O interesse pelo projeto aumentou.

In Portuguese, it is very common to use the definite article with nouns, even where English would often leave it out.

So o interesse literally means the interest. In this sentence, it refers to the interest being shown in the project in general.

Portuguese often sounds more natural with the article here:

• O interesse pelo projeto aumentou.

If you removed o, the sentence would usually sound less natural in this context.

Pelo is a contraction of:

• por
  • o = pelo

So:

• pelo projeto = for the project, in the project, or more naturally in English here, in the project

In this sentence, interesse por is the usual structure for interest in something:

• interesse por música = interest in music
• interesse pelo projeto = interest in the project

This is just a difference in how the two languages build the expression.

English says:

• interest in something

Portuguese says:

• interesse por algo

So you should learn interesse por as a fixed pattern.

Examples:

• Tenho interesse por história.
• Ela mostrou interesse pelo curso.

Even though English uses in, Portuguese normally uses por.

In modern standard Portuguese spelling, both Portugal and Brazil now write:

• projeto

Older European Portuguese spelling often had:

• projecto

That older spelling is no longer standard after the spelling reform. So projeto is the normal current spelling.

Aumentou is the pretérito perfeito simples of aumentar.

That tense is often used for a completed change or event in the past:

• aumentou = increased

Depending on context, English might translate it as:

• increased
• has increased

So:

• O interesse pelo projeto aumentou.
  can mean:
• Interest in the project increased.
• Interest in the project has increased.

English chooses between those two based on context; Portuguese can use aumentou for both kinds of situation.

The infinitive is aumentar, meaning to increase or to raise depending on context.

Here it is in the 3rd person singular:

• aumentou = it increased

The subject is o interesse, which is singular, so the verb is singular too.

A few forms:

• aumentar = to increase
• aumento = I increase
• aumenta = he/she/it increases
• aumentou = he/she/it increased

Because the subject is singular:

• O interesse = the interest

That whole noun phrase is one singular thing, so the verb must also be singular:

• O interesse ... aumentou

If the subject were plural, the verb would change:

• Os custos aumentaram. = The costs increased.
• As dúvidas aumentaram. = The doubts increased.

Yes. Portuguese allows more word-order flexibility than English.

Both are correct:

• O interesse pelo projeto aumentou.
• Aumentou o interesse pelo projeto.

The first version is more neutral and straightforward.
The second can sound a bit more stylistic or can put slightly more focus on the fact that it increased.

For a learner, the original order is the safest default.

Yes, you may hear interesse no projeto, and many speakers use it. But interesse por is the more traditional and widely taught structure for interest in something.

So:

• O interesse pelo projeto aumentou. = very standard
• O interesse no projeto aumentou. = also possible in real usage

If you are learning and want a reliable choice, use interesse por.

Not always. Interesse can have a few related meanings, such as:

• curiosity or attention
• personal benefit
• financial interest or stake

But in this sentence, it clearly means:

• interest in the sense of attention, enthusiasm, or engagement

So O interesse pelo projeto aumentou means that more people became interested in the project, or that interest in it grew.

In European Portuguese, some unstressed vowels are reduced, so it may sound less like the spelling than you expect.

A rough pronunciation guide is:

• O interesse pelo projeto aumentou
• approximately: u inte-RESH sə PEH-lu pru-ZHE-tu aw-men-TOH

A few notes:

• o at the beginning often sounds like u
• interesse has stress on re
• j in projeto sounds like the s in measure
• final ou in aumentou sounds like oh

This is only an approximation, but it reflects European Portuguese better than a Brazilian-style reading.

The contractions of por with the definite article are:

• por + o = pelo
• por + a = pela
• por + os = pelos
• por + as = pelas

Examples:

• interesse pelo projeto
• interesse pela ideia
• interesse pelos livros
• interesse pelas aulas

These contractions are very common and important to learn early.

You have a few options:

• O interesse pelo projeto aumenta.
  = The interest in the project increases / is increasing (more general or habitual)

• O interesse pelo projeto está a aumentar.
  = The interest in the project is increasing
  This is very common in European Portuguese for an action in progress.

• O interesse pelo projeto tem aumentado.
  = The interest in the project has been increasing
  This suggests growth over a period of time.

In European Portuguese, estar a + infinitive is the normal everyday way to express be doing:

• está a aumentar = is increasing