If you put a kilogram of water at 32F and a kilogram of ice at 32F in a perfectly insulated container at standard pressure and exactly 32F, the water won't freeze while the ice melts.
This is not correct. In a
theoretical perfectly insulated container (which in reality does not exist), the mass of ice and water would remain relatively constant. Think of it like this, it takes heat to raise the temperature of ice and eventually melt it. Let's say you have a gram of ice at -3 °C. Raising the temp (i.e. increasing the enthalpy) to -2 °C requires an input of roughly 1 calorie of heat. Raising it to -1 °C requires another 1 calorie. And raising it to 0 °C requires yet another. But to change the state from a gram of ice at 0 °C to a gram of water at 0 °C requires an additional 80 calories.
But while we're talking about the miracle substance water, the most spectacular property of all is that water expands when it freezes. If this were not the case, then our oceans, rivers, and lakes would be mostly ice from the bottom up.