A ketrec nyitva van.

Hungarian has two forms of the definite article:

• a before a consonant sound
• az before a vowel sound

Since ketrec begins with k, you say a ketrec.

So:

• a ketrec
• az ablak

Both a and az mean the.

Ketrec is the basic dictionary form of the noun. Here it is:

• singular
• nominative
• with no grammatical gender

Hungarian does not have masculine, feminine, and neuter nouns the way many European languages do. So ketrec does not change based on gender, because Hungarian nouns simply do not work that way.

Nyitva is a special form built from the verb nyit meaning to open.

In sentences like this, nyitva expresses a state: something is in an open condition.

So nyitva van is a very common Hungarian pattern meaning something like:

• is open
• is in an opened state

You will see similar forms with other verbs too, for example:

• zárva van = is closed / is locked
• csukva van = is shut

That is a very common question.

Hungarian often does leave out van in the present tense in 3rd person when the predicate is a noun or adjective:

• A ketrec kicsi. = The cage is small.
• A fiú tanár. = The boy is a teacher.

But nyitva belongs to a different kind of structure. With these state expressions, van is normally used in a full standard sentence:

• A ketrec nyitva van.

You may sometimes hear shortened, more colloquial versions such as:

• Nyitva a ketrec.

But the full standard form is nyitva van.

Yes, A ketrec nyitott is also correct.

The difference is mainly one of nuance:

• nyitott is more like a regular adjective: open
• nyitva van emphasizes the current state of being open

In many everyday situations, both are natural and close in meaning.

A rough way to think about it is:

• nyitott = open
• nyitva van = is open, is standing open, has been left open

That said, the distinction is not always sharp, and native speakers often choose whichever sounds more natural in context.

Yes, both are possible.

Hungarian word order is flexible, but it is not random. It often reflects topic and focus.

• A ketrec nyitva van.
  This is a neutral statement. A ketrec is the topic, and the rest says something about it.

• Nyitva van a ketrec.
  This puts more attention on the fact that it is open.

There is also:

• A ketrec van nyitva.

That sounds contrastive, roughly like:

• It is the cage that is open
• The cage is the thing that is open

So the basic sentence is the most neutral version.

Usually no, not in normal everyday speech.

If you mean a specific cage, Hungarian normally wants the definite article:

• A ketrec nyitva van.

Without the article, it sounds unnatural in an ordinary sentence. A bare noun like that might only sound acceptable in very special styles, such as notes, labels, headlines, or very fragmented speech.

A rough pronunciation guide is:

• a ≈ a short o-like sound
• ketrec ≈ KET-rets
• nyitva ≈ NYIT-vaw
• van ≈ vawn

A few important details:

• ny is like the ny in canyon, or Spanish ñ
• the final c in ketrec is pronounced ts
• Hungarian stress is normally on the first syllable of each word

A more technical pronunciation is approximately:

• [ɒ ˈkɛtrɛt͡s ˈɲitvɒ vɒn]

You would say:

A ketrecek nyitva vannak.

Changes:

• ketrec → ketrecek = cages
• van → vannak = are
• nyitva stays the same
• the article is still a even though the noun is plural

So:

• A ketrec nyitva van. = The cage is open.
• A ketrecek nyitva vannak. = The cages are open.

You would say:

A ketrec nincs nyitva.

In the present tense, the negative of van in 3rd person singular is nincs.

So:

• van = is
• nincs = is not

Examples:

• A ketrec nyitva van. = The cage is open.
• A ketrec nincs nyitva. = The cage is not open.

For plural:

• A ketrecek nincsenek nyitva. = The cages are not open.

Yes, you may also hear:

A ketrec ki van nyitva.

Here, ki is a verbal prefix that often adds the idea of something being opened up or opened out.

This version can sound a bit more like:

• has been opened
• is opened up
• is open because someone opened it

In many contexts, nyitva van and ki van nyitva are both possible, but ki van nyitva can feel a little more explicitly connected to the action of opening.