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**Summary:**group theory, number theory, commutative algebra, topology, calculus, linear algebra

**Remark:**new solution manual (01/20-06/20) attached

https://www.physicsforums.com/threads/solution-manuals-for-the-math-challenges.977057/

**1.**Given a group ##G## then the intersection of all maximal subgroups of ##G## is called Frattini subgroup ##\Phi(G)##. If ##G## hasn't a maximal subgroup, we set ##\Phi(G)=G.## Show that ##\Phi(G) \trianglelefteq G## is a normal subgroup, and that ##\Phi(G)## is nilpotent in case ##G## is finite.

**2.**The ##n##-th Fermat number ##F_n=2^{2^n}+1## is prime for ##n\in \mathbb{N}## if and only if ##3^{(F_n-1)/2}\equiv -1 \mod F_n.

## ##3## is a primitive root modulo ##F_n## in this case.

**3.**(solved by @The Fez ) Show that none of the numbers

$$11\, , \,111\, , \,1111\, , \,11111\, , \,111111\, , \,\ldots$$

can be written as a sum of two squares.

**4.**(solved by @julian ) Let ##G=\langle a,b\,|\,a^p=b^q=1,(aba)=b^r,a^s=b^t\rangle## be a group of order twelve which operates on ##\mathbb{R}^4## by

$$

a.v=\dfrac{1}{2}\cdot \begin{bmatrix}1&\sqrt{3}&0&0\\-\sqrt{3}&1&0&0\\0&0&1&-\sqrt{3}\\0&0&\sqrt{3}&1\end{bmatrix}.v, \quad b.v=\begin{bmatrix}0&0&1&0\\0&0&0&1\\-1&0&0&0\\0&-1&0&0\end{bmatrix}.v.

$$

**a.)**Determine the group ##G## and its presentation ##(p,q,r,s,t)##.

**b.)**Which group is

$$

H=\langle a,b\,|\,a^6=b^2=1,(aba)=b \rangle \,?

$$

**c.)**The above groups are obviously not Abelian. There is another non Abelian group ##L## of order twelve. Which one and what is ##(p,q,r,s,t)## in that case?

**5.**Let ##A## be an associative, finite dimensional algebra with ##1## over a field ##\mathbb{F},## ##M\neq 0## an ##A##-module, and ##0\neq P\subseteq A_A## a submodule of ##A## as right ##A##-module. Show that

**a.)**##M## is irreducible if and only if ##0## and ##1## are the only idempotent elements of the endomorphism ring ##\operatorname{End}_A(M).##

**b.)**##P## is a direct summand of ##A_A## if and only if there is an idempotent element ##e \in A## such that ##P=eA.##

**6.**(solved by @julian ) We consider the topological space ##\mathbb{C}_\infty =\mathbb{C}\cup \{\infty\}## equipped with distance

$$

\chi(x,y) :=\begin{cases}

\dfrac{\|x-y\|_2}{\sqrt{1+\|x\|_2^2}\sqrt{1+\|y\|_2^2}} &\text{ if } x,y\neq \infty \\[10pt]

\dfrac{1}{\sqrt{1+\|x\|_2^2}}&\text{ if }x\neq \infty,y=\infty \\[10pt]

\dfrac{1}{\sqrt{1+\|y\|_2^2}}&\text{ if }x= \infty,y\neq\infty \\[10pt]

0&\text{ if }x=y=\infty

\end{cases}

$$

Show that ##\chi## defines a metric such that ##\mathcal{C}:=(\mathbb{C}_\infty,\chi)## is a compact topological space.

**7.**

**a.)**(solved by @benorin ) Calculate ##\displaystyle{\int_{|z|=5} \; \dfrac{e^z}{z^2+\pi^2}\,dz}\,.##

**b.)**(solved by @benorin ) Determine all ##z\in \mathbb{C}## such that ##f(z)=e^{z^7(\sin z)^{16}} +\bar{z}^2## is complex differentiable.

**8.**(solved by @etotheipi ) Calculate

$$

\int_1^\infty \dfrac{1 + x^2 - 2 x^2 \log(x)}{x (1 + x^2)^2}\;dx

$$

**9.**(solved by @julian ) Determine the square root and the inverse matrix of

$$

A=\begin{pmatrix}5&-4&2\\ -4&7&-8\\ 1&-4&6\end{pmatrix}

$$

**Hint:**What is the dimension of the simple Lie algebra whose Cartan matrix is ##\sqrt{A}##?

**10.**Let ##R## be a commutative ring with ##1##. We define the nilradical ##N(R) =N\subseteq R## as intersection of all prime ideals of ##R##, and the Jacobson radical ##J(R)=J## as intersection of all maximal ideals.

**a.)**(solved by @julian ) Show that ##N(R)## contains exactly all nilpotent Elements of ##R##.

**b.)**Assume ##R## is Artinian. Show that all prime ideals are maximal, hence ##N(R)=J(R)## in an Artinian ring.

**c.)**Assume ##R## is Artinian. Show that ##N(R)## is a nilpotent Ideal.

**d.)**Give an example of ##N(R)\neq J(R)## if ##R## is not Noetherian and thus not Artinian either.

**High Schoolers only**

11.(solved by @Schalk21 ) Prove that the geometric mean of two numbers is less or equal the arithmetic mean of these numbers by three different methods (e.g.: geometric, algebraic, optimization).

11.

**12.**(solved by @Schalk21 ) Calculate the formula for the tangent at the unit circle at ##p=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)## by three different methods, or better points of view.

**13.**We are looking for the number ##n=abc##, where ##a## is the maximal number of rotations which are necessary to solve Rubik's cube out of any state, ##b## is the largest natural number of Chicken McNuggets which cannot be bought by the usual box sizes of ##6,9## or ##20##, and ##c## is the smallest three digit emirp number.

**14.**Show that the following linear equation system with variables ##x_1,\ldots,x_n## has always a unique solution:

\begin{align*}

x_1&= 2x_{n-m+1}+3x_{n-m+2}+b_1 \\

x_2&= 4x_{n-m+2}+9x_{n-m+3}+b_2 \\

\ldots &\qquad \ldots \\

x_{m-1}&=2^{m-1}x_{n-1}+3^{m-1}x_{n}+b_{m-1}\\

x_m&= 2^{m}x_{n} +b_m \\

x_{m+1}&=b_{m+1} \\

\ldots &\qquad \ldots \\

x_n&= b_n

\end{align*}

for all positive integers ##1\leq m < n## and any real numbers ##b_1,\ldots,b_n.##

**15.**Calculate the following derivatives:

**a.)**(solved by @Mayhem ) ##\dfrac{dy}{dx}## if ##y=1+y^x##

**b.)**##\dfrac{dy}{dx}## and ##\dfrac{d^2y}{dx^2}## if ##y=x+\log y##

**c.)**##\left. \dfrac{dy}{dx}\right|_{x=1}## and ##\left. \dfrac{d^2y}{dx^2}\right|_{x=1}## if ##x^2-2xy+y^2+x+y-2=0##

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