i think you can answer these from the definition of a mode. have a go and post your best guess.
my guesses are,
if no value repeats itself, then no mode
if one value is repeated throughout, that itself is the mode
if all have same frequency, then no mode
yes a data set with 201 values, and 100 pairs of same values, and one single value, will have a 100 modes, a multimodal data!
Am i right?