: he/she goes home.

Hungarian often uses the present tense for actions that are:

• habitual
• general
• scheduled
• immediate or very likely in context

So here it works naturally as a present-tense statement about what happens as a result of being so tired.

Hungarian can express future time with present tense plus context, or with fog:

• hazamegy = goes home / will go home depending on context
• haza fog menni = will go home

In your sentence, the plain present is completely natural.

Fáradt is simply the adjective tired.

In this sentence it works as the predicate adjective:

• A barátom fáradt. = My friend is tired.

Hungarian adjectives do not need to change here for gender or number the way they do in some other languages. Hungarian has no grammatical gender, so the same form works for he, she, or it.

So:

• a fiú fáradt = the boy is tired
• a lány fáradt = the girl is tired

Yes, that is also grammatical.

Both of these are possible:

• A vizsga után a barátom olyan fáradt, hogy azonnal hazamegy.
• A barátom a vizsga után olyan fáradt, hogy azonnal hazamegy.

The difference is mainly about what comes first in the sentence and what feels like the starting point.

• A vizsga után... puts the time frame first: After the exam...
• A barátom... puts my friend first as the topic.

Hungarian word order often reflects discourse structure more than strict grammatical rules. Both versions are natural, but the original sentence foregrounds after the exam.

Yes. Vizsga is a general word for exam or test, especially in a formal sense.

Depending on context, it could mean:

• a school exam
• a university exam
• a driving exam
• another formal examination

If the context is already known, a vizsga simply means the exam.