Learn Hardcore Portuguese (Portugal): A inscrição para o exame está aberta esta semana. - Registration for the exam is open this week.

A inscrição = the registration
English can say: “Registration for the exam is open this week” (no “the”).
Portuguese usually needs the article here, because we are talking about a specific registration process (for a particular exam this time).

Saying just Inscrição para o exame está aberta… sounds incomplete and unnatural in standard Portuguese. The article a must agree with the feminine noun inscrição.

Why is inscrição feminine? How do I know which article to use?

Noun gender is mostly arbitrary and must be memorized, but there are some tendencies:

Nouns ending in -ção (like inscrição, informação, profissão) are almost always feminine.
Feminine singular article: a
Feminine plural article: as

So:

a inscrição – the registration
as inscrições – the registrations

You can’t say o inscrição; that would be grammatically wrong because inscrição is feminine.

Could I say As inscrições para o exame estão abertas esta semana instead? Is that more natural?

Yes, that’s very common and often sounds more natural:

As inscrições para o exame estão abertas esta semana.
“Exam registrations are open this week.”

Difference in meaning:

A inscrição está aberta… – focuses on the registration process as a single thing.
As inscrições estão abertas… – focuses on the individual registrations people can make; this is the usual way to talk about an “enrolment period”.

In everyday use, speakers often prefer the plural with this meaning.

Why is it para o exame instead of do exame? What’s the difference between para and de here?

Both are possible, but they emphasise slightly different ideas:

para o exame = “for the exam”
- Focuses on the purpose of the registration: you are signing up in order to take that exam.
do exame = “of the exam”
- Focuses on the relationship or association: the registration of that exam (more like “the exam’s registration”).

So:

A inscrição para o exame está aberta…
Stresses: you’re registering to sit this exam.
A inscrição do exame está aberta…
Stresses: the exam’s registration (still correct, just a slightly different angle).

In practice, para o exame is very natural and common when you think of registration as something you do in order to take the exam.

Why do we use está aberta with estar, not é aberta with ser?

Because we’re talking about a temporary state, not a permanent characteristic.

estar = temporary situation, condition, result of an action
ser = inherent or permanent characteristic, definition

Here:

A inscrição … está aberta esta semana.
→ It is open (for now), only during this week.

If you said é aberta, it would sound like you’re describing a general characteristic, e.g.
A inscrição é aberta a todos os alunos. – “Registration is open to all students.”
(“Open to all students” = a general rule, not a temporary state.)

So está aberta is the correct choice for “is open (this week)”.

What is aberta exactly? Is it an adjective or a verb form?

Aberta is the feminine singular form of the adjective/past participle aberto (“open”).

Here it works like an adjective describing inscrição:

A inscrição (feminine singular)
está aberta (adjective agrees in gender and number)

Agreement patterns:

A inscrição está aberta. (feminine singular)
As inscrições estão abertas. (feminine plural)
O prazo está aberto. (masculine singular)
Os prazos estão abertos. (masculine plural)

So you must match aberto/aberta/abertos/abertas with the noun.

Can I move esta semana to the beginning or middle of the sentence? Does the meaning change?

You can move esta semana without changing the basic meaning:

A inscrição para o exame está aberta esta semana.
Esta semana, a inscrição para o exame está aberta.
A inscrição para o exame, esta semana, está aberta. (less common, a bit more marked)

All mean essentially: “This week the registration is open.”

The differences are stylistic:

Putting esta semana at the start (Esta semana, …) emphasises the time frame.
The original word order is neutral and very natural.

Avoid putting esta semana between está and aberta (e.g. está esta semana aberta), which sounds awkward.

Why is there no article before esta semana (why not *a esta semana)?

Because esta already acts like a determiner (“this”), so you don’t add another article.

esta semana = this week
essa semana = that week
a semana = the week

You normally choose either a demonstrative (este/esta/esse/essa/aquele) or a definite article (o/a), not both:

esta semana ✅
a semana ✅
a esta semana ❌ (wrong in this meaning)

Can I contract para o to pro in European Portuguese, like in Brazilian Portuguese?

In speech, yes, you will hear something like pr’o exame (/pɾu ˈzɐm(ɨ)/) in informal European Portuguese.

In standard writing, especially anything formal (emails to a university, notices, etc.), you should write the full form:

para o exame (standard, correct in writing)
pr’o exame / pro exame (informal, mainly in speech; avoid in formal writing)

Is this sentence used the same way in European and Brazilian Portuguese?

Yes, the sentence is perfectly understandable and acceptable in both varieties:

A inscrição para o exame está aberta esta semana.

Minor notes:

Pronunciation will differ, of course.
In both varieties, you’ll also very often see the plural:
As inscrições para o exame estão abertas esta semana.

But in terms of grammar and vocabulary, this sentence works for both European and Brazilian Portuguese.