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2018-03-26 Advanced

         

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

Find the remainder when ϕ(22018+1)\phi\big(2^{2018} + 1\big) is divided by 4036.


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

ABCABC is an equilateral triangle with side length 1. Square DEFGDEFG is placed in triangle ABCABC such that points E,E, F,F, and GG are on segments AB,AB, BC,BC, and CA,CA, respectively.

What is the length of the perpendicular from point DD to segment BC?BC?

(xyyyxyyyz)(abc)=(abc)\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,zx, y, z such that any Pythagorean triple (a,b,c)(a, b, c) produces another Pythagorean triple (a,b,c).\big(a', b', c'\big).

Find the sum of such x,y,x, y, and z.z.

1eln(x)+12(ln(x)+13(ln(x)+14(ln(x)+15())))dx=?\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 mm sits on a horizontal, frictionless surface some distance \ell from a wall. Another block of mass MM collides from the right with velocity V.V. Depending on that ratio x=Mm,x =\frac Mm, the blocks will collide a total of nn times.

Find limxnx.\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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