apps no longer run on snow leopard

Reinstalling OS X Without Erasing the Drive

Make sure you have a current backup before proceeding.

Boot to the Recovery HD: Restart the computer and after the chime press and hold down the COMMAND and R keys until the menu screen appears.

Repair the Hard Drive and Permissions: Upon startup select Disk Utility from the main menu. Repair the Hard Drive and Permissions as follows.

When the recovery menu appears select Disk Utility and press the Continue button. After Disk Utility loads select the indented Macintosh HD entry from the the left side list. Click on the First Aid tab, then click on the Repair Disk button. If Disk Utility reports any errors that have been fixed, then re-run Repair Disk until no errors are reported. If no errors are reported click on the Repair Permissions button. Wait until the operation completes, then quit Disk Utility and return to the main menu.

Reinstall OS X: Select Reinstall OS X and click on the Continue button.

Note: You will need an active Internet connection. I suggest using Ethernet if possible because it is three times faster than wireless and more reliable.

Sorry, those were the wrong instructions. Here they are for Snow Leopard:

Reinstall OS X without erasing the drive

Make sure you have a current backup before proceeding.

1. Repair the Hard Drive and Permissions

Boot from your Snow Leopard Installer disc. After the installer loads select your language and click on the Continue button. When the menu bar appears select Disk Utility from the Utilities menu. After DU loads select your hard drive entry (mfgr.'s ID and drive size) from the the left side list. In the DU status area you will see an entry for the S.M.A.R.T. status of the hard drive. If it does not say "Verified" then the hard drive is failing or failed. (SMART status is not reported on external Firewire or USB drives.) If the drive is "Verified" then select your OS X volume from the list on the left (sub-entry below the drive entry,) click on the First Aid tab, then click on the Repair Disk button. If DU reports any errors that have been fixed, then re-run Repair Disk until no errors are reported. If no errors are reported click on the Repair Permissions button. Wait until the operation completes, then quit DU and return to the installer.

If DU reports errors it cannot fix, then you will need Disk Warrior and/or Tech Tool Pro to repair the drive. If you don't have either of them or if neither of them can fix the drive, then you will need to reformat the drive and reinstall OS X.

2. Reinstall Snow Leopard

If the drive is OK then quit DU and return to the installer. Proceed with reinstalling OS X. Note that the Snow Leopard installer will not erase your drive or disturb your files. After installing a fresh copy of OS X the installer will move your Home folder, third-party applications, support items, and network preferences into the newly installed system.

Download and install Mac OS X 10.6.8 Update Combo v1.1.

Another possible cause - expired certificate.

About the Mac App Store Update for OS X Snow Leopard - Apple Support

Make sure App Store is updated via "Software Update..." on Apple menu.

The updater is also available at the Download page, Apple - Support - Downloads as below.

Mac App Store Update for OS X Snow Leopard