Proof that any integer $z>1$ can be written as $2x+y$, where $x>y$

Question

Imagine a multiple choice questionnaire with 3 choices $a, b,$ and $c$. At the end the sums of each choice are tallied. It seems it's always possible to have a tie for first, as long as the total number of questions n is greater than 1. For example:

$n=2: 1 a, 1 b, 0 c$

$n=3: 1 a, 1 b, 1 c$

$n=4: 2 a, 2 b, 0 c$

$n=5: 2 a, 2 b, 1 c$

$n=6: 2 a, 2 b, 2 c$

$n=7: 3 a, 3 b, 1 c$

etc...

This question equates to asking whether any integer $z>1$ can be written as $2x+y$, where $x$ and $y$ are integers and $x$ is greater than or equal to $y$. Intuitively this seems to be true, but is there a rigorous proof?

Answer 1

Any even number can be written as 2*x, where x is an integer +0 (=y). Any odd number can be written as an even number+1.