try slaving your hdd to another computer and scan it to see if it really is because of virus.
while you wait for the scan result, try to boot up you pc to see if it can stay up long, better if you use live cd to test the environment...

my guess:
- try to boot up your pc, it is to see if the hardware is the problem. if your pc cannot stay long than the problem is the hardware (probably the psu or the dust, try cleaning/disassemble)...
- av scan the hdd if it is in fact a problem of virus corrupting the system, if so repair it after cleaning the virus.
- try also scanning your hdd for any badsector...

I agree its a good idea to perform a full antivirus and adware scan.
Have you tried starting it in safe mode to see if it stays on long enough to do this?

I've booted in safe mode, but the computer only boots about once out of every 5 times, and in that one time, it still turns off in safe mode in the middle of a virus scan. Someone else has told me it's a motherboard problem, could that be it?

there could be many thing:
software wise (OS, application, malware, etc)
Hardware wise (heat, cold, dust, electricity, dust, etc)

to see if it is indee hardware:
go to bios, check the temperature s it over heat (usually over 60-70 celcius)
still in bios, see the electricity output, is it not fluctuated much (over 0.25-0.35 volt).

if it is about heat, you must realize that probably some of your components might be burn, check if the tube thing in your mb not "pregnant".
the solution to over heat is just give space for each of your components (hdd, cards, etc) and add some fan with good inflow-outflow management.
if it is about electricity, the best thing is to buy ups, but if you can't afford it try using stabilizer with motor. try also check your psu, whether it is a good brand (like simbadda, enlight, etc). because the cheap psu is not stable on its dc output.
also check out your ram, try to clean its gold bar, if it is dusty.
for the hdd integration try to do scandisk on it in case you'll have new bad sector.
also check out your vga, nowadays vga need more electricity when in full power than the older generation. that can also make your psu unable to support the electricity output. the solution is just to have the higher watt psu.

for the software test.
try to take off your hdd and make it a slave to other hdd in other computer and scan it using the latest updated av database.

Did your friend install any drivers or software?
I would start by searching for those and uninstalling them or going back to a safe point in system restore.