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It's time for an intermediate math challenge! If you find the problems difficult to solve don't be disappointed! Just check our other basic level math challenge thread!

1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored.

2) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common knowledge to all mathematicians". Whether the latter is satisfied will be decided on a case-by-case basis.

3) If you have seen the problem before and remember the solution, you cannot participate in the solution to that problem.

4) You are allowed to use google, wolframalpha or any other resource. However, you are not allowed to search the question directly. So if the question was to solve an integral, you are allowed to obtain numerical answers from software, you are allowed to search for useful integration techniques, but you cannot type in the integral in wolframalpha to see its solution.

##1.## (solved by @julian ) Let ##A## be an ##\text{n x n}## matrix that is

where ##k \geq 2##. Prove that:

##\prod_{i = 1}^k \det\Big(A +x_i I\Big) \geq \det\Big(A +\big( \prod_{i = 1}^k x_i I \big)^\frac{1}{k}\Big)^k## ##\space##

##2.## (solved by @Biker ) Solve ##\mathcal{I}=\int_{-1}^0 \,x \cdot \sqrt{x^2+x+1} \,dx## . Hint: Use ##\cosh^2x - \sinh^2 x=1## . ##\space##

##3.## (solved by @nuuskur ) Show that there are infinitely many prime numbers of the form ##4k + 3##, ##k \in \mathbb{N} - \{0\}## ##\space##

##4.## (solved by @lpetrich ) What's the 100th digit after the decimal point of ##(1 + \sqrt{3})^{5000}## ##\space##

##5.## (solved by @lpetrich ) Given the differential operators ## D_n := x^n \cdot \dfrac{d}{dx}\,\,(n \in \mathbb{Z})## on smooth real valued functions ##\mathcal{C}^\infty(\mathbb{R})## .

Determine for which subsets ##L \subseteq \mathbb{Z}## the set ##\{D_n \,\vert \,n \in L\}## is a basis for a finite dimensional Lie algebra and which Lie algebra is it. ##\space##

##6.## (solved by @lpetrich, @julian ) Calculate the integral ## I = \int_{0}^{\pi} \sin^{2n} x dx## , ##n \in \mathbb{N}## ##\space##

##7.## (solved by @julian ) Solve ##\sum_{k=1}^\infty \dfrac{1}{k \binom{2k}{k}}## . ##\space##

##8.## (solved by @julian ) Find the coordinates of the center of gravity for the arc of the curve ##y = a \cosh(\frac{x}{a})## for ##-a \leq x \leq a## ##\space##

##9.## (resolved in post #62) For a given real Lie algebra ##\mathfrak{g}## , we define

##\mathfrak{A(g)} = \{\,\alpha \, : \,\mathfrak{g}\longrightarrow \mathfrak{g}\,\,: \,\,[\alpha(X),Y]=-[X,\alpha(Y)]\text{ for all }X,Y\in \mathfrak{g}\,\}##

the set of

a) Show that ##\mathfrak{A(g)}\subseteq \mathfrak{gl}(g)## is a Lie subalgebra in the Lie algebra of all linear transformations of ##\mathfrak{g}## with the commutator as Lie product: ##[\alpha, \beta]= \alpha \beta -\beta \alpha## .

b) Show that ##\mathfrak{g} \ltimes \mathfrak{A(g)}## is a semidirect product (##\,\mathfrak{A(g)}## is the ideal ) given by

##[X,\alpha]:=[\operatorname{ad}X,\alpha]=\operatorname{ad}X\,\alpha - \alpha\,\operatorname{ad}X##

c) Show that for all ##\alpha \in \mathfrak{A(g)}## and ##X,Y,Z \in \mathfrak{g}##

##[α(X),[Y,Z]]+[α(Y),[Z,X]]+[α(Z),[X,Y]]=0## ##\space##

##10.## (solved by @julian ) Calculate the integral ##\int_{c}^{} \frac{e^{kz}}{z}dz## where ##c## is the circle ## z = e^{i\theta}## with ##-\pi \leq \theta \leq \pi## and then prove that ##\int_{-\pi}^{\pi} e^{k\cos\theta} \cos(k\sin\theta)d\theta = 2\pi## ##\space##

**RULES:**1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored.

2) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common knowledge to all mathematicians". Whether the latter is satisfied will be decided on a case-by-case basis.

3) If you have seen the problem before and remember the solution, you cannot participate in the solution to that problem.

4) You are allowed to use google, wolframalpha or any other resource. However, you are not allowed to search the question directly. So if the question was to solve an integral, you are allowed to obtain numerical answers from software, you are allowed to search for useful integration techniques, but you cannot type in the integral in wolframalpha to see its solution.

**5) Mentors, advisors and homework helpers are kindly requested not to post solutions, not even in spoiler tags, for the challenge problems, until 16th of each month. This gives the opportunity to other people including but not limited to students to feel more comfortable in dealing with / solving the challenge problems. In case of an inadvertent posting of a solution the post will be deleted by**@fresh_42##1.## (solved by @julian ) Let ##A## be an ##\text{n x n}## matrix that is

__real skew symmetric__, and with positive numbers ## \{x_1, ..., x_k\}##where ##k \geq 2##. Prove that:

##\prod_{i = 1}^k \det\Big(A +x_i I\Big) \geq \det\Big(A +\big( \prod_{i = 1}^k x_i I \big)^\frac{1}{k}\Big)^k## ##\space##

**(by**@StoneTemplePython**)**##2.## (solved by @Biker ) Solve ##\mathcal{I}=\int_{-1}^0 \,x \cdot \sqrt{x^2+x+1} \,dx## . Hint: Use ##\cosh^2x - \sinh^2 x=1## . ##\space##

**(by**@fresh_42**)**##3.## (solved by @nuuskur ) Show that there are infinitely many prime numbers of the form ##4k + 3##, ##k \in \mathbb{N} - \{0\}## ##\space##

**(by**@QuantumQuest**)**##4.## (solved by @lpetrich ) What's the 100th digit after the decimal point of ##(1 + \sqrt{3})^{5000}## ##\space##

**(by**@StoneTemplePython**)**##5.## (solved by @lpetrich ) Given the differential operators ## D_n := x^n \cdot \dfrac{d}{dx}\,\,(n \in \mathbb{Z})## on smooth real valued functions ##\mathcal{C}^\infty(\mathbb{R})## .

Determine for which subsets ##L \subseteq \mathbb{Z}## the set ##\{D_n \,\vert \,n \in L\}## is a basis for a finite dimensional Lie algebra and which Lie algebra is it. ##\space##

**(by**@fresh_42**)**##6.## (solved by @lpetrich, @julian ) Calculate the integral ## I = \int_{0}^{\pi} \sin^{2n} x dx## , ##n \in \mathbb{N}## ##\space##

**(by**@QuantumQuest**)**##7.## (solved by @julian ) Solve ##\sum_{k=1}^\infty \dfrac{1}{k \binom{2k}{k}}## . ##\space##

**(by**@fresh_42**)**##8.## (solved by @julian ) Find the coordinates of the center of gravity for the arc of the curve ##y = a \cosh(\frac{x}{a})## for ##-a \leq x \leq a## ##\space##

**(by**@QuantumQuest**)**##9.## (resolved in post #62) For a given real Lie algebra ##\mathfrak{g}## , we define

##\mathfrak{A(g)} = \{\,\alpha \, : \,\mathfrak{g}\longrightarrow \mathfrak{g}\,\,: \,\,[\alpha(X),Y]=-[X,\alpha(Y)]\text{ for all }X,Y\in \mathfrak{g}\,\}##

the set of

*antisymmetric transformations*of ##\mathfrak{g}##. Remember that a real Lie algebra is a real vector space equipped with a multiplication for which holds- anti-commutativity: ##[X,X]=0##
- Jacobi-identity: ##[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0##

a) Show that ##\mathfrak{A(g)}\subseteq \mathfrak{gl}(g)## is a Lie subalgebra in the Lie algebra of all linear transformations of ##\mathfrak{g}## with the commutator as Lie product: ##[\alpha, \beta]= \alpha \beta -\beta \alpha## .

b) Show that ##\mathfrak{g} \ltimes \mathfrak{A(g)}## is a semidirect product (##\,\mathfrak{A(g)}## is the ideal ) given by

##[X,\alpha]:=[\operatorname{ad}X,\alpha]=\operatorname{ad}X\,\alpha - \alpha\,\operatorname{ad}X##

c) Show that for all ##\alpha \in \mathfrak{A(g)}## and ##X,Y,Z \in \mathfrak{g}##

##[α(X),[Y,Z]]+[α(Y),[Z,X]]+[α(Z),[X,Y]]=0## ##\space##

**(by**@fresh_42**)**##10.## (solved by @julian ) Calculate the integral ##\int_{c}^{} \frac{e^{kz}}{z}dz## where ##c## is the circle ## z = e^{i\theta}## with ##-\pi \leq \theta \leq \pi## and then prove that ##\int_{-\pi}^{\pi} e^{k\cos\theta} \cos(k\sin\theta)d\theta = 2\pi## ##\space##

**(by**@QuantumQuest**)**
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