If two waves with equal amplitudes and wavelengths travel through a medium in such a way that a particular particle of the medium is at the crest of one wave and at the trough of the other wave at the same time, what will happen to that particle?

The particle won't move. this is because at the point where te particle is at, the waves interfere destructively as the trough of one wave and the crest of another are out of phase. the amplitude of the resultant wave is zero, hence there is no energy transmitted to the particle and it does not move.