It's possible you have a faulty power brick. Is your console the old fat 360 or the slim ones? I'm not familiar with the 360 slim consoles, so I'm not sure if it's the same. Take a look at the power brick light when you turn it on. Does the light go from amber (off) to green (on) and amber again? I had an issue with my power brick, which I confirmed when I brought my console to a shop and tested with another power brick. The TS in the thread below had a power brick issue, similar to the one I was also facing.

https://forum.lowyat.net/index.php?showtopic=3620757&hl=