HMMT February

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 $A+B-2 C=2 \cdot\left(3^{2017}-2^{2017}\right)$.