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**Summary:**Lie Algebras, Commutative Algebra, Ordering, Differential Geometry, Algebraic Geometry, Gamma Function, Calculus, Analytic Geometry, Functional Analysis, Units.

**1.**Prove that all derivations ##D:=\operatorname{Der}(L)## of a semisimple Lie algebra ##L## are inner derivations ##M:=\operatorname{ad}(L).##

**2.**Give four possible non-isomorphic meanings for the notation ##\mathbb{Z}_p.##

**3.**(solved by @Office_Shredder ) Let ##T\subseteq (\mathbb{Z}_+^n,\preccurlyeq )## with the partial natural ordering. Then there is a finite subset ##S\subseteq T## such that for every ##t\in T## exists a ##s\in S## with ##s\preccurlyeq t.##

$$

\alpha \preccurlyeq \beta \Longleftrightarrow \alpha_i \leq \beta_i \text{ for all }i=1,\ldots,n

$$

**4. (a)**Solve the following linear differential equation system:

\begin{align*}

\dot{y}_1(t)=11y_1(t)-80y_2(t)\;&\wedge\;\dot{y}_2(t)=y_1(t)-5y_2(t)\\

y_1(0)=0\;&\wedge\;y_2(0)=0

\end{align*}

**(b)**Which solutions do ##y_1(0)=\pm \varepsilon\;\wedge\;y_2(0)=\pm \varepsilon## have?

**(c)**How does the trajectory for ##y_1(0)=0.001\;\wedge\;y_2(0)=0.001## behave for ##t\to \infty##?

**(d)**What will change if we substitute the coefficient ##-80## by ##-60##?

**(e)**Calculate (approximately) the radius of the osculating circle at ##t=\pi/12## for both trajectories with initial condition ##\mathbf{y}(0)=(-1,1).##

**5.**(solved by @bpet ) Consider the ideal ##I =\langle x^2y+xy ,xy^2+1 \rangle \subseteq \mathbb{R}[x,y] ## and compute a reduced Gröbner basis to determine the number of irreducible components of the algebraic variety ##V(I).##

**6.**(solved by @benorin ) Define the complex function gamma function as

$$

\Gamma(z):=\lim_{n \to \infty}\dfrac{n!\,n^z}{z(z+1)\cdot\ldots\cdot(z+n)}

$$

and prove

**(a)**##\Gamma(z)=\displaystyle{\int_0^\infty}e^{-t}\,t^{z-1}\,dt\;;\quad\mathfrak{R}(z)>0##

**(b)**##\Gamma(z)^{-1}=e^{\gamma z}z\,\displaystyle{\prod_{n=1}^\infty}\left(1+\dfrac{z}{n}\right)e^{-\frac{z}{n}}##

where ##\gamma :=\displaystyle{\lim_{n \to \infty}\left(1+\dfrac{1}{2}+\dfrac{1}{3}+\ldots+\dfrac{1}{n}-\log(n)\right)}## is the Euler-Mascheroni constant.

**7.**(solved by @benorin) Let ##u\, : \,[0,1]\times [a,b]\longrightarrow \mathbb{C}## be a continuous function, such that the partial derivative in the first coordinate exists everywhere and is continuous. Define

$$

U(\lambda):=\int_a^b u(\lambda,t)\,dt\; , \;V(\lambda):=\int_a^b \dfrac{\partial u}{\partial \lambda}(\lambda,t)\,dt.

$$

Show that ##U## is continuously differentiable and ##U'(\lambda)=V(\lambda)## for all ##0\leq \lambda\leq 1.##

**8.**A cardiod is defined as the trace of a point on a circle that rolls around a fixed circle of the same size without slipping.

It can be described by ##(x^{2}+y^{2})^{2}+4x(x^{2}+y^{2})-4y^{2}\,=\,0## or in polar coordinates by ##r(\varphi )=2(1-\cos \varphi ).##

Show that:

**(a)**(solved by @etotheipi ) Given any line, there are exactly three tangents parallel to it. If we connect the points of tangency to the cusp, the three segments meet at equal angles of ##2\pi/3\,.##

**(b)**(solved by @etotheipi ) The length of a chord through the cusp equals ##4.##

**(c)**(solved by @etotheipi ) The midpoints of chords through the cusp lie on the perimeter of the fixed generator circle (black one in the first picture).

**(d)**(solved by @etotheipi ) Calculate length, area and curvature.

**9.**(solved by @nuuskur ) Let ##A## be a complex Banach algebra with ##1##. Prove that the spectrum

$$

\sigma(a)=\{\lambda\in \mathbb{C}\,|\,\lambda\cdot 1-a \text{ is not invertible }\} \subseteq \{\lambda\in \mathbb{C}\,|\,|\lambda|\leq \|a\| \}

$$

for any ##a\in A## is not empty, bounded and closed.

**10.**

**(a)**Determine all primes which occur as orders of an element from ##G:=\operatorname{SL}_3(\mathbb{Z}).##

**(b)**(solved by @Office_Shredder ) Let ##I \trianglelefteq R## be a two-sided ideal in a unitary ring with group of unities ##U##. Show by two different methods that

$$

M:=\{u\in U\,|\,u-1\in I\} \trianglelefteq U

$$

is a normal subgroup.

**High Schoolers only (until 26th)**

11.(solved by @Not anonymous ) If ##a,b,c## are real numbers such that ##a+b+c=2## and ##ab+ac+bc=1,## show that ##0\leq a,b,c\leq \dfrac{4}{3}.##

11.

**12.**Determine all pairs ##(m,n)## of (positive) natural numbers such that ##2022^m-2021^n## is a square.

**13.**(solved by @archaic , @Not anonymous )

**(a)**Prove for any ##n\in \mathbb{N},\,n\geq 4##

$$

Q(n):=\dfrac{4^2-9}{4^2-4}\cdot\dfrac{5^2-9}{5^2-4}\cdot\ldots\cdot\dfrac{n^2-9}{n^2-4}>\dfrac{1}{6}.

$$

**(b)**Does the above statement still hold, if we replace ##1/6## on the right hand side by ##0.1667\,?##

**14.**(solved by @Not anonymous ) Determine all pairs ##(x,y)\in \mathbb{R}^2## such that

\begin{align*}

5&=\sqrt{1+x+y}+\sqrt{2+x-y}\\

2-x+y&=\sqrt{18+x-y}

\end{align*}

**15.**(solved by @Not anonymous ) Given a real, continuous function ##f:\mathbb{R}\longrightarrow \mathbb{R}## such that ##f(f(f(x)))=x.##

Prove that ##f(x)=x## for all ##x\in\mathbb{R}.##

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