At least when I was studying there (late 90-s), all entrance exam problems were supposed to be solved without any calculus. The motivation given was that algebra and geometry give enough variety of material for hard and beautiful problems and high school calculus is a joke so they would have to teach it again from scratch.
This problem can be done without any derivatives, actually. First observe that since (x-y)^2=(y-x)^2, we have |F(x)-F(y)|<=(x-y)^2. Given x and y with y>x, divide up the interval between them into x_0=x, x_1, x_2, ..., x_n=y evenly spaced. Then applying the observation above and the triangle inequality, since F(y)-F(x) is the sum of the F(x_(i+1))-F(x_i), we have |F(y)-F(x)|<=n*((x-y)/n)^2=(x-y)^2/n. Since n can be arbitrarily large, |F(y)-F(x)| is smaller than any positive number and hence 0, so F(y)=F(x).
Note that 2 here can be replaced with any number greater than 1; this is actually a well-known fact, that any Hölder-continuous function on the reals with exponent greater than 1 is constant. But I suppose it would not be well-known to high-school students! To be honest, I mostly only know it because of the old legend about the student who... well, here's a link: http://mathoverflow.net/questions/53122/mathematical-urban-l...
