18. A flea is initially at the origin of the Cartesian plane and can move to integer coordinate points by choosing one of the following three moves each time:

- from the point $(x, y)$ it jumps to the point $(x, y+5)$;
- from the point $(x, y)$ it jumps to the point $(x-2, y-3)$;
- from the point $(x, y)$ it jumps to the point $(x+4, y-9)$.

How many paths, realizable by the flea with its moves, lead from the origin $(0,0)$ to the point $(0,2016)$?
(A) none
(B) precisely 1
(C) a number between 5 and 20
(D) a number between 20 and 100
(E) infinitely many
Problem 18. The answer is $(\mathbf{A})$.

If the flea makes $a$ moves of the first type, $b$ of the second, and $c$ of the third, the total displacement in the horizontal direction will be $4 c-2 b$. We want to move from the point $(0,0)$ to the point $(0,2016)$, and therefore $4 c-2 b$ must be equal to 0, from which $b=2 c$. The total displacement in the vertical direction will then be $5 a-3 b-9 c=5 a-6 c-9 c=5(a-3 c)$, which is necessarily a multiple of 5. It will therefore be impossible to start from the point $(0,0)$ and reach $(0,2016)$, since the required vertical displacement is 2016, which is not a multiple of 5.

(Problem proposed by C. Casamento Tumeo)