# Problems of the Week

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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?

A ball with zero initial velocity falls from a height of $\frac{R}{n}$ near the vertical axis of symmetry on a concave spherical surface of radius $R$. Assuming that the collision is elastic, it is observed that the second impact of the ball is at the lowest point of the spherical surface. Determine the value of $n$ to the nearest integer.

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