I'm experiencing a problem with cumulative voting. Here is my reasoning:

Suppose there are n directors up for election, of which 11 will be chosen. There are 5M shares outstanding, and you possess S of them. What is S if you have enough to be certain to win 2 directors?

Begin by dividing your n*S shares among your 2 favorite directors, resulting in n*S/2 per director.

Your opponent (for simplicity let's assume you have only one) has the rest of the shares (5M-S) which amounts to n*(5M-S) votes. He must allocate at least n*S/2 votes to 11 other directors in order to compete with your vote and win. This will only happen if he has enough votes to do so:

n * (5M - S) > 11 * (n*S/2)

Solving for S:

S < 769,230.

So if you are to win you need at least 769,231 shares, which is wrong. Can someone explain to me the idea behind the formula given? I have read the Jacob and Rachel dialogue in the homework but I do not know how to extend it to 2 candidates aside from the reasoning i have given.

Your opponent wants to stop you from getting 2 directors - ie he doesn't care if you get one director.  So he divides his vote among 10 people (other than your 2 candidates), not 11.