NL-NWO 2017 Problem 1

We consider positive integers written down in the (usual) decimal system. Within such an integer, we number the positions of the digits from left to right, so the leftmost digit (which is never a $0$ ) is at position $1$ . An integer is called even-steven if each digit at an even position (if there is one) is greater than or equal to its neighbouring digits (if these exist). An integer is called oddball if each digit at an odd position is greater than or equal to its neighbouring digits (if these exist). For example, $3122$ is oddball but not even-steven, $7$ is both even-steven and oddball, and $123$ is neither even-steven nor oddball. (a) Prove: every oddball integer greater than $9$ can be obtained by adding two oddball integers. (b) Prove: there exists an oddball integer greater than $9$ that cannot be obtained by adding two even-steven integers.