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2018-03-26 Advanced

         

\(\phi(n)\) is the number of positive integers less than \(n\) that are relatively prime to \(n.\)

Find the remainder when \(\phi\big(2^{2018} + 1\big)\) is divided by 4036.


Bonus: Generalize for the remainder when \(\phi(2^n+1)\) is divided by \(2n.\)

\(ABC\) is an equilateral triangle with side length 1. Square \(DEFG\) is placed in triangle \(ABC\) such that points \(E,\) \(F,\) and \(G\) are on segments \(AB,\) \(BC,\) and \(CA,\) respectively.

What is the length of the perpendicular from point \(D\) to segment \(BC?\)

\[\begin{pmatrix}x & y & y \\ y & x & y \\ y & y & z \end{pmatrix}\begin{pmatrix}a \\ b \\ c \end{pmatrix} = \begin{pmatrix}a' \\ b' \\ c' \end{pmatrix}\] If the above equation holds true, then there exist coprime positive integers \(x, y, z\) such that any Pythagorean triple \((a, b, c)\) produces another Pythagorean triple \(\big(a', b', c'\big).\)

Find the sum of such \(x, y,\) and \(z.\)

\[\int_{1}^{e}\ln (x)+\frac{1}{2}\left ( \ln (x)+\frac{1}{3}\Big ( \ln (x)+\tfrac{1}{4}\big( \ln (x)+\tfrac{1}{5} {\small(\cdots)}\, \big) \Big ) \right )\, dx = \, ?\]

A block of mass \(m\) sits on a horizontal, frictionless surface some distance \(\ell\) from a wall. Another block of mass \(M\) collides from the right with velocity \(V.\) Depending on that ratio \(x =\frac Mm,\) the blocks will collide a total of \(n\) times.

Find \[\displaystyle \lim_{x\rightarrow \infty} \frac{n}{\sqrt{x}}.\]

Assume that all collisions are elastic and that the motion is in 1-dimension.

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