HMMT February, 2016

Solution:
For each $f, g \in X$, we define
$$d(f, g):=\min _{0 \leq i \leq 2016}(\max (f(i), g(i)))-\max _{0 \leq i \leq 2016}(\min (f(i), g(i)))$$
Thus we desire $\max _{g \in X} d(f, g)=2015$.

First, we count the number of functions $f \in X$ such that
$$\exists g: \min _{i} \max \{f(i), g(i)\} \geq 2015 \text{ and } \exists g: \min _{i} \max \{f(i), g(i)\}=0 .$$
That means for every value of $i$, either $f(i)=0$ (then we pick $g(i)=2015$) or $f(i) \geq 2015$ (then we pick $g(i)=0$). So there are $A=3^{2017}$ functions in this case.

Similarly, the number of functions such that
$$\exists g: \min _{i} \max \{f(i), g(i)\}=2016 \text{ and } \exists g: \min _{i} \max \{f(i), g(i)\} \leq 1$$
is also $B=3^{2017}$.

Finally, the number of functions such that
$$\exists g: \min _{i} \max \{f(i), g(i)\}=2016 \text{ and } \exists g: \min _{i} \max \{f(i), g(i)\}=0$$
is $C=2^{2017}$.

Now $A+B-C$ counts the number of functions with $\max _{g \in X} d(f, g) \geq 2015$ and $C$ counts the number of functions with $\max _{g \in X} d(f, g) \geq 2016$, so the answer is $A+B-2C=2 \cdot\left(3^{2017}-2^{2017}\right)$.