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**Summary::**Functional Analysis, Algebras, Measure Theory, Differential Geometry, Calculus, Optimization, Algorithm, Integration. Lie Algebras.

**1.**(solved by @julian ) Let ##(a_n)\subseteq\mathbb{R}## be a sequence of real numbers such that ##a_n \leq n^{-3}## for all ##n\in \mathbb{N}.## Given the family ##\mathcal{A}## of functions ##f_n\, : \,[0,1]\longrightarrow \mathbb{R}## defined by ##f_n(x)=\sum_{k=n}^\infty a_k\sin(kx)## for ##n\in \mathbb{N},## show that every sequence ##(g_n)\subseteq\mathcal{A}## contains a subsequence ##(g_{n_k})## which converges uniformly on ##[0,1].##

**2. (a)**Show that if a ##*##-algebra ##A## admits a complete ##C^*##-norm, then it is the only ##C^*##-norm on ##A##.

**(b)**Let ##A## be a ##*##-algebra. Show that there is a ##*##-isomorphism ##M_n(\Bbb{C}) \otimes A \cong M_n(A)##.

**(c)**Deduce that the ##C^*##-algebra ##M_n(\Bbb{C})## is nuclear for all ##n \geq 1##. (MQ)

**3.**Give an example of a Riemann integrable function that is not Borel measurable. (MQ)

**4.**Let ##C## be the Cantor set. Show that ##\frac{1}{2}C + \frac{1}{2}C = [0,1]##. Deduce that the sum of sets of measure ##0## must not have measure ##0##. (MQ)

**5.**Let ##\pi\, : \,\mathbb{R}^n\longrightarrow \mathbb{T}^n## be the canonical projection and ##f:=\pi|_{[0,1]^n}## its restriction on the closed unit cube. Show with the help of ##f\, : \,[0,1]^n\longrightarrow\mathbb{T}^n##, that a quotient map in general doesn't have to be open.

**6.**Let ##D=\{\,z\in \mathbb{C}\, : \,|z|<1\,\}## be the complex open unit disk and let ##0<a<1## be a real number. Suppose ##f\, : \,D\longrightarrow \mathbb{C}## is a holomorphic function such that ##f(a)=1## and ##f(-a)=-1.##

**(a)**Prove that ##\sup_{z\in D}\{|f(z)|\}\geq \dfrac{1}{a}.##

**(b)**Prove that if ##f## has no root, then ##\sup_{z\in D}\{|f(z)|\}\geq \exp\left(\dfrac{1-a^2}{4a}\,\pi\right).##

**7.**(solved by @julian )

**(a)**Let ##0<p\leq a,b,c,d,e \leq q## and show that

$$

(a+b+c+d+e)\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}+\dfrac{1}{e}\right) \leq 25+6\left(\sqrt{\dfrac{p}{q}}-\sqrt{\dfrac{q}{p}}\right)^2.

$$

**(b)**This is a special case of a general inequality. Which is the general case and how is it proven?

**8.**Let ##n>1## be an integer. There are ##n## lamps ##L_0,\ldots,L_{n-1}## arranged in a circle. Each lamp is either ON ##(1)## or OFF ##(0)##. A sequence of steps ##S_0,\ldots,S_i,\ldots## is carried out. Step ##S_j## affects the state of ##L_j## only (leaving the states of all other lamps unaltered) as follows:

If ##L_{j-1}## is ON, ##S_j## changes the state of ##L_j## from ON to OFF or from OFF to ON.

If ##L_{j-1}## is OFF, ##S_j## leaves the state of ##L_j## unchanged.

The lamps are labeled modulo ##n##, that is ##L_{-1}=L_{n-1}, L_0=L_n,## etc. Initially all lamps are ON.

Show that

**(a)**(solved by @Jarvis323 ) there is a positive integer ##M(n)## such that after ##M(n)## steps all the lamps are ON again;

**(b)**if ##n=2^k,## then all lamps are ON after ##(n^2-1)## steps;

**(c)**if ##n=2^k+1,## then all lamps are ON after ##(n^2-n+1)## steps.

**9.**(solved by @Fred Wright ) The pseudosphere is the rotational surface of the tractrix, e.g. parameterized by

$$

f\, : \,\mathbb{R}^2\longrightarrow \mathbb{R}^3\; , \;f(x,y)=\begin{bmatrix} \cos (y)/ \cosh (x)\\ \sin (y)/ \cosh (x)\\x- \tanh (x)\end{bmatrix}.

$$

Show that the pseudosphere has a constant negative Gauß curvature.

**10.**Let ##\mathfrak{g}## be a Lie algebra with trivial center ##\mathfrak{Z(g)}=\{0\}## over a field of characteristic not equal two and

\begin{align*}

\mathfrak{A(g)}&=\{\varphi:\mathfrak{g}\stackrel{linear}{\longrightarrow}\mathfrak{g}\,|\,[\varphi(X),Y]=[\varphi(Y),X]\text{ for all }X,Y\in \mathfrak{g}\}\\

&=\operatorname{lin}\{\alpha,\beta\neq 0\,|\,[\alpha,\beta]=\alpha\beta-\beta\alpha=\beta\}

\end{align*}

Show that image ##\operatorname{im}\beta## and kernel ##\operatorname{ker}\beta## of ##\beta## are ideals in ##\mathfrak{g}.##

**Hint:**##\mathfrak{A(g)}## is a ##\mathfrak{g}##-module by ##X.\varphi =[\operatorname{ad}X,\varphi].##

**High Schoolers only**

11.(solved by @etotheipi ) Let ##I## and ##J## be bounded open intervals in ##\Bbb{R}## with ##I \cap J \neq \emptyset## and the length of ##J## is greater than the length of ##I##. Show that ##I \subseteq 3*J##, where ##3*J## is the interval with length ##3## times the length of ##J## and with the same centre as ##J##. (MQ)

11.

**12.**Given a positive integer ##n##. Assume that ##4^n## and ##5^n## start with the same digit in the decimal system.

Show that this digit has to be ##2## or ##4.##

**13.**A parcel service charges a price proportional to the sum height plus length plus width per box.

Could it be, that there is a case where it is cheaper to put a more expensive package into a cheaper box?

**14.**Let ##a## be a positive integer and ##(a_n)_{n\in \mathbb{N}_0}## the sequence defined by

$$

a_0:=1\; , \;a_n:=a+\prod_{k=0}^{n-1}a_k \quad (n\geq 1)

$$

**(a)**There are infinitely many primes which divide at least one number of the sequence.

**(b)**There is a prime which does not divide any of the numbers in the sequence

**15.**Let ##a,b,c## be positive real numbers such that ##a+b+c+2=abc.##

Show that ##(a+1)(b+1)(c+1)\geq 27.## Under which condition does equality hold?

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