# Problems of the Week

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A monkey is typing a string of random letters, with all letters equally (and independently) likely to be typed.

Which word is more likely to appear first in the string, heart or earth?

Take a point $$P$$ on the graph of $$y=2x^2,$$ and draw a line going through $$P$$ parallel to the $$y$$-axis. Then call the red area $$A:$$ the area bounded by this line, the graph of $$y=x^2,$$ and the graph of $$y=2x^2.$$

Now, draw a line going through $$P$$ parallel to the $$x$$-axis. Then call the blue area $$B:$$ the area bounded by this line, the graph of $$y=2x^2,$$ and the graph of $$y=f(x).$$

This function $$f(x)$$ is continuous and displays the unique property that for every point $$P$$ on $$y = 2x^2,$$ the two areas $$A$$ and $$B$$ are equal.

Find $$f(x)$$ and evaluate $$f(12).$$

An infinitely thin, $$k\text{ cm}$$-long squeegee begins to slide down from the upper-left corner of a $$k\text{ cm} \times k\text{ cm}$$ square window. Its other end simultaneously slides toward the lower-right corner of the window, with the upper end kept in contact with the left side of the window.

If $$k=\sqrt{\dfrac{2016}{\pi }}$$, what is the area of the window $$\big($$in $$\text{cm}^2\big)$$ cleaned by the squeegee?

Note: you may end up with an integral expression that's difficult to evaluate analytically. Feel free to finish the job using the code environment below:

import math
Python 3

Evaluate the following integral:

$\large \int_{ - \infty}^{\infty} \frac{\cos x}{1+x^2} dx.$

In the country of Brilliantia, there are $$n > 60$$ airports built recently such that the distances between any two airports are all different from one another.

A plane will always fly from an airport to the nearest airport. Let an origin of an airport $$\text{A}$$ be another airport that flies planes to $$\text{A}$$.

What is the maximum number of origins an airport can have?

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