I'm thinking that in theory you can have any molecular formula for Compound A that is an even multiple of the empirical formula C3H6O. For example,
X2 A = C6H12O2 B = C6H10O2 A has 1 OH and 1 C=O
X4 A = C12H24O4 B = C12H20O4 A has 2 OH and 2 C=O
X6 A = C18H36O6 B = C18H30O6 A has 3 OH and 3 C=O
etc.
If you know that only one equivalent of HNO3 reacts with Compound A to form Compound B (assuming the stoichiometry is 1-to-1), then only one OH group can be present which would narrow the possibilities to the C6H12O2 formula.
Many cyclic structures with formula C6H12O2 containing an alcohol group are possible. For instance, imagine a 6-membered heterocyclic ring containing one oxygen atom. Put a methyl group and and OH group on the ring somewhere, and that opens up a whole new can of worms! 
Steve
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