MHB Is $x$ a perfect square?

[anemone]

Prove that $x=\underbrace{1\cdots1}_{2012}\underbrace{5\cdots5}_{2011}6$ is a perfect square.

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[anemone]

Congratulations to the following members for their correct solutions::)

1. soroban
2. kaliprasad
3. lfdahl

Solution from soroban:

Spoiler

We have 2011 5's, moved one place to the left:
$\quad 5\cdot\frac{10^{2011}-1}{9}\cdot 10$

We have 2012 1's, moved 2012 places to the left:
$\quad \frac{10^{2012}-1}{9}\cdot 10^{2012}$

Hence,

$\begin{align*}\:x \;&=\;\frac{10^{2012}-1}{9}\cdot10^{2012} + 5\frac{10^{2011}-1}{9}\cdot 10 + 6\\&=\frac{(10^{2012})^2+4(10^{2012})+4}{9}\\&=\left(\frac{10^{2012}+2}{3}\right)^2\end{align*} $