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This month's challenges will be my last thread of this kind for a while. Call it a creative break. Therefore, we will have a different format this month. I will post one problem a day, like an advent calendar, only for the entire month. I will try to post the questions as close as possible to 0:00 GMT.

I(a):=\int_0^1 \left(\dfrac{\log x}{a+1-x}-\dfrac{\log x}{a+x}\right)\,dx \; ; \; a\in \mathbb{C}\backslash[-1,0]

$$

$$

\mathfrak{A(g)}=\{\alpha \in \mathfrak{gl(g)}\,|\,[\alpha (X),Y]+[X,\alpha (Y)]=0\text{ for all }X,Y\in\mathfrak{g}\}

$$

is a Lie algebra. Determine ##\mathfrak{A(B)}## for the two-dimensional non-abelian Lie algebra ##\mathfrak{B}##.

\int_0^{\frac{\pi}{4}} \log(1+\tan x)\,dx

$$

$$

\psi(\varepsilon ,(x,y)):=\left(\dfrac{x}{1-\varepsilon x},\dfrac{y}{1-\varepsilon x}\right)

$$

defined on

$$

U=\left\{(\varepsilon ,(x,y))\,|\,\varepsilon <\dfrac{1}{x}\text{ for }x>0\text{, or }\varepsilon >\dfrac{1}{x}\text{ for }x<0\right\}\subseteq \mathbb{R}\times \mathbb{R}^2

$$

Show that ##\psi## defines a local group action of ##G## on the manifold ##M.## Does it have a global counterpart on ##\mathbb{R}^2##?

$$

\prod_{n=2}^\infty \left(1-\dfrac{1}{n}\right)\; , \;\prod_{n=3}^\infty \left(1-\dfrac{4}{n^2}\right)

$$

$$

\mathfrak{H}=\left\{\begin{bmatrix}

0&x_1&x_3\\0&0&x_2\\0&0&0

\end{bmatrix}\, : \,x_1,x_2,x_3\in \mathbb{R}\right\}.

$$

Calculate ##\exp(H)## for a matrix ##H\in \mathfrak{H}##.

\int_{-\infty }^{\infty }\dfrac{|\sin(\alpha x)|}{1+x^2}\,dx\; , \; \alpha >0

$$

$$

p\,|\,n\Longrightarrow p\,|\,\left(\dfrac{n}{p}-1\right)

$$

E.g. ##30=2\cdot 3\cdot 5## is such a number, since ##2\,|\,14\, , \,3\,|\,9\, , \,5\,|\,5.##

Show that ##n## is square-free (all prime factors have exponent ##1##), and no semiprime (product of exactly two primes).

$$

\mathbb{Q} \subseteq \mathbb{Q}\left(\sqrt[3]{\dfrac{9+\sqrt{69}}{18}}+\sqrt[3]{\dfrac{9-\sqrt{69}}{18}}\right).

$$

\begin{align*}

f\, : \,X &\longrightarrow \mathcal{P}(X)\\

x&\longmapsto \{x\}\\[6pt]

g\, : \,\mathcal{P}(X)\times \mathcal{P}(X)&\longrightarrow \mathcal{P}(X)\\

(A,B)&\longmapsto A\cup B

\end{align*}

and decide whether they are injective, surjective, and calculate the fiber (pre-image) of the empty set.

\begin{align*}

x &\equiv 2 \mod 3\\

x &\equiv 3 \mod 4\\

x &\equiv 2 \mod 5

\end{align*}

$$

A_1=\begin{bmatrix}

0&0&0\\0&1&0\\0&0&-1

\end{bmatrix}\, , \,A_2=\begin{bmatrix}

0&0&1\\-1&0&0\\0&0&0

\end{bmatrix}\, , \,A_3=\begin{bmatrix}

0&1&0\\0&0&0\\-1&0&0

\end{bmatrix}

$$

Calculate its center ##\mathfrak{Z(g)}=\{X\,|\,[A_i,X]=0\,(i=1,2,3)\},## its commutator subalgebra ##[\mathfrak{g},\mathfrak{g}],## and a Cartan subalgebra ##\mathfrak{h}##.

$$

\int_a^b f(x)g(x)\,dx = f(\xi)\int_a^b g(x)\,dx

$$

$$

x^2+\left(y+\dfrac{1}{\sqrt{3}}\right)^2=\dfrac{4}{3}\,.

$$

The point ##P:=\left(\dfrac{1}{\sqrt{3}},1-\dfrac{1}{\sqrt{3}}\right)## lies on this segment. Calculate the height ##h## of the circle segment, and ##|AP|+|PB|.##

$$

\operatorname{ker} (\varphi) \cap \operatorname{im}(\varphi) =\{0\} \Longleftrightarrow \operatorname{ker}(\varphi \circ \varphi )=\operatorname{ker}(\varphi )

$$

$$

\int_A \langle F,n \rangle\,d^2r

$$

for the vector field ##F\, : \,\mathbb{R}^3\rightarrow \mathbb{R}^3## defined by ##F(x,y,z)=(xz,yz,123).##

.

**1.**(solved by @Infrared ) Let ##G## be a group with ##3129## elements. Prove it is solvable.**2.**$$I(a):=\int_0^1 \left(\dfrac{\log x}{a+1-x}-\dfrac{\log x}{a+x}\right)\,dx \; ; \; a\in \mathbb{C}\backslash[-1,0]

$$

**3.**(solved by @Infrared ) Let ##\mathfrak{g}## be a Lie algebra over a field of characteristic not ##2.## Prove that$$

\mathfrak{A(g)}=\{\alpha \in \mathfrak{gl(g)}\,|\,[\alpha (X),Y]+[X,\alpha (Y)]=0\text{ for all }X,Y\in\mathfrak{g}\}

$$

is a Lie algebra. Determine ##\mathfrak{A(B)}## for the two-dimensional non-abelian Lie algebra ##\mathfrak{B}##.

**4.**(solved by @Infrared ) Show that a path-connected set is connected but not vice versa and not necessarily simply connected.**5.**(solved by @julian ) $$\int_0^{\frac{\pi}{4}} \log(1+\tan x)\,dx

$$

**6.**(solved by @TeethWhitener and @PeroK ) There are currently about ##7,808,000,000## people on earth. If we would enumerate them all, how many of them would have a prime number?**7.**Let ##M=\mathbb{R}^2## and ##G=\mathbb{R}## and consider the map$$

\psi(\varepsilon ,(x,y)):=\left(\dfrac{x}{1-\varepsilon x},\dfrac{y}{1-\varepsilon x}\right)

$$

defined on

$$

U=\left\{(\varepsilon ,(x,y))\,|\,\varepsilon <\dfrac{1}{x}\text{ for }x>0\text{, or }\varepsilon >\dfrac{1}{x}\text{ for }x<0\right\}\subseteq \mathbb{R}\times \mathbb{R}^2

$$

Show that ##\psi## defines a local group action of ##G## on the manifold ##M.## Does it have a global counterpart on ##\mathbb{R}^2##?

**8.**Give an example of a ring and a maximal ideal that isn't a prime ideal.**9.**Let ##U,V\subseteq \mathbb{C}## open sets, ##\varphi \, : \,U\longrightarrow V## a holomorphic function, and ##\gamma \, : \,[0,1]\longrightarrow U## a closed, smooth path. Show that if ##\gamma ## is ##0##-homologue in ##U,## then ##\varphi \circ \gamma ## is ##0##-homologue in ##V.##**10.**(solved by @julian ) Examine convergence:$$

\prod_{n=2}^\infty \left(1-\dfrac{1}{n}\right)\; , \;\prod_{n=3}^\infty \left(1-\dfrac{4}{n^2}\right)

$$

**11.**(solved by @TeethWhitener ) The Heisenberg algebra can be viewed as$$

\mathfrak{H}=\left\{\begin{bmatrix}

0&x_1&x_3\\0&0&x_2\\0&0&0

\end{bmatrix}\, : \,x_1,x_2,x_3\in \mathbb{R}\right\}.

$$

Calculate ##\exp(H)## for a matrix ##H\in \mathfrak{H}##.

**12.**(solved by @julian ) $$\int_{-\infty }^{\infty }\dfrac{|\sin(\alpha x)|}{1+x^2}\,dx\; , \; \alpha >0

$$

**13.**(solved by @fishturtle1 ) Show that ##(n-1)!\equiv -1 \mod n## holds if and only if ##n## is prime. Determine the first two primes for which even ##(p-1)!\equiv -1 \mod p^2## holds.**14.**(solved by @QuantumSpace ) Determine all possible topologies on ##X:=\{a,b\}##, and which of them are homeomorphic. Give an example of a topological space with more than one element such that all sequences converge.**15.**(solved by @fishturtle1 ) Explain the difference between ##\mathbb{Z}_2\times \mathbb{Z}_3## and ##\mathbb{Z}_2 \ltimes \mathbb{Z}_3\,.## Is there also a group ##\mathbb{Z}_2 \rtimes \mathbb{Z}_3\,?##**16.**Show that ##16## and ##33## are Størmer numbers, but no number ##N:=2n^2>2## can be one, e.g. ##32.##**17.**(solved by @fishturtle1 ) Consider a number ##n## which is not a prime and$$

p\,|\,n\Longrightarrow p\,|\,\left(\dfrac{n}{p}-1\right)

$$

E.g. ##30=2\cdot 3\cdot 5## is such a number, since ##2\,|\,14\, , \,3\,|\,9\, , \,5\,|\,5.##

Show that ##n## is square-free (all prime factors have exponent ##1##), and no semiprime (product of exactly two primes).

**18.**Prove that path integrals in ##\mathbb{R}^n## over gradient vector fields depend only on starting and endpoint, and not on the path itself.**19.**(solved by @julian ) Let ##P_0=0,P_1=1,P_n=2P_{n-1}+P_{n-2}## for all ##n\in \mathbb{N},n\geq2##. Determine a closed form for ##P_n.##**20.**Find the irreducible minimal polynomial for$$

\mathbb{Q} \subseteq \mathbb{Q}\left(\sqrt[3]{\dfrac{9+\sqrt{69}}{18}}+\sqrt[3]{\dfrac{9-\sqrt{69}}{18}}\right).

$$

**21.**Show that the embedding ##\mathbb{S}^1\longrightarrow \mathbb{R}^2-\{0\}## is a homotopy equivalence, and that ##\mathbb{R}\longrightarrow \mathbb{R}^2-\{0\}## defined by ##x\mapsto (x,1)## is none.**22.**(solved by @fishturtle1 ) Let ##\emptyset\neq X## be a set, ##\mathcal{P}(X)## its power set. Consider the following mappings\begin{align*}

f\, : \,X &\longrightarrow \mathcal{P}(X)\\

x&\longmapsto \{x\}\\[6pt]

g\, : \,\mathcal{P}(X)\times \mathcal{P}(X)&\longrightarrow \mathcal{P}(X)\\

(A,B)&\longmapsto A\cup B

\end{align*}

and decide whether they are injective, surjective, and calculate the fiber (pre-image) of the empty set.

**23.**(solved by @TeethWhitener ) Find the smallest positive integer ##x## that solves\begin{align*}

x &\equiv 2 \mod 3\\

x &\equiv 3 \mod 4\\

x &\equiv 2 \mod 5

\end{align*}

**24.**Let ##\vec{u},\vec{v},\vec{w}## be three different coplanar vectors of equal length, originating at a point ##O.## Their endpoints define a triangle ##\triangle UVW##. How can the barycenter ##S## be found?**25.**Is a partially differentiable function ##f\, : \,\mathbb{R}^2\rightarrow \mathbb{R}## at some point ##x_0## also continuous at ##x_0##?**26.**Let ##\mathfrak{g}## be the real Lie algebra generated by$$

A_1=\begin{bmatrix}

0&0&0\\0&1&0\\0&0&-1

\end{bmatrix}\, , \,A_2=\begin{bmatrix}

0&0&1\\-1&0&0\\0&0&0

\end{bmatrix}\, , \,A_3=\begin{bmatrix}

0&1&0\\0&0&0\\-1&0&0

\end{bmatrix}

$$

Calculate its center ##\mathfrak{Z(g)}=\{X\,|\,[A_i,X]=0\,(i=1,2,3)\},## its commutator subalgebra ##[\mathfrak{g},\mathfrak{g}],## and a Cartan subalgebra ##\mathfrak{h}##.

**27.**(solved by @julian ) Let ##f\, : \,[a,b]\longrightarrow \mathbb{R}## be a continuous function and ##g\, : \,[a,b]\longrightarrow \mathbb{R}## integrable with ##g(x)\geq 0## for all ##x\in [a,b]##. Then there is a ##\xi\in [a,b]## such that$$

\int_a^b f(x)g(x)\,dx = f(\xi)\int_a^b g(x)\,dx

$$

**28.**Consider the circle segment above ##A=(-1,0)## and ##B=(1,0)## of$$

x^2+\left(y+\dfrac{1}{\sqrt{3}}\right)^2=\dfrac{4}{3}\,.

$$

The point ##P:=\left(\dfrac{1}{\sqrt{3}},1-\dfrac{1}{\sqrt{3}}\right)## lies on this segment. Calculate the height ##h## of the circle segment, and ##|AP|+|PB|.##

**29.**(solved by @ergospherical and @PeroK ) Let ##\varphi :V\longrightarrow V## a linear mapping. Prove$$

\operatorname{ker} (\varphi) \cap \operatorname{im}(\varphi) =\{0\} \Longleftrightarrow \operatorname{ker}(\varphi \circ \varphi )=\operatorname{ker}(\varphi )

$$

**30.**(solved by @ergospherical ) Let ##A## by a cylindric surface (without base or cover) that rotates around the ##z##-axis and stands on the plane ##\{z=0\}##, with radius ##R>0## and height ##h>0##. Give a parameterization and calculate the surface integral$$

\int_A \langle F,n \rangle\,d^2r

$$

for the vector field ##F\, : \,\mathbb{R}^3\rightarrow \mathbb{R}^3## defined by ##F(x,y,z)=(xz,yz,123).##

**31.**Let ##\mathcal{P}## be a finite set of points in a plane, that are not all collinear. Then there is a straight, that contains exactly two points..

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