My IBM T22 laptop won`t power on. When I push the power button
the light flashes on but goes out. Laptop does not power up.

Try taking the battery out, cleaning the metal terminals and putting it back in. Even if you're plugging it into a wall socket, it needs this battery connection to work.

Cleaning the battery connections did not help. The battery is only
3 months old, so they should be pretty clean anyway.

I'm not very knowledgeable about laptops, I only mentioned the terminals thing because I've heard others have this problem.

If you can borrow a friend's (or computer store's) battery, and find that works you've found your problem. A laptop battery should last from 1 to 3 years though, so this is an unlikely cause in your situation.

I assume you charged the battery (joke, sorry but I had to say it).

Sorry I can't be more help. Someone else will have a better idea.

Does the power charger have a light and is it on? These units are often the culprits.

if it's not a power issue, then it's inside the laptop and unless it's in warranty, it's a lot of hassle.