Every man, every woman, and every child comes across to me as plural:
Every man, every woman, and every child in this line are required to sign the forms.
but it's for a strange reason. Because every man, woman, and child is such a familiar phrase, or at least the expected way to say it, the fact that it's reworded as three every phrases suggests that you're trying to emphasize three distinct groups rather than treat all the men, women, and children indifferently. This would call for a plural verb.
On the other hand, a singular verb can easily make the listener hear the triple-every phrase as singular. A listener can still understand the phrase as defining a single category to which the predicate will apply individually to each member, the same as the version with only one every.
Here are some examples that require the plural:
In this kit, every wheel and every axle are mutually compatible.
Every husband and every wife are to meet in the gymnasium.
Here are some examples that favor the plural, although the singular could work:
Every child and every parent are requested to attend the PTA meeting.
Every conversation and every semester are agony.
The triple-every phrase in the question sounds like it's trying to be heard like the comparable phrases in the previous two sentences (that is, as things that are very different).
Only the singular can work here:
Every nation, every province, and every city has its own government.
Every invention, every work of art, and every scientific theory is a product of human imagination.
The singular is by far the most common form because every means to apply some predicate to the individual members of the category, and that's usually still what you want when you combine every phrases with and. The exceptions happen when the speaker wants the listener to hear the different every phrases as separate in some important way.
However, on a written examination, you should ignore all that and just choose the "singular" answer. Above, I tried to explain the patterns and pressures that influence how a fluent speaker understands or chooses the verb. That kind of thing is too subtle for a person to include in a written examination, or for a machine to consider when grading it. There is a commonly taught "rule" that every is always singular no matter what. That's what an examination will test you on, even though in real life you find that the rule is not entirely true.