Этот знак принадлежит офису.

Because знак is a masculine singular noun, and этот has to agree with it in gender, number, and case.

• этот = masculine singular
• эта = feminine singular
• это = neuter singular
• эти = plural

So:

• этот знак = this sign
• эта книга = this book
• это письмо = this letter

Знак is in the nominative case because it is the subject of the sentence — the thing that belongs to someone or something.

In Этот знак принадлежит офису, the structure is:

• Этот знак = the subject
• принадлежит = belongs
• офису = to the office

Because the verb принадлежать (to belong) requires the thing it belongs to to be in the dative case.

So:

• офис = nominative
• офису = dative

This is the same pattern you see in sentences like:

• Книга принадлежит студенту. = The book belongs to the student.
• Дом принадлежит семье. = The house belongs to the family.

A useful rule to remember is:

• принадлежать + dative

Because офиса is the genitive form, not the dative form.

With принадлежать, Russian uses the dative, not the genitive:

• correct: принадлежит офису
• incorrect: принадлежит офиса

English uses to in belongs to the office, and in Russian that idea is expressed by the dative case.

Принадлежит means belongs or belongs to.

It is the 3rd person singular form of принадлежать:

• я принадлежу = I belong
• ты принадлежишь = you belong
• он / она / оно принадлежит = he / she / it belongs
• мы принадлежим = we belong
• вы принадлежите = you belong
• они принадлежат = they belong

In this sentence, знак is singular, so the verb is принадлежит.

Yes, normally принадлежать takes the dative case for the owner or possessor.

Examples:

• Эта машина принадлежит моему брату. = This car belongs to my brother.
• Квартира принадлежит компании. = The apartment belongs to the company.
• Эти документы принадлежат директору. = These documents belong to the director.

So in your sentence, офису is exactly what you should expect.

Yes. Russian word order is more flexible than English word order.

The neutral version is:

• Этот знак принадлежит офису.

But you could also say:

• Офису принадлежит этот знак.
• Принадлежит офису этот знак. (less neutral, more marked)

The basic meaning stays the same, but the emphasis changes.

• Этот знак принадлежит офису focuses naturally on this sign
• Офису принадлежит этот знак puts more emphasis on the office

Russian has no articles like a or the.

So знак by itself could mean a sign or the sign, depending on context.
In this sentence, этот makes it definite in the sense of this sign.

So:

• знак = a sign / the sign
• этот знак = this sign

Russian usually leaves definiteness to context, word order, or words like этот.

Usually yes, офис means office in the sense of a workplace or office space.

But learners should know that English office can mean different things, and Russian may use different words depending on context:

• офис = office workplace
• кабинет = office/room, especially a person’s office
• контора = office/firm, sometimes informal or old-fashioned

So офису here most naturally means to the office as an organization or place.

You simply add не before the verb:

• Этот знак не принадлежит офису.

That is the normal way to negate the sentence.

Yes. Russian often expresses possession in other ways, and принадлежать can sound a bit formal or explicit.

For example:

• Это знак офиса. = This is the office’s sign / a sign of the office.
• Это офисный знак. = This is an office sign.
• У офиса есть знак. = The office has a sign.

But these are not exactly identical:

• Этот знак принадлежит офису clearly states ownership
• знак офиса can mean the office’s sign or a sign associated with the office
• у офиса есть знак just says the office has one

So принадлежит is a very clear ownership verb.

The main word stress is:

• э́тот
• знак
• принадлежи́т
• о́фису

A rough pronunciation guide:

• э́тот ≈ EH-tuht
• знак ≈ znak
• принадлежи́т ≈ pree-nud-lizh-EET
• о́фису ≈ OH-fee-su

The most important stress point for learners is принадлежи́т.

No. It can also be used more broadly for belonging in a figurative or abstract sense.

For example:

• Эта идея принадлежит профессору. = This idea belongs to the professor.
• Это право принадлежит всем. = This right belongs to everyone.
• Он принадлежит к этому поколению. = He belongs to this generation.

But note that belong to a group often uses a different structure:

• принадлежать к + dative

So:

• Книга принадлежит студенту. = The book belongs to the student.
• Он принадлежит к элите. = He belongs to the elite.

These are related but slightly different patterns.