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**Summary::**Group Theory, Lie Algebras, Number Theory, Manifolds, Calculus, Topology, Differential Equations.

**1.**(solved by @Infrared ) Suppose that ##G## is a finite group such that for each subgroup ##H## of ##G## there exists a homomorphism ##\varphi \,:\, G \longrightarrow H## such that ##\varphi(h)=h## for all ##h \in H##. Show that ##G## is a product of groups of prime order.

**2.**(solved by @StoneTemplePython and by @julian ) Let ##G## be a finite group that operates on a set ##X##. Then the number of orbits is

$$

|X/G|=\dfrac{1}{|G|}\sum_{g\in G}|X^g|

$$

where ##X^g=\{\,x\in X\, : \,g.x=x\,\}## are the fixed points in ##X\,.##

**3.**(solved by @martinbn ) Prove that there is a Lie algebra isomorphism ##\mathfrak{g} \hookrightarrow \mathfrak{gl(g)}## if ##\mathfrak{g}## is a semisimple Lie algebra. Is this also a necessary condition?

**4.**(solved by @julian ) If ##n>1## is a square-free natural number, prove for all ##k>1##

$$

\sum_{d|n} \sigma(d^{k-1})\,\varphi(d)=n^k

$$

Remark: ##\varphi## is Euler's phi-function and ##\sigma(m)## the sum of divisors of ##m##.

**5.**(solved by @martinbn ) Show that

$$

M:=\{\,x\in \mathbb{R}^3\,|\,x_1+x_2+x_3=0\, , \,x_1^2+2x_2^2+x_3^2-2x_2(x_1+x_3)=9\,\} \subseteq \mathbb{R}^3

$$

is a manifold, and determine the tangent space ##T_pM## and the normal space ##N_pM## at ##p=(2,-1,-1)\in M\,.##

**6.**(solved by @Infrared ) Two persons ##P## and ##Q## play the following game:

##P## starts by selecting exactly one real value for ##a,b,## or ##c## in the equation

$$

x^3+ax^2+bx+c=0

$$

Then ##Q## does the same for one of the remaining coefficients before ##P## finally chooses the last value. ##P## wins if and only if the equation has three different real roots. Is there a winning strategy for one of the players?

**7.**What are the composition factors of ##\rm GL(2,\mathbb{F}_{19})\,?##

**8.**(solved by @Infrared ) Show that any group ##G## of order ##70## has always a normal subgroup of order ##5##.

**9.**(solved by @Infrared ) Let ##X,Y,Z## be topological spaces, ##X## covering compact (not necessarily Hausdorff), and ##Z## Hausdorff. Let ##g\, : \,X\longrightarrow Y## be continuous, and ##h\, : \,X \longrightarrow Z## surjective and continuous.

Show that the following statements are equivalent:

- ##g(x)=g(x')## for all ##x,x'\in X## with ##h(x)=h(x').##
- There is a continuous function ##f\, : \,Z\longrightarrow Y## with ##g=f\circ h\,.##
- There is a unique continuous function ##f\, : \,Z\longrightarrow Y## with ##g=f\circ h\,.##

**10.**(solved by @The Fez ) Given ## y'''=y''+y'-y\,.## Determine a fundamental system, and solve the initial value problem ##y(0)=1\, , \,y'(0)=0\, , \,y''(0)=3\,.##

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

11.(solved by @Not anonymous ) Assume we have put a Cartesian coordinate system on France and got the following positions:

11.

Paris ##(0,0)##, Lyon ##(3,-8)## and Marseille ##(4,-12)##. Look up the definitions and calculate the distance between Lyon and Marseille according to

- the Euclidean metric.
- the maximum metric.
- the French railway metric.
- the Manhattan metric.
- the discrete metric.

**12.**(solved by @kshitij ) Consider the ellipse in the first quarter of a Cartesian coordinate system

$$

\dfrac{(x-2)^2}{4}+\left(y-1\right)^2=1

$$

and rotate it such that the coordinate axes are always tangents to the ellipse.

Which locus describes the center of the ellipse during a full rotation?

**13.**(solved by @kshitij and @Not anonymous ) Maximize ##f(x,y,z)=4x^2y^2-(x^2+y^2-z^2)^2## under the conditions ##x+y+z=c## and ##x,y,z > 0.##

**14.**(solved by @kshitij , shorter solution possible) Let ##\triangle ABC## be a triangle and ##s## a straight which intersects all three sides (or their prolongations), say ##X \in BC\, , \,Y\in CA\, , \,Z\in AB## are the intersection points. Prove

$$

\overline{AZ}\cdot \overline{BX}\cdot \overline{CY}=\overline{AY}\cdot \overline{BZ}\cdot \overline{CX}

$$

**15.**(solved by @Shreya ) Otto von Guericke, who invented the air pump, led an experiment in Berlin in 1654. Two groups of eight horses tried in vain to pull apart two bronze hemispheres between which a vacuum was created. Assume that the radius ##R## of the hemispheres is so thin that we can neglect the difference between inner and outer radius.

Show that the force required to tear apart the hemispheres is ## F=\pi R^2 \cdot \Delta p ## where ##\Delta p## is the difference of air pressure within and outside the hemispheres. Next assume ##R=30\,\rm cm##, the inner pressure of ##0.1\,\rm bar## and the outer pressure of ##1.013\,\rm bar\,## Which force had each group of horses to apply in order to separate the hemispheres?

**P.s.:**These are easy questions this month. Please solve them in detail and with an explanation.

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