Hello. I will really appreciate some help from a native speaker of English.

Which is correct, "this" or "that",(maybe, both?)in the following context:
1)He signed this/that (?) contract last week.
2)This/that(?)letter was prepared yesterday.
Do "past" words, such as "yesterday", "last ...", mean that "that" rather than "this" is to be used?
Thank you very much for your help.

Yes.

"This" means something close at hand or something you were just talking about. "That" means something farther away or in the past.

This contract is either one you're looking at or that you've been discussing. That contract is one that isn't present or that you had discussed in the past.

Hello! I'd be happy to help you with your question.

In the given context, both "this" and "that" can be used, depending on the speaker's perspective and the intended meaning.

When choosing between "this" and "that," it's important to consider the proximity of the object being referred to. Generally, "this" is used when something is close to the speaker or happening in the present moment, while "that" is used when something is further away from the speaker or happened in the past.

Let's take a closer look at your examples:

1) "He signed this/that(?) contract last week."
You can use either "this" or "that" in this sentence, depending on how you want to convey the meaning. If the contract is physically present or recently discussed, you can use "this." On the other hand, if the contract is not physically present or the act of signing it happened in the past, you can use "that."

Example using "this": "He signed this contract last week, and it's binding for the next five years."
Example using "that": "He signed that contract last week, and now we need to review its terms."

2) "This/that(?) letter was prepared yesterday."
Again, the choice between "this" and "that" depends on the speaker's perspective and the intended meaning. If the letter is physically close or recently mentioned, you can use "this." However, if the letter is not physically close or the act of preparation occurred in the past, you can use "that."

Example using "this": "This letter was prepared yesterday, and it requires your immediate attention."
Example using "that": "That letter was prepared yesterday, and we have already addressed its contents."

To sum up, "past" words such as "yesterday" or "last" do not necessarily dictate the use of "that" over "this." The choice between the two depends on the proximity and timeframe being referred to, as well as the speaker's intended meaning.

I hope this explanation helps! If you have any further questions, feel free to ask.