His story says that "the Doors of Acceptance glowed and slowly swung open as soon as it began."

What do you think happened in that short amount of time?

Personally I think Jax said passing through and got in.

Jax inputed Up, Up, Down, Down, Left, Right, Left, Right, B, A, Start and proceeded into the League of Legends

I think Jax had already been accepted. It says that he had been placed at the top of the list of candidates by High Councilor Reginald Ashram. To say that it was simply a formality wouldn't be terribly inaccurate. This case is further put forward by Renekton's judgement. Though one could argue that he was not judged because of his madness, it seems far more likely that he was simply accepted because of the circumstances that surround his summoning. Summoners had already targeted Renekton to be within the League, had planned to take him as their first off-world summon but failed and then found him again, ready as ever to fight. To say his rage was the reaosn he was not judged is a little bias. I suppose dit all depends if we ever get any insight into the judgements for other ill-minded champions such as Fiddlesticks or Cho'Gath.