Có bao nhiêu cặp số nguyên dương $\left( x,y \right)$ thỏa mãn...

Điều kiện: $x+3>0\Leftrightarrow x>-3$.
Ta có:
$\begin{aligned}
& {{2}^{y+2}}-{{\log }_{2}}\left( x+3 \right)\le \dfrac{x-3-2y}{2} \\
& \Leftrightarrow {{2}^{y+3}}-2{{\log }_{2}}\left( x+3 \right)\le x+3-2\left( y+3 \right) \\
& \Leftrightarrow {{2}^{y+3}}+2\left( y+3 \right)\le x+3+2{{\log }_{2}}\left( x+3 \right) \\
\end{aligned}$;
Đặt ${{\log }_{2}}\left( x+3 \right)=t\Rightarrow x+3={{2}^{t}}$. Khi đó:
${{2}^{y+3}}+2\left( y+3 \right)\le {{2}^{t}}+2t\Leftrightarrow f\left( y+3 \right)\le f\left( t \right)$ với $f\left( t \right)={{2}^{t}}+2t\Rightarrow f'\left( t \right)={{2}^{t}}.\ln 2+2>0,\forall t$, do đó: $f\left( y+3 \right)\le f\left( t \right)\Leftrightarrow y+3\le t\Leftrightarrow y+3\le {{\log }_{2}}\left( x+3 \right)$.
Suy ra $x+3\ge {{2}^{y+3}}\Rightarrow x\ge {{2}^{y+3}}-3\Rightarrow {{2}^{y+3}}-3\le 1000\Rightarrow y\le {{\log }_{2}}1003-3\Leftrightarrow y\le 6$.
+) $y=1\Rightarrow x\ge 13\Rightarrow x\in \left\{ 13;...;999 \right\}:987$ cặp.
+) $y=2\Rightarrow x\ge 29\Rightarrow x\in \left\{ 29;...;999 \right\}:971$ cặp.
+) $y=3\Rightarrow x\ge 29\Rightarrow x\in \left\{ 61;...;999 \right\}:939$ cặp.
+) $y=4\Rightarrow x\ge 125\Rightarrow x\in \left\{ 125;...;999 \right\}:875$ cặp.
+) $y=5\Rightarrow x\ge 253\Rightarrow x\in \left\{ 253;...;999 \right\}:747$ cặp.
+) $y=5\Rightarrow x\ge 509\Rightarrow x\in \left\{ 509;...;999 \right\}:491$ cặp.
Vậy có $5010$ cặp.