Grammar: A page of multiplication problems [sit|sits] in front of me

A page of multiplication problems sits in front of me.

The subject of your sentence is 'page'. What type of page? A page of multiplication problems. Still, the subject is 'page', which is singular.

To agree with the singular subject 'page', the verb 'to sit' must be conjugated as 'sits'.

The reason 'sit' sounds correct to you is its close proximity to 'multiplication problems', a plural noun. Don't let that confuse you. The subject is 'page'. Whether it's a page of multiplication problems or a page of recycled peanut butter the subject 'page' remains singular and thus the conjugation of 'to sit' as 'sits' remains unchanged.

You're quite right that both are perfectly valid and make sense.

Page can be seen here as a collective noun (a word that describes a group, category, or other type of measurement applied to a following noun phrase). The most common pattern with collective nouns is that in cases like

a [collection] of [things]

– verbal agreement can be based on the plurality of either the collective noun or the noun ‘measured’.

When the verb agrees with the collective noun, the sense is that the subject is perceived as a single unit:

A pack of cigarettes is lighter than a tennis ball.

Here, we're interested in the weight of the pack as a whole, not the individual cigarettes.

When the verb agrees with the noun being ‘measured’, the focus is on the individuality and plurality of that noun:

A pack of wolves make their way through the woods, sniffing out their prey.

Here, the focus is on each individual wolf acting individually. The pack is not just one thing, it's so many individual things acting more or less in unison.

In your example, both versions are equally plausible:

A page of multiplication problems sits before me.

– means that there is a page in front of you. That page is filled with multiplication problems.

A page of multiplication problems sit before me.

– means that some multiplication problems (individual, different ones) sit in front of you, and they are collected on a page.

There are some collective nouns where only one type of verbal agreement is possible. For example, using a lot of [things] with singular agreement is highly unusual except when lot means ‘shipment, consignment’. Your example, however, is not one of these.