My 2005 Toyota Camry has intermittent starting problems.

The next time it just gives you the buzz sound and doesn't start, do this: Have someone hold the key to the ON position like they are starting the car. As they hold the key, you rap on the starter with a hammer. If the car suddenly starts, you will need to replace your starter. If after rapping on the starter and it still will not turn over, I would test or replace the starter relay. Post back and keep us updated.

I tried the way you explained, asked my friend to hold the key to ON position and I knocked the starter, it did not work. So, towed it to garage (4th time to a new place) and they replaced the starter. It started fine for a week and issue reappeared. This time the same repairer came to my place and took the car, after having inspection for one full day, they came to the conclusion as MAIN RELAY. They replaced it, since last 3 weeks there is no issue.

I have chased starter problems, replaced the starter, battery, etc. It was the negative battery cable the whole time. It did not look old at all, the issue was that if was corroded (under the insulation)