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**Questions**

1.(solved by @Pi-is-3 )The maximum value of ##f## with ##f(x) = x^a e^{2a - x}## is minimal for which values of positive numbers ##a## ?

1.

**2.**(solved by @KnotTheorist ) Find the equation of a curve such that ##y''## is always ##2## and the slope of the tangent line is ##10## at the point ##(2,6)##.

**3.**(solved by @nuuskur ) Let ##F_q## be the finite field with ##q## elements and ##n \geq 1## with ##(n,q)=1##. What is the smallest field extension of ##F_q## containing an ##n##-th root of unity?

**4.**(solved by @nuuskur ) Show that there is no infinite countable ##\sigma##-algebra.

**5.**(solved by @nuuskur ) Classify all groups of order ##6,8,## and ##12##.

**6.**(solved by @nuuskur ) Let ##G## be a group of order ##2m## with ##m>1## an odd number. Show that ##G## is not simple, i.e. has at least one non trivial normal subgroup.

**7.**

**(a)**(solved by @nuuskur ) Show that a group ##G## is abelian if and only the map ##g \mapsto g^{-1}## is a group homomorphism.

**7. (b)**(solved by @nuuskur ) Suppose that ##G## is a finite group and that we have ##\sigma \in \operatorname{Aut}(G)## with the neutral element ##e## as unique fixed point. If ##\sigma^2=\operatorname{id}_G##, prove that ##G## is abelian.

**Hint:**Prove that every element of ##G## can be written in the form ##x^{-1} \sigma(x)## and apply ##\sigma## to such an expression.

(Credits will only be given for solving both subquestions.)

**8.**(solved by @Pi-is-3 ) Show that there is no triple ##(a,b,c) \in \mathbb{Z}^3\setminus \{(0,0,0)\}## such that ##a^2 + b^2 = 3 c^2##.

**Hint**: Find all squares in ##\mathbb{Z}/4\mathbb{Z}##.

**9.**(solved by @nuuskur ) Three identical airplanes start at the same time at the vertices of an equilateral triangle with side length ##L##. Let's say the origin of our coordinate system is the center of the triangle. The planes fly at a constant speed ##v## above ground in the direction of the clockwise next airplane. How long will it take for the planes to reach the same point, and which are the flight paths?

**10.**(solved by @nuuskur ) The Schwarzian derivative of a holomorphic function ##f## is given by

$$

S_f(z)=\{\,f,z\,\} := \dfrac{d}{dz}\left( \dfrac{f^{''}(z)}{f^{'} (z)} \right)-\dfrac{1}{2} \left( \dfrac{f^{''}(z)}{f^{'}(z)} \right)^2= \dfrac{f^{'''}(z)}{f^{'} (z)}- \dfrac{3}{2} \left( \dfrac{f^{''}(z)}{f^{'}(z)} \right)^2

$$

Prove a chain rule for the Schwarzian derivative and show that

$$

\{\,f,z\,\} <0 \,\wedge \,\{\,h,z\,\}<0 \Longrightarrow \{\,f\circ h,z\,\}<0

$$

Schwarzian derivatives are used in dynamical systems to investigate attractors, in flows of surfaces, or in the theory of Schwarz-Christoffel mappings.

**11.**(solved by @archaic ) Show geometrically that ##\sin^{-1}(\frac{x}{\sqrt{x^2 + 1}}) = \tan^{-1} x## for ##x \geq 0##.

**12.**(solved by @archaic and @Pi-is-3 ) Give an example of a continuous function ##f(x)## on ##[0,1]## for which the conclusion of Rolle's theorem fails.

Rolle's theorem: If ##f## is differentiable on ##(0,1)## and ##f(0)=f(1)## then there is a point ##c\in (0,1)## such that ##f'(c)=0##.

**13.**(solved by @etotheipi ) David drives to work every working day by car. Outside towns he drives at an average speed of ##180\,\text{km/h}##. On the

##10\,\text{km}## in town, he drives at an average speed of ##40\,\text{km/h}##. As a result, he is often too fast and gets a ticket. Meanwhile he has realized that things can not go on like this and he decides to reduce his average speed by ##20\,\text{km/h}## in town as well as outside. How long is his way to work, if this reduces his average speed by ##40\,\text{km/h}## on total?

**14.**Show that ##2x^6+3y^6=z^3## has no other rational solutions than ##x=y=z=0\,.##

**15.**Let ##x,yz \in \mathbb{R}-\{\,0\,\}## such that

$$

x+\dfrac{y}{z}=2\;\; , \;\;y+\dfrac{z}{x}=2\;\; , \;\;z+\dfrac{x}{y}=2

$$

Show that ##s:=x+y+z## can only have the values ##3## or ##7##.

You do not need to solve the equation system.

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