Offer Received On Rental #1 (Updated)

I received an offer today on rental #1. I'm not going to list the details until we get a signed contract (or until we walk away from negotiations). This is simply because I know there are people who read this blog that know where this property is located and I don't want to give away too much during the negotiating period. I will say however, that the offer was for below my asking price, but it was for as-is condition. That's a big plus for me, given that the property isn't in the best neighborhood and seems to have a habit of falling into disrepair rather quickly. Still, their offer price was too long for me to consider, so I'm going to counter-offer. My counter-offer is still below the max amount the buyers are approved for. Another plus is that they want to close in less than a month. The as-is clause and less than a month till close makes me believe this is an investor making the offer, not someone who will live in the place.

Update: I got a fax of the offer this evening (I just had an email from the agent before) and, if I read it correctly, it looks like they accepted my counter-offer of $55,000. Strangely, the agent never faxed me the full contract for the offer. All I have is the addendum page and the page with the price, which was changed to show $55,000 and which I initialed. I'll get all the pages before I sign anything.

One very important fact the agent did not tell me: this is an all-cash offer! When the agent told me the buyer was approved to $57,000, she was incorrect. The letter from the bank, which I got a copy of, actually says the buyer has $57,000 on deposit with them. The contract addendum states this is a cash sale, the sale is as-is although the buyer can still inspect the property before close of escrow, and that I will pay property taxes through the close of escrow date. Note sure why the agent didn't think to tell me this was a cash sale. It's obviously a big plus for the buyer. Escrow is to close on July 2.