Start in the Library or Develop module of LightRoom and make sure that the image you want is either showing in Loupe View or is highlighted in Grid view.
Hit the "File" tab at the top of the screen and then hit "Export". You will now be faced with a screen on which to enter information about the new exported file. I suggest exporting to "Desktop", renaming to "Custom Name" and then setting the Custom Text to be whatever you want for a title. In the file settings box choose "JPEG" as the format and "sRGB" as the Color Space. Check the "Limit File Size" box and set it to "4,000" K (this is 4MB).
In "Image Sizing" check the "Resize to Fit" box and enter 1920 for "W" and 1200 for "H" and set "pixels" in the units box. Resolution doesn't matter.
The other sections don't really matter, but in the Output Sharpening section you can "Sharpen for screen" by a "Standard" amount if you wish.
Now hit "Export" at the bottom of the page and you have an image on the desktop that is perfectly sized for our competitions. The original image, the one in Lightroom, is unchanged.