While installing Linux Mint 14.1 MATE you will get to the question where it asks you how it should install--use the entire disk, install alongside exisiting operating systems, overwrite an older version (this may possibly be shown, or not), or something else. Pick something else so you can do a manual set up of which partitions to use. Pick /dev/sda6 and select to format it as Ext4 and to mount it as /. Pick your swap partition and select to use it as swap (no need to format). Make sure also to select install the boot loader GRUB to /dev/sda6, and not to /dev/sda as probably will be the default. If you do it like this, the GRUB that was installed by Linux Mint 14.1 Cinnamon stays on /dev/sda and it will see your new Linux Mint 14.1 MATE installation and add it to the GRUB menu.
Linux Mint 14 Nadia / 64-bit / Cinnamon