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Tom prefers A over B and B over C. Also, Tom is indifferent between a lottery in which he gets C with probability p and A with probability 1-p and a lottery in which he gets B with a probability p and C with probability 1-p. The value of p in both lotteries is the same. For what values of p would Tom's preference be rational in the sense of von Neumann-Morgenstern's expected utility theory?

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Use the fact that the von Neumann-Morgenstern utility is only determined up to a scale factor and an additive constant. Thus, you can arbitrarily choose the utilities of two of the three outcomes as values that are easy to calculate with -- say, 0 and 1. Then only the utility of the third outcome remains as a variable. Then you can use the equation from the given indifference to determine that variable as a function of $p$, and find the values of $p$ which make the order of the three outcomes come out right.

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