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2017-10-23 Advanced

         

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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