Science

First I have to know where this question came from, to know what this question expect me to answer, in order for me to answer.
If it is a F.1 - F.3 (maybe F.5) question, the answer would no doubt be:
A. the cold water will be frozen first
The heat capacity of hot water is greater than that of the cold water, and to a home freezer, which is not enough to cool them instantly, need some time for the cooling. So the answer 'the cold water will be frozen first' is intuitive. Although intuitive answer may not be the right answer, still teachers will accept that answer.
If this is a 'higher level' question, the answer, would be different, is 'it depends'. Because although the specific heat capacity of hot water is bigger than that of the cold water, the temperature of the home freezer is just a bit lower than 0 deg C, so the heat transfer rate of the 80 deg C liquid is almost 4 times than that of the 20 deg C liquid. The heat transfer rate is the heart of this question.
For the glass of cold water, ice first appears on the surface, followed by the bottom and sides. This has two effects, first, the surface of water is no longer in contact with the open air. This prevents heat loss through evaporation. The second effect is that as the water freezes, the solutes in the water move toward the unfrozen part, a process known as zone refining. Because of this, the unfrozen water begins to have an increasing concentration of solutes which lowers the freezing point of the water.

圖片參考：http://www.lancs.ac.uk/ug/thompsom/cold-ice-Final-sm-medium.jpg

Cold water


For the glass of hot water, there are convection currents which move heat from the bottom of the glass to the top. i.e. carry heat to the surface. At the top, energy is used up as some of the water evaporates. Because of the rising heat, the warm water does not form the layer of ice on the top from the start. This allows the convection currents to continue to draw heat to the surface where it is used up as the water evaporates. Also, the warm water has lost some solutes whose solubility decreases with increasing temperature, due to the heating process. Thus the concentration of solutes is less than the cold water. It has an increasing concentration of solutes as the water freezes and the process of conducting heat away from the water is hindered by the formation of ice which surround the water.

圖片參考：http://www.lancs.ac.uk/ug/thompsom/warm-ice-final-sm-medium.jpg

Warm water


The phenomenon of hot water freezes before cold water is called the Mpemba effect. There have been many attempts to explain the Mpemba effect, yet there is still no accepted theory.

Conclusion
"At first sight, the question is quite simple: 'Does hot water freeze faster
than cold?' However, a little thought shows that this formulation is not adequate. If the initially hotter water is at 99.9999°C, and the initially colder water is at 0.0001°C, then the initially cold water is just seconds away from freezing, and it is clear that the hot water cannot possibly overtake it." - Monwhea Jeng
The quote above sums up very well the reason why it is so difficult to find a solution to the problem. This is because the question allows for such a wide range of parameters and this in turn leads to a wide variety of results. So, in order to answer the question, you first need to define it specifically. However, any experiments to prove this would only be valid for that set of parameters. So, there doesn't appear to be any fundamental reason that this effect occurs, however this does explain why so many people have had such different results!
I hope this can help your understanding. :)