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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 PP on the graph of y=2x2,y=2x^2, and draw a line going through PP parallel to the yy-axis. Then call the red area A:A: the area bounded by this line, the graph of y=x2,y=x^2, and the graph of y=2x2.y=2x^2.

Now, draw a line going through PP parallel to the xx-axis. Then call the blue area B:B: the area bounded by this line, the graph of y=2x2,y=2x^2, and the graph of y=f(x).y=f(x).

This function f(x)f(x) is continuous and displays the unique property that for every point PP on y=2x2,y = 2x^2, the two areas AA and BB are equal.

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

An infinitely thin, k cmk\text{ cm}-long squeegee begins to slide down from the upper-left corner of a k cm×k cmk\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=2016πk=\sqrt{\dfrac{2016}{\pi }}, what is the area of the window (\big(in cm2)\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
You need to be connected to run code

Evaluate the following integral:

cosx1+x2dx.\large \int_{ - \infty}^{\infty} \frac{\cos x}{1+x^2} dx.

In the country of Brilliantia, there are n>60n > 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 A\text{A} be another airport that flies planes to A\text{A}.

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


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