Upgrading OS X will not fix something that's already broken.
Boot from your original System Install DVD: insert the disc and restart while holding the ckey.
At the Mac OS X Utilities screen, select the Utilities menu and run Disk Utility.
Select your startup disk - it is usually the first icon at the upper left of the Disk Utility window. At the bottom of the window you should see this
What does it say next to "SMART Status"?
Next select the startup volume - it is usually named "Macintosh HD". The bottom of the window will look like this
What are the numbers next to "Capacity", "Available", and "Used"? Only the numbers before the ones in parentheses are necessary.
Next, click the Repair Disk button. This will take a few minutes.
Report any errors it finds.
Upgrading generally does not fix problems like freezing unless the software you are using is only compatible with the update. You've basically just traded one problem for more problems. The key to get out of this loop is to identify what applications you do use, and if there are drivers or system software that might not work with your operating system. If there aren't and booting into safe mode with the shift key doesn't help, you may have a hardware problem. Presumably since you have an Intel Core 2 Duo with 10.5, you still have the original discs? If you do, run the hardware test identified in this article:
Thank you for your impressive answer.
I can't tell if it helps because I do not have any disc, I downloaded on line...
I have looked at the authorizations issues though, and had tons of repairs to do.
I did everything, but did not save what had to be done (I might be able to find them though).
Nevertheless, without discs...
Why don't you have the original discs? You buy the computer from a nonauthorized source? They must give you the discs or call Applecare for them if lost. Otherwise who knows what software they left behind that could be harmful. If you lost the discs, call AppleCare, or try to track them down. 10.5.8 as an update is available to update 10.5 to 10.5.7 online, but the erase and install version of 10.5 is only available on disc legally. Illegal sources may have malware if obtained online. We can't help you there.
I was answering John regarding the use of the 10.6.8 disc.
I have my OS X 10.3.6 CDs and the 10.6.3 but I downloaded the 10.6.8 online from Apple.
Now, when I enter the Install disc2 (10.3.6), the computer sends a message saying it can't open it because it doesn't support it any more.The hardware test is on the disc2 and it seems I can't get it start doing what you suggested I do with the link provided.
A Mac OS X 10.3.6 CD will not function on an Intel iMac. You must use the original DVD that shipped with your Mac to run Apple Hardware Test. Use the one with the words "Apple Hardware Test" or "AHT version xxx" printed on it. There is no other option outside of a visit to the Genius Bar.
If you no longer have that DVD then contact Apple to purchase a replacement. It may be a challenge to get them to understand what you need. If you are fortunate enough to speak with someone knowledgeable, be ready with your Mac's serial number because the disc is specific to that model Mac.
To run Disk Utility, you may use your Mac OS X 10.6.3 "Snow Leopard" installation DVD. Insert the disk and restart your Mac while holding the c key. This will attempt to boot from the optical drive.
If for some reason that does not work and the Mac boots normally, use Startup Manager: hold an option key while you start the Mac and select the Snow Leopard DVD.
When you see the following screen, select Disk Utility and follow my earlier instructions to repair the disk:
Also report the information regarding SMART status (if applicable) and the amount of free space available. The symptoms you describe are consistent with insufficient disk space, or corruption that may or may not be fixable with Disk Utility.