Existe un voto nulo.

Both can express existence, but they are not identical in tone.

• Hay is the most common everyday way to say there is / there are.
• Existe is a little more formal or precise, closer to there exists.

So:

• Hay un voto nulo. = the most natural everyday phrasing
• Existe un voto nulo. = correct, but slightly more formal, technical, or emphatic

In election-related language, existe can sound like part of a report or official statement.

Existe is the third-person singular present indicative of the verb existir.

Its infinitive is existir = to exist.

Because the noun phrase is singular — un voto nulo — the verb is singular too:

• Existe un voto nulo.
• Existen votos nulos.

This is different from hay, which stays the same in singular and plural:

• Hay un voto nulo.
• Hay votos nulos.

Because es and existe do different jobs.

• Es un voto nulo describes or identifies something: It is an invalid vote or This is an invalid vote.
• Existe un voto nulo states that such a vote is present or exists: There is an invalid vote.

So if you are pointing at a ballot and describing it, you could say:

• Este voto es nulo.

But if you are saying that among the votes, one invalid vote exists, you use:

• Existe un voto nulo.
• More commonly: Hay un voto nulo.

Because un is the form used before a masculine singular noun.

• un voto
• un problema
• un libro

Uno is usually used when the number stands alone, not directly before a noun.

Compare:

• Hay un voto nulo.
• Hay solo uno.

So here, because voto is a masculine singular noun, un is the correct form.

In Spanish, most descriptive adjectives normally come after the noun.

So:

• voto nulo
• coche rojo
• casa grande

Putting nulo before the noun would sound unusual here and not like the normal election term. In this context, voto nulo is the standard expression.

So for a learner, the safest pattern is:

• noun + adjective

unless you learn a specific reason for changing the order.

Yes. In Spain, voto nulo is a standard election term.

It refers to a ballot that is considered invalid, usually because it was filled in incorrectly or contains something that makes it not count as a valid vote.

A very important contrast in Spain is:

• voto nulo = invalid vote
• voto en blanco = blank vote

These are not the same thing.

A voto en blanco is intentionally left blank and is counted in a specific way.
A voto nulo is invalid and does not count as a valid choice.

It is correct, but a native speaker would often say:

• Hay un voto nulo.

That sounds more natural in ordinary conversation.

Existe un voto nulo is more likely in:

• formal speech
• written reports
• official or analytical contexts
• situations where the existence of the vote is being stressed

So the sentence is good Spanish, but hay is more common in casual use.

Usually, no.

If you are talking about one invalid vote, the normal form is:

• Existe un voto nulo.

Without the article, Existe voto nulo, the sentence sounds unnatural in most normal contexts.

Spanish usually wants the article here because voto is a countable noun. You are referring to one instance of that noun, so un is expected.

A simple approximation for an English speaker is:

• existe ≈ ek-SEES-teh
• voto nulo ≈ BOH-toh NOO-loh

The stress is on:

• ex-IS-te
• VO-to
• NU-lo

A few pronunciation notes for Spain Spanish:

• v sounds very similar to b
• the final e in existe is pronounced, not silent
• all vowels are clear and short: e, i, e

With existir, the verb changes to match the plural noun:

• Existe un voto nulo.
• Existen votos nulos.

So both the noun and adjective also become plural:

• voto nulo
• votos nulos

But if you use hay, the verb does not change:

• Hay un voto nulo.
• Hay votos nulos.