Source: 2017 Canadian Open Math Challenge, Problem C1
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For a positive integer $n$, we define function $P(n)$ to be the sum of the digits of $n$ plus the number of digits of $n$. For example, $P(45) = 4 + 5 + 2 = 11$. (Note that the first digit of $n$ reading from left to right, cannot be $0$).
$\qquad$(a) Determine $P(2017)$.
$\qquad$(b) Determine all numbers $n$ such that $P(n) = 4$.
$\qquad$(c) Determine with an explanation whether there exists a number $n$ for which $P(n) - P(n + 1) > 50$.
### Part (a)
1. Calculate the sum of the digits of \(2017\):
   \[
   2 + 0 + 1 + 7 = 10
   \]
2. Determine the number of digits in \(2017\):
   \[
   \text{Number of digits} = 4
   \]
3. Apply the function \(P(n)\):
   \[
   P(2017) = 10 + 4 = 14
   \]

The final answer is \(\boxed{\text{No}}\)