Dilemma /(?)/

Di·lem·ma

Dilemma

n.
  1. An argument which presents an antagonist with two or more alternatives, but is equally conclusive against him, whichever alternative he chooses. (Logic)
  2. A state of things in which evils or obstacles present themselves on every side, and it is difficult to determine what course to pursue; a vexatious alternative or predicament; a difficult choice or position.
    A strong dilemma in a desperate case! To act with infamy, or quit the place.

Phrases & Compounds
Horns of a dilemma
alternatives, each of which is equally difficult of encountering.