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Summary: Group Theory, Integrals, Representation Theory, Iterations, Geometry, Abstract Algebra, Linear Algebra.1. Integrate
$$
\int_{0}^\infty \int_{0}^\infty e^{-\left(x+y+\frac{\lambda^3 }{xy}\right)} x^{-\frac{2}{3}}y^{-\frac{1}{3}}\,dx\,dy
$$
2. (solved by @Infrared , basic solution still possible ) Let ##F_n## be the free group of rank ##n## with generators ##\{w_1,\ldots,w_n\}.## Then
$$
\prod_{i=1}^m w_{a_i}^{b_i} \in [F_n,F_n] \Longleftrightarrow \forall_{k=1}^m \;\sum_{a_i=k}b_i =0
$$
3. (solved by @julian ) Calculate
$$
\int_0^\pi \int_0^\pi \int_0^\pi \dfrac{1}{1-\cos x\,\cos y\,\cos z}\,dx\,dy\,dz
$$
4. (solved by @fishturtle1 ) Let ##G## be a finite group, ##\mathbb{K}## a field such that ##\operatorname{char}(\mathbb{K})\nmid |G|,## and ##(\rho,V)## and ##(\tau,W)## linear representations of ##G## over ##\mathbb{K}.## The ##\mathbb{K}-##linear mapping
\begin{align*}
\operatorname{Sym}\, : \,\operatorname{Hom}_\mathbb{K}(V,W)&\longrightarrow
\operatorname{Hom}_\mathbb{K}(V,W)\\
\varphi &\longmapsto \operatorname{Sym}(\varphi)=\dfrac{1}{|G|}\sum_{g\in G}\tau(g)\circ\varphi \circ\rho(g^{-1})
\end{align*}
is a projection onto the subspace
$$
\operatorname{Hom}_\mathbb{K}((\rho,V),(\tau,W))=\{\vartheta :V\longrightarrow W\,|\,\forall_{g\in G}\, : \,\tau(g)\circ\vartheta\circ \rho(g^{-1})=\vartheta \}
$$
of ## \operatorname{Hom}_\mathbb{K}(V,W).## Determine and prove (mention) all five claims.
5. Let ##f(x)=x^3-\dfrac{49}{6}x^2+\dfrac{39}{2}x-\dfrac{31}{3}##. Prove that there are at least one ##a,b## such that ##f^2(a)=a\, , \,f(a)\neq a ## and ##f^4(b)=b\, , \,f^k(b)\neq b\;(k<4)## where ##f^n:=f\circ f^{n-1}, f^1=f.##
Is this true for every even power?
6. (solved by @etotheipi ) Prove the equivalence of the theorem of Pythagoras with the following transversal theorem about isosceles triangles:
Given an isosceles triangle ##\triangle ABC## with baseline ##\overline{AB}\subseteq g,## peak ##C,## i.e. ##|AC|=|BC|,## and ##g## the straight along the baseline. Moreover let ##P\in g## be an arbitrary point. Then
\begin{align*}
|CP|^{2}&=|CA|^{2}+|PA|\cdot |PB|\text{ if }P\not \in \overline{AB}\\
|CP|^{2}&=|CA|^{2}-|PA|\cdot |PB|\text{ if }P \in \overline{AB}
\end{align*}
7. (solved by @julian , another solution via MVT still possible ) Let ##\alpha ## be an algebraic number of degree ##n\geq 1.## Then there is a real number ##c>0## such that for all ##\mathbb{Q}\ni\dfrac{p}{q}\neq \alpha ##
$$
\left|\alpha -\dfrac{p}{q}\right|\geq \dfrac{c}{q^n}
$$
8. (solved by @etotheipi ) Let ##a_{n+1}=2+\sqrt{4+a_n}\, , \,a_0\geq -4\,,## be a sequence of real numbers. Determine - if existent - its limit in dependence of the initial value ##a_0,## and show that ##a_n\in [2,5]## in cases where ##a_0\in[-4,5],## and ##a_n\geq 5## in cases where ##a_0\geq 5## ##(n\in \mathbb{N}).##
9. (solved by @etotheipi ) Calculate center, foci, semi-axis, and area of the maximal inscribed ellipse of the triangle ##(1,1),(5,2),(3,6).##
10. (solved by @julian ) Let ##A,B\in \mathbb{M}(n,\mathbb{F})## be two square ##n\times n ## matrices over a field ##\mathbb{F}.## Show that the minimal polynomials of ##AB## and ##BA## are the same in case ##A## is regular. Is it true as well, if ##A## is singular?
High Schoolers only (until 26th)11. (solved by @kshitij ) For which positive real numbers ##\mathbb{R}\ni a,b>0 ## does
$$
f(a,b)=\dfrac{a^4}{b^4}+\dfrac{b^4}{a^4}-\dfrac{a^2}{b^2}-\dfrac{b^2}{a^2}+\dfrac{a}{b}+\dfrac{b}{a}
$$
assume a minimal value, and which one?
12. (solved by @kshitij ) Find all pairs ##(x,y)## of integers such that
$$
y^2=x\cdot (x+1)\cdot (x+7)\cdot (x+8)
$$
13. (solved by @kshitij ) Show that
$$
\underbrace{\left|x-\dfrac{\sin(x)(14+\cos(x))}{9+6\cos(x)}\right|}_{=:f(x)}\leq 10^{-4}\text{ for } x\in \left[0,\dfrac{\pi}{4}\right]
$$
You may use ##\pi =3.14159+\delta \, , \,\sqrt{2}=1.41421+\varepsilon ## with ##\delta,\varepsilon \in \left(0,10^{-5}\right).##
14. (solved by @kshitij ) If ##f(x)=a_nx^n+\ldots+a_1x+a_0\in \mathbb{R}[x]## is a real polynomial of degree ##n## which doesn't have real zeros, and ##h\in\mathbb{R}## a real number, then
$$
F(x):=f(x)+h\cdot f'(x)+h^2\cdot f''(x)+\ldots+h^n\cdot f^{(n)}(x)
$$
doesn't have real zeros either.
15. (solved by @kshitij ) Solve the following real equations system:
\begin{align*}
x+y&=az\\
x-y&=bz\\
x^2+y^2&=cz
\end{align*}
$$
\int_{0}^\infty \int_{0}^\infty e^{-\left(x+y+\frac{\lambda^3 }{xy}\right)} x^{-\frac{2}{3}}y^{-\frac{1}{3}}\,dx\,dy
$$
2. (solved by @Infrared , basic solution still possible ) Let ##F_n## be the free group of rank ##n## with generators ##\{w_1,\ldots,w_n\}.## Then
$$
\prod_{i=1}^m w_{a_i}^{b_i} \in [F_n,F_n] \Longleftrightarrow \forall_{k=1}^m \;\sum_{a_i=k}b_i =0
$$
3. (solved by @julian ) Calculate
$$
\int_0^\pi \int_0^\pi \int_0^\pi \dfrac{1}{1-\cos x\,\cos y\,\cos z}\,dx\,dy\,dz
$$
4. (solved by @fishturtle1 ) Let ##G## be a finite group, ##\mathbb{K}## a field such that ##\operatorname{char}(\mathbb{K})\nmid |G|,## and ##(\rho,V)## and ##(\tau,W)## linear representations of ##G## over ##\mathbb{K}.## The ##\mathbb{K}-##linear mapping
\begin{align*}
\operatorname{Sym}\, : \,\operatorname{Hom}_\mathbb{K}(V,W)&\longrightarrow
\operatorname{Hom}_\mathbb{K}(V,W)\\
\varphi &\longmapsto \operatorname{Sym}(\varphi)=\dfrac{1}{|G|}\sum_{g\in G}\tau(g)\circ\varphi \circ\rho(g^{-1})
\end{align*}
is a projection onto the subspace
$$
\operatorname{Hom}_\mathbb{K}((\rho,V),(\tau,W))=\{\vartheta :V\longrightarrow W\,|\,\forall_{g\in G}\, : \,\tau(g)\circ\vartheta\circ \rho(g^{-1})=\vartheta \}
$$
of ## \operatorname{Hom}_\mathbb{K}(V,W).## Determine and prove (mention) all five claims.
5. Let ##f(x)=x^3-\dfrac{49}{6}x^2+\dfrac{39}{2}x-\dfrac{31}{3}##. Prove that there are at least one ##a,b## such that ##f^2(a)=a\, , \,f(a)\neq a ## and ##f^4(b)=b\, , \,f^k(b)\neq b\;(k<4)## where ##f^n:=f\circ f^{n-1}, f^1=f.##
Is this true for every even power?
6. (solved by @etotheipi ) Prove the equivalence of the theorem of Pythagoras with the following transversal theorem about isosceles triangles:
Given an isosceles triangle ##\triangle ABC## with baseline ##\overline{AB}\subseteq g,## peak ##C,## i.e. ##|AC|=|BC|,## and ##g## the straight along the baseline. Moreover let ##P\in g## be an arbitrary point. Then
\begin{align*}
|CP|^{2}&=|CA|^{2}+|PA|\cdot |PB|\text{ if }P\not \in \overline{AB}\\
|CP|^{2}&=|CA|^{2}-|PA|\cdot |PB|\text{ if }P \in \overline{AB}
\end{align*}
7. (solved by @julian , another solution via MVT still possible ) Let ##\alpha ## be an algebraic number of degree ##n\geq 1.## Then there is a real number ##c>0## such that for all ##\mathbb{Q}\ni\dfrac{p}{q}\neq \alpha ##
$$
\left|\alpha -\dfrac{p}{q}\right|\geq \dfrac{c}{q^n}
$$
8. (solved by @etotheipi ) Let ##a_{n+1}=2+\sqrt{4+a_n}\, , \,a_0\geq -4\,,## be a sequence of real numbers. Determine - if existent - its limit in dependence of the initial value ##a_0,## and show that ##a_n\in [2,5]## in cases where ##a_0\in[-4,5],## and ##a_n\geq 5## in cases where ##a_0\geq 5## ##(n\in \mathbb{N}).##
9. (solved by @etotheipi ) Calculate center, foci, semi-axis, and area of the maximal inscribed ellipse of the triangle ##(1,1),(5,2),(3,6).##
10. (solved by @julian ) Let ##A,B\in \mathbb{M}(n,\mathbb{F})## be two square ##n\times n ## matrices over a field ##\mathbb{F}.## Show that the minimal polynomials of ##AB## and ##BA## are the same in case ##A## is regular. Is it true as well, if ##A## is singular?
High Schoolers only (until 26th)11. (solved by @kshitij ) For which positive real numbers ##\mathbb{R}\ni a,b>0 ## does
$$
f(a,b)=\dfrac{a^4}{b^4}+\dfrac{b^4}{a^4}-\dfrac{a^2}{b^2}-\dfrac{b^2}{a^2}+\dfrac{a}{b}+\dfrac{b}{a}
$$
assume a minimal value, and which one?
12. (solved by @kshitij ) Find all pairs ##(x,y)## of integers such that
$$
y^2=x\cdot (x+1)\cdot (x+7)\cdot (x+8)
$$
13. (solved by @kshitij ) Show that
$$
\underbrace{\left|x-\dfrac{\sin(x)(14+\cos(x))}{9+6\cos(x)}\right|}_{=:f(x)}\leq 10^{-4}\text{ for } x\in \left[0,\dfrac{\pi}{4}\right]
$$
You may use ##\pi =3.14159+\delta \, , \,\sqrt{2}=1.41421+\varepsilon ## with ##\delta,\varepsilon \in \left(0,10^{-5}\right).##
14. (solved by @kshitij ) If ##f(x)=a_nx^n+\ldots+a_1x+a_0\in \mathbb{R}[x]## is a real polynomial of degree ##n## which doesn't have real zeros, and ##h\in\mathbb{R}## a real number, then
$$
F(x):=f(x)+h\cdot f'(x)+h^2\cdot f''(x)+\ldots+h^n\cdot f^{(n)}(x)
$$
doesn't have real zeros either.
15. (solved by @kshitij ) Solve the following real equations system:
\begin{align*}
x+y&=az\\
x-y&=bz\\
x^2+y^2&=cz
\end{align*}
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