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An English class use this as the topic for students.

What social problems exist in big cities

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Typically, to construct a wh-question, one needs to use an auxiliary verb, such as "do/does", which the quoted sentence doesn't have one. So, is the sentence grammatically correct?

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The rule "to construct a question, one needs to use an auxiliary verb, such as 'do/does'" does not apply when what is being asked is the subject, because, if what is asked is the subject, it is obviously not possible to have subject-auxiliary inversion (where would we place the auxiliary if the subject is in front position?).

Therefore, questions asking for the subject and introduced by who, what, which, whose, how much/many are followed by the verb as it will appear in the answer:

  • What social problems exist in big cities?
  • Problems such as pollution, crime and unemployment exist in big cities.

  • Which represents the worst problem?

  • Pollution represents the worst problem.

  • How many poeople live in big cities?

  • Millions of people people live in big cities.

  • Who lives in a big city?

  • I live in a big city.

By the way, in the source you just quoted you can read:

Warning:

When what, who, which or whose is the subject or part of the subject, we do not use the auxiliary. We use the word order subject + verb:

What fell off the wall? Which horse won? Who bought this? Whose phone rang?

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(1) When wh-word is the subject or part of the subject of the question,

Wh-word [subject] + verb

e.g., [Who] wrote 'The Merchant of Venice'?

[What social problems] exist in big cities?

The bracketed part is the subject.

(2) When wh-word is not the subject,

Wh-word + Auxiliary verb + Subject + Main Verb ...

e.g., What do you want?

(Subject = You, Verb = want, Object = what)

Where will you go?

(Subject = you, Verb = will go, Adverbial = Where).

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