How to find output of ideal op-amp system?
This is a good problem because you actually have to think to solve it.  +1 to your professor.

It's hard to see, but I'm assuming the negative inputs of the opamps are at top, and the positive ones at bottom.

The first thing to do is to look at the overall topology and what is going on.  You should immediately see that the first opamp by itself is a inverting amplifier, and the second a non-inverting amplifier.

The gains should also be obvious if you know anything about basic opamp circuits.  The first one has a gain of -2, and the second of 3.  Therefore, without the global feedback, the two-opamp chain has a gain of -6.

Now consider what happens when some positive feedback is added around a amplifier.  Surely hysteresis was covered in your class.  You should be able to see what the two hysteresis thresholds are.  Once you know those, you can make a plot of input voltage as a function of output voltage.

Since these are ideal opamps, assume the output goes fully to one power rail or the other when the two inputs aren't equal.