Hon vill vara med i tävlingen.

Because hon is the subject form of she.

• hon = she
• henne = her

In Hon vill vara med i tävlingen, she is the one doing the wanting, so Swedish uses hon.

Compare:

• Hon vill vara med. = She wants to join / take part.
• Jag ser henne. = I see her.

In Swedish, verbs do not change according to the subject the way they do in English.

So:

• jag vill = I want
• du vill = you want
• hon vill = she wants
• vi vill = we want

The form stays vill for all persons.

Because vill is a modal verb, and modal verbs in Swedish are normally followed by the bare infinitive.

So you say:

• Hon vill vara med.
• Jag kan simma.
• Du måste gå.

Not:

• Hon vill att vara med ❌

This is similar to English want to in meaning, but the structure is different here. Swedish just uses vill + infinitive.

Vara med is a very common expression. In this sentence it means:

• to take part
• to join in
• to participate
• literally, to be with / be included

So Hon vill vara med i tävlingen means that she wants to be part of the competition.

Be aware that vara med can have slightly different meanings depending on context:

• Jag vill vara med. = I want to join in.
• Är du med? = Are you following? / Do you understand?
• Hon är med i laget. = She is on the team.

Yes. A more formal or precise verb is delta = to participate.

So you could also say:

• Hon vill delta i tävlingen.

That is very natural too.

The difference is mainly tone:

• vara med = more everyday, common, conversational
• delta = a bit more formal or explicit

Because with tävling in the sense of participating in a competition, Swedish normally uses i.

So:

• vara med i tävlingen
• delta i tävlingen

This is similar to English in the competition.

You may see på with some events or locations, but with tävling meaning the contest itself, i is the usual choice when talking about taking part.

Tävlingen is the definite form of tävling.

• en tävling = a competition
• tävlingen = the competition

In Swedish, the definite article is often added as a suffix at the end of the noun.

So instead of a separate word like English the, Swedish often attaches it:

• bok = book
• boken = the book
• tävling = competition
• tävlingen = the competition

Because Swedish usually puts the definite article onto the noun itself.

So English:

• the competition

becomes Swedish:

• tävlingen

A separate definite word like den/det/de is used in some other structures, especially with adjectives:

• tävlingen = the competition
• den stora tävlingen = the big competition

So in this sentence, no extra word for the is needed.

The basic structure is:

• Hon = subject
• vill = finite verb
• vara med = infinitive phrase
• i tävlingen = prepositional phrase

So the sentence pattern is roughly:

Subject + finite verb + infinitive + rest

This is very normal Swedish word order in a main clause.

Because Swedish often uses multi-word expressions where English uses a single verb.

Here:

• vara med literally contains vara = to be
• but together vara med functions like join in / take part

So even if English uses one word such as participate, Swedish may prefer the phrase vara med.

Yes, definitely. That is why context matters.

Depending on the situation, vara med can mean:

• join in
• be included
• be present
• be involved
• understand / follow

Examples:

• Jag vill vara med. = I want to join in.
• Alla barn får vara med. = All children may join / be included.
• Är du med? = Do you follow?
• Hon är med i klubben. = She is in the club.

In your sentence, the phrase i tävlingen makes the meaning clearly take part in the competition.

Then you would use the indefinite form:

• Hon vill vara med i en tävling. = She wants to take part in a competition.

Compare:

• i tävlingen = in the competition
• i en tävling = in a competition

So the noun form changes depending on whether it is definite or indefinite.