<s>[INST] If $x$ and $y$ are integers such that $5 \mid x^2 - 2xy - y$ and $5 \mid xy - 2y^2 - x$, prove that $5 \mid 2x^2 + y^2 + 2x + y$. 
Given that $x$ and $y$ are integers satisfying $5 \mid x^2 - 2xy - y$ and $5 \mid xy - 2y^2 - x$, prove that $5 \mid 2x^2 + y^2 + 2x + y$.

I have provided a (dumbfounding) solution down below if you want to check out. There should be simpler solutions, I believe so.
 [/INST] 1) When $S=x^2-2xy-y$ and $T = xy-2y^2-x$, then $$ 5|S-T  =
(x-y)(x-2y+1) \ (a)$$
 and $$ 5| S+T =  (x-2y-1)(x+y)\ (b)
$$
Hence I know that $(x-2y+1)-(x-2y-1)=2$, by mathlove's comment. 
Hence at least one of $(x-2y+1),\ (x-2y-1)$ can not be divided by $5$. So by $(a),\ (b)$ we have $5|x^2-y^2$.
2) When $U=2x^2+y^2+2x+y$, then $$ U+S+2T= 3(x^2-y^2) $$
So we complete the proof.
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