Exporting to JPEG without losing quality! Urgernt!

For this project, use RGB. If your images are already cmyk, see if you can find original RGB files, if they exist, and use those. RGB jpegs will be slightly smaller than CMYK, and starting with RGB will likely give you better color than converting to RGB from CMYK (save for web or convert to profile in Photoshop). I would also submit a nice PDF with the jpeg, just in case the contest rules are outdated (lets hope so).

You can’t fit ten gallons of water in a five gallon jug. If you need amazing quality, forget about file size. If you need under 2 MB, then expect to lose some quality.

Back up. Rather than telling us you need something so specific (raster image in a certain format with a specific size limitation) tell us what you are trying to do with it. Another format or process may work better for you.

This sounds like a contradiction in terms.

By saving as JPEG you are automatically losing quality – that's what JPEG compression does. It's a lossy kind of compression.

LZW tiff is not lossy but usually doesn't compress images as much as JPEG.

I'm not exactly sure what it is you are trying to do, but have you explored all possibilities in Acrobat?

For example, Smallest File Size will only compress placed images while leaving vector work unscathed.