Learn Russian: Мой друг часто пытается знакомить меня с одними богатыми предпринимателями. - My friend often tries to introduce me to some rich entrepreneurs.

While один normally means 'one', in the plural it translates to 'some' or 'a certain'. By adding одними, the speaker implies that the friend has a specific group of rich entrepreneurs in mind, rather than just any random rich businesspeople. This is called the indefinite specific use.

Why is the imperfective verb знакомить used instead of the perfective познакомить?

The adverb часто (often) shows that the friend's attempts to introduce the speaker happen repeatedly. Repeated or habitual actions require the imperfective aspect, even after a verb like пытаться (to try).

Why are the entrepreneurs in the instrumental case (с одними богатыми предпринимателями)?

The verb знакомить (to introduce) requires the accusative case for the person being introduced (here, меня) and the preposition с followed by the instrumental case for the people they are being introduced to.

How does один decline when used in the plural like this?

When used to mean 'some' or 'certain' with plural nouns, it declines very similarly to a plural adjective. Its plural nominative is одни, and since it follows the preposition с here, it takes the instrumental plural form одними.