Rút gọn A

To simplify the expression

\[
A = \frac{1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{2011} + \frac{1}{2012}}{\frac{2013}{1} + \frac{2014}{2} + \ldots + \frac{4024}{2012} - 2012},
\]

we first analyze the numerator and the denominator separately.

### Numerator

The numerator is

\[
S = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{2012}.
\]

This is the harmonic sum \( H_{2012} \).

### Denominator

The denominator can be simplified as follows:

\[
D = \frac{2013}{1} + \frac{2014}{2} + \ldots + \frac{4024}{2012} - 2012.
\]

We can express \( D \) as:

\[
D = \sum_{k=1}^{2012} \frac{2012 + k}{k} - 2012.
\]

This can be broken into two parts:

\[
D = \sum_{k=1}^{2012} \frac{2012}{k} + \sum_{k=1}^{2012} 1 - 2012,
\]

which simplifies to:

\[
D = 2012 H_{2012} + 2012 - 2012 = 2012 H_{2012}.
\]

### Final Simplification

Now substituting back into our expression for \( A \):

\[
A = \frac{H_{2012}}{2012 H_{2012}}.
\]

We simplify to:

\[
A = \frac{1}{2012}.
\]

Thus, the final result is

\[
\boxed{\frac{1}{2012}}.
\]