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Apple has stopped updating Leopard 10.5 many years ago. It is considered obsolete and vulnerable to malware and viruses out there. All the latest anti virus applications out there are not compatible with 10.5. It is very unlikely to be a virus but must be something like File system corruption, issue with hard drive etc
I'd recommend to upgrade to 10.6, so you will be able to use latest Firefox on it. Latest Safari and Chrome requires 10.9 or later version of OS X.
What are the hardware specs of your Mac ? i.e Processor and Memory ?
If you are using an Intel Mac, then I'd strongly recommend to upgrade to 10.6 Snow Leopard atleast. To upgrade to 10.6 Snow Leopard, you need to buy the Snow Leopard disc. You can buy it from here --> http://www.apple.com/shop/product/MC573Z/A/mac-os-x-106-snow-leopard
When you get the disc, just put it on your Mac and follow the onscreen instructions to upgrade. After upgrading to Snow Leopard, you can upgrade to 10.11 El Capitan from App Store if your Mac is compatible,. El Capitan is FREE to upgrade, but that requires 10.6 Snow Leopard minimum.
You will be able to install latest Firefox on Snow Leopard, but not Chrome or Safari FYI as they require 10.9 or later version. The security updates of Snow Leopard are better compared to 10.5.8
Re the unresponsive issue, it must be something related to corruption in system files, failing hard drive. So we can try some basic troubleshooting steps.
Go to Applications > Utilities folder and open Disk Utility. In Disk Utility, select your hard disk in left column (Macintosh HD), then Repair Permission. After that click Verify Disk and if asked to repair, click Repair Disk.
Also reset the PRAM and NVRAM of your Mac (to clear temporary memory and speed things up)
Shut down your Mac
Restart and immediately hold down Command-Option-P-R keys at the same time
Hold the keys down until the computer restarts and you hear the startup sound for two or three times.
Release the keys and let the Mac boot up.
Let me know how it goes please.