Putnam 2018 B1

Let $\mathcal{P}$ be the set of vectors defined by $$\mathcal{P} = \left\{ \left. \begin{pmatrix} a \\ b \end{pmatrix} \right| 0 \leq a \leq 2, 0 \leq b \leq 100, \text{ and } a,b \in \mathbb{Z} \right\}.$$ Find all $\mathbf{v} \in \mathcal{P}$ such that the set $\mathcal{P} \setminus \{ \mathbf{v} \}$ obtained by omitting vector $\mathbf{v}$ from $\mathcal{P}$ can be partitioned into two sets of equal size and equal sum.