Google "search engine" and you'll see ads for Bing. I don't know what their monthly spend is, but I see Bing ads all the time when searching on Google for a variety of keywords.
EDIT: Here's a chart that claims to show the top spenders on AdWords, and there aren't any search engines I can see:
I see no Bing....but I love that adage chart. Thanks.
Btw, I don't think it makes sense for Bing to buy the keyword search engine. I can't see how that makes any sense. Unless consumers are REALLLLY dumb. Then again, I might be expecting too much.
I have seen Bing ads on other search queries...that was the one I could think of that gives me a Bing ad 100% of the time. I think it makes sense to advertise for the keyword "search engine" because you may catch people who are looking to leave Google but don't know the alternatives. Seeing a Bing ad might be what lures a loyal user. And it could be that the naive user is the one who is more likely to click on ads once there.
That second link pre-dates Bing's launch, I believe. You'll see Yahoo and Ask.com in the top 15. Much of the top 20 are product, travel or local search engines.