Assume a spherical brain, in a straw man
Sure, let's say your country is populated by robots, all the same model (including firmware revision), with the same IQ. The median and the average are the same, as are the maxima and minima. It's a pretty artificial example, though.
If your sample is more varied, but the distribution is symmetrical about the mean, then the median and average (mean) can be the same, and therefore 50% would be below the average. But again, those are pretty special circumstances but will be true of IQ in human populations if the scores are scaled/weighted in order to adhere to such a distribution. They seem to be scaled such the mean is 100, but I've seen no claim to symmetry.
In general, given a statistical distribution of real world populations (which is the category of the statistic that was challenged), the claim that 50% will be below the median holds. Not so for the mean.