Only One Judge Has Signed On For 'American Idol'

"American Idol" is in a pickle -- the judges start their on-air auditions in New York City in 2 weeks, and only one judge has signed a deal ... sources connected with the show tell TMZ.

Our sources tell us ... only Mariah Carey has inked a deal ... and a good deal it is at $18 mil.

We know although producers are set on Nicki Minaj and Keith Urban ... they have not fully come to terms on the deal and nothing is signed.

And it gets worse. We're told "A.I." honchos want 4 judges on the panel this year, because they don't think 2 women and a man will work. And they have not picked the 4th.

But wait ... there's more. Randy Jackson -- who is being kicked upstairs to mentor -- has not signed his deal either.