Does there exist a function \(f: \mathbb{R} \rightarrow \mathbb{R}\) that is **continuous** on *exactly* the rational numbers?

Note: The following function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is **discontinuous** on exactly the rational numbers:

\(\hspace{1.0cm}\) Index all the rational numbers using the bijection function \( a: \mathbb{N} \rightarrow \mathbb{Q} \).

\(\hspace{1.0cm}\) Define \( f: \mathbb{R} \rightarrow \mathbb{R} \) as \(\displaystyle f(x) = \sum_{ a(n) < x } 2^{-n }. \)

Let \(ABCDE\) be an equilateral convex pentagon such that \(\angle ABC=136^\circ\) and \(\angle BCD=104^\circ\). What is the measure (in degrees) of \(\angle AED\)?

Note: An equilateral polygon is a polygon whose sides are all of the same length. It does not imply that all the internal angles are equal, nor that the polygon is cyclic.

A parallel plate capacitor, with plates \(\textbf{A}\) and \(\textbf{B}\) of equal dimensions \(t \times L\) at a distance of \(L\), is filled with square tiles of dielectric to make a chess-board-like capacitor, as shown in the picture above.

Dielectric constant of the dark tile is \(\sigma_1,\) that of the light tile is \(\sigma_2,\) and \(\epsilon_0\) is the permittivity of free space. All the dielectric tiles are square cuboids of thickness \(t\).

Find the capacitance of this capacitor.

\(\)

**Details and Assumptions:**

- Assume that the electric field varies between the plates like an ideal parallel plate capacitor.

How many ways are there to tile a \(4\times6\) rectangle with twelve \(1\times2\) dominoes?

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