John Nash was a famous mathematician. Here is his example of market inefficiency:
Take the case of two gas stations being built in a small town made up of a single, one-mile long Main Street.
Town planners agree the gas stations should be placed at one-quarter and three-quarter mile marks, so that no one in town has to drive more than a quarter mile to fill up. And since residents are distributed evenly along Main Street, both stations would share exactly half the business in town. But try explaining that to the station owners. The owner who should build at the one-quarter mile mark knows people at his end of town will never go to the competing station because it's too far away. So he'd want to build closer to the center of town to dip into his competitor's mid-town market. Of course, the other owner is equally wily, and he, too, edges his station closer to the center of town. Game theory tells us — and an astute business sense dictates — that the two gas stations will both end up on the same corner in the exact center of Main Street. The equilibrium solution of the gas station game is clearly not the most efficient. While the stations still share half the town's business, people on the edge of town have to drive farther to get gas under equilibrium than under the town planners' solution.
But neither owner would have it any other way, because being located in the center of town is the most secure solution for each. This equilibrium is an example of free market inefficiency — and a critique of Adam Smith's invisible hand.
More...I'm sure your professor will say that's not a real world example; but then, mathematics doesn't deal with the real world, it deals with models.
I've read that there is a mathematical proof that contradicts the free market assumption that if all individuals act in their own interest, then the result is optimal for society; but, I haven't been able to find such a proof.