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Mathematics
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Math Proof Training and Practice
Math Challenge - December 2021
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[QUOTE="fresh_42, post: 6570285, member: 572553"] This month's challenges will be my last thread of this kind for a while. Call it a creative break. Therefore, we will have a different format this month. I will post one problem a day, like an advent calendar, only for the entire month. I will try to post the questions as close as possible to 0:00 GMT. [B]1.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/#post-6571475'][COLOR=rgb(184, 49, 47)]solved by[/COLOR][/URL] [USER=467682]@Infrared[/USER] ) Let ##G## be a group with ##3129## elements. Prove it is solvable. [B]2. [/B]$$ I(a):=\int_0^1 \left(\dfrac{\log x}{a+1-x}-\dfrac{\log x}{a+x}\right)\,dx \; ; \; a\in \mathbb{C}\backslash[-1,0] $$ [B]3.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/#post-6571475'][COLOR=rgb(184, 49, 47)]solved by[/COLOR][/URL] [USER=467682]@Infrared[/USER] ) Let ##\mathfrak{g}## be a Lie algebra over a field of characteristic not ##2.## Prove that $$ \mathfrak{A(g)}=\{\alpha \in \mathfrak{gl(g)}\,|\,[\alpha (X),Y]+[X,\alpha (Y)]=0\text{ for all }X,Y\in\mathfrak{g}\} $$ is a Lie algebra. Determine ##\mathfrak{A(B)}## for the two-dimensional non-abelian Lie algebra ##\mathfrak{B}##. [B]4.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/#post-6571475'][COLOR=rgb(184, 49, 47)]solved by[/COLOR][/URL] [USER=467682]@Infrared[/USER] ) Show that a path-connected set is connected but not vice versa and not necessarily simply connected. [B]5.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/#post-6571781'][COLOR=rgb(184, 49, 47)]solved by[/COLOR][/URL] [USER=142346]@julian[/USER] ) $$ \int_0^{\frac{\pi}{4}} \log(1+\tan x)\,dx $$ [B]6.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-3#post-6578681'][COLOR=rgb(184, 49, 47)]solved by[/COLOR][/URL] [USER=511972]@TeethWhitener[/USER] [URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-3#post-6579534'][COLOR=rgb(184, 49, 47)]and[/COLOR][/URL] [USER=493650]@PeroK[/USER] ) There are currently about ##7,808,000,000## people on earth. If we would enumerate them all, how many of them would have a prime number? [B]7.[/B] Let ##M=\mathbb{R}^2## and ##G=\mathbb{R}## and consider the map $$ \psi(\varepsilon ,(x,y)):=\left(\dfrac{x}{1-\varepsilon x},\dfrac{y}{1-\varepsilon x}\right) $$ defined on $$ U=\left\{(\varepsilon ,(x,y))\,|\,\varepsilon <\dfrac{1}{x}\text{ for }x>0\text{, or }\varepsilon >\dfrac{1}{x}\text{ for }x<0\right\}\subseteq \mathbb{R}\times \mathbb{R}^2 $$ Show that ##\psi## defines a local group action of ##G## on the manifold ##M.## Does it have a global counterpart on ##\mathbb{R}^2##? [B]8. [/B]Give an example of a ring and a maximal ideal that isn't a prime ideal. [B]9.[/B] Let ##U,V\subseteq \mathbb{C}## open sets, ##\varphi \, : \,U\longrightarrow V## a holomorphic function, and ##\gamma \, : \,[0,1]\longrightarrow U## a closed, smooth path. Show that if ##\gamma ## is ##0##-homologue in ##U,## then ##\varphi \circ \gamma ## is ##0##-homologue in ##V.## [B]10.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-2#post-6574855'][COLOR=rgb(184, 49, 47)]solved [/COLOR][/URL][URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-2#post-6574904'][COLOR=rgb(184, 49, 47)]by[/COLOR][/URL] [USER=142346]@julian[/USER] ) Examine convergence: $$ \prod_{n=2}^\infty \left(1-\dfrac{1}{n}\right)\; , \;\prod_{n=3}^\infty \left(1-\dfrac{4}{n^2}\right) $$ [B]11.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-3#post-6578681'][COLOR=rgb(184, 49, 47)]solved by[/COLOR][/URL] [USER=511972]@TeethWhitener[/USER] ) The Heisenberg algebra can be viewed as $$ \mathfrak{H}=\left\{\begin{bmatrix} 0&x_1&x_3\\0&0&x_2\\0&0&0 \end{bmatrix}\, : \,x_1,x_2,x_3\in \mathbb{R}\right\}. $$ Calculate ##\exp(H)## for a matrix ##H\in \mathfrak{H}##. [B]12.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/#post-6574673'][COLOR=rgb(184, 49, 47)]solved by[/COLOR][/URL] [USER=142346]@julian[/USER] ) $$ \int_{-\infty }^{\infty }\dfrac{|\sin(\alpha x)|}{1+x^2}\,dx\; , \; \alpha >0 $$ [B]13.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-2#post-6575271'][COLOR=rgb(184, 49, 47)]solved[/COLOR][/URL] [URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-2#post-6575520'][COLOR=rgb(184, 49, 47)]by[/COLOR][/URL] [USER=606256]@fishturtle1[/USER] ) Show that ##(n-1)!\equiv -1 \mod n## holds if and only if ##n## is prime. Determine the first two primes for which even ##(p-1)!\equiv -1 \mod p^2## holds. [B]14.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-3#post-6577356'][COLOR=rgb(184, 49, 47)]solved by[/COLOR][/URL] [USER=688197]@QuantumSpace[/USER] ) Determine all possible topologies on ##X:=\{a,b\}##, and which of them are homeomorphic. Give an example of a topological space with more than one element such that all sequences converge. [B]15.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-2#post-6577113'][COLOR=rgb(184, 49, 47)]solved by[/COLOR][/URL] [USER=606256]@fishturtle1[/USER] ) Explain the difference between ##\mathbb{Z}_2\times \mathbb{Z}_3## and ##\mathbb{Z}_2 \ltimes \mathbb{Z}_3\,.## Is there also a group ##\mathbb{Z}_2 \rtimes \mathbb{Z}_3\,?## [B]16.[/B] Show that ##16## and ##33## are Størmer numbers, but no number ##N:=2n^2>2## can be one, e.g. ##32.## [B]17.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-3#post-6578725'][COLOR=rgb(184, 49, 47)]solved by[/COLOR][/URL] [USER=606256]@fishturtle1[/USER] ) Consider a number ##n## which is not a prime and $$ p\,|\,n\Longrightarrow p\,|\,\left(\dfrac{n}{p}-1\right) $$ E.g. ##30=2\cdot 3\cdot 5## is such a number, since ##2\,|\,14\, , \,3\,|\,9\, , \,5\,|\,5.## Show that ##n## is square-free (all prime factors have exponent ##1##), and no semiprime (product of exactly two primes). [B]18.[/B] Prove that path integrals in ##\mathbb{R}^n## over gradient vector fields depend only on starting and endpoint, and not on the path itself. [B]19.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-3#post-6577520'][COLOR=rgb(184, 49, 47)]solved by[/COLOR][/URL] [USER=142346]@julian[/USER] ) Let ##P_0=0,P_1=1,P_n=2P_{n-1}+P_{n-2}## for all ##n\in \mathbb{N},n\geq2##. Determine a closed form for ##P_n.## [B]20.[/B] Find the irreducible minimal polynomial for $$ \mathbb{Q} \subseteq \mathbb{Q}\left(\sqrt[3]{\dfrac{9+\sqrt{69}}{18}}+\sqrt[3]{\dfrac{9-\sqrt{69}}{18}}\right). $$ [B]21.[/B] Show that the embedding ##\mathbb{S}^1\longrightarrow \mathbb{R}^2-\{0\}## is a homotopy equivalence, and that ##\mathbb{R}\longrightarrow \mathbb{R}^2-\{0\}## defined by ##x\mapsto (x,1)## is none. [B]22.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-3#post-6580897'][COLOR=rgb(184, 49, 47)]solved by[/COLOR][/URL] [USER=606256]@fishturtle1[/USER] ) Let ##\emptyset\neq X## be a set, ##\mathcal{P}(X)## its power set. Consider the following mappings \begin{align*} f\, : \,X &\longrightarrow \mathcal{P}(X)\\ x&\longmapsto \{x\}\\[6pt] g\, : \,\mathcal{P}(X)\times \mathcal{P}(X)&\longrightarrow \mathcal{P}(X)\\ (A,B)&\longmapsto A\cup B \end{align*} and decide whether they are injective, surjective, and calculate the fiber (pre-image) of the empty set. [B]23.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-3#post-6580476'][COLOR=rgb(184, 49, 47)]solved[/COLOR][/URL] [URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-3#post-6583713'][COLOR=rgb(184, 49, 47)]by[/COLOR][/URL] [USER=511972]@TeethWhitener[/USER] ) Find the smallest positive integer ##x## that solves \begin{align*} x &\equiv 2 \mod 3\\ x &\equiv 3 \mod 4\\ x &\equiv 2 \mod 5 \end{align*} [B]24.[/B] Let ##\vec{u},\vec{v},\vec{w}## be three different coplanar vectors of equal length, originating at a point ##O.## Their endpoints define a triangle ##\triangle UVW##. How can the barycenter ##S## be found? [B]25.[/B] Is a partially differentiable function ##f\, : \,\mathbb{R}^2\rightarrow \mathbb{R}## at some point ##x_0## also continuous at ##x_0##? [B]26. [/B]Let ##\mathfrak{g}## be the real Lie algebra generated by $$ A_1=\begin{bmatrix} 0&0&0\\0&1&0\\0&0&-1 \end{bmatrix}\, , \,A_2=\begin{bmatrix} 0&0&1\\-1&0&0\\0&0&0 \end{bmatrix}\, , \,A_3=\begin{bmatrix} 0&1&0\\0&0&0\\-1&0&0 \end{bmatrix} $$ Calculate its center ##\mathfrak{Z(g)}=\{X\,|\,[A_i,X]=0\,(i=1,2,3)\},## its commutator subalgebra ##[\mathfrak{g},\mathfrak{g}],## and a Cartan subalgebra ##\mathfrak{h}##. [B]27.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-3#post-6580346'][COLOR=rgb(184, 49, 47)]solved by[/COLOR][/URL] [USER=142346]@julian[/USER] ) Let ##f\, : \,[a,b]\longrightarrow \mathbb{R}## be a continuous function and ##g\, : \,[a,b]\longrightarrow \mathbb{R}## integrable with ##g(x)\geq 0## for all ##x\in [a,b]##. Then there is a ##\xi\in [a,b]## such that $$ \int_a^b f(x)g(x)\,dx = f(\xi)\int_a^b g(x)\,dx $$ [B]28.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-3#post-6601629'][COLOR=rgb(184, 49, 47)]solved by[/COLOR][/URL] [USER=664857]@Not anonymous[/USER] ) Consider the circle segment above ##A=(-1,0)## and ##B=(1,0)## of $$ x^2+\left(y+\dfrac{1}{\sqrt{3}}\right)^2=\dfrac{4}{3}\,. $$ The point ##P:=\left(\dfrac{1}{\sqrt{3}},1-\dfrac{1}{\sqrt{3}}\right)## lies on this segment. Calculate the height ##h## of the circle segment, and ##|AP|+|PB|.## [B]29.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-3#post-6580648'][COLOR=rgb(184, 49, 47)]solved by[/COLOR][/URL] [USER=691758]@ergospherical[/USER] [URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-3#post-6580717'][COLOR=rgb(184, 49, 47)]and[/COLOR][/URL] [USER=493650]@PeroK[/USER] ) Let ##\varphi :V\longrightarrow V## a linear mapping. Prove $$ \operatorname{ker} (\varphi) \cap \operatorname{im}(\varphi) =\{0\} \Longleftrightarrow \operatorname{ker}(\varphi \circ \varphi )=\operatorname{ker}(\varphi ) $$ [B]30.[/B] ([URL='https://www.physicsforums.com/threads/math-challenge-december-2021.1009717/page-4#post-6583810'][COLOR=rgb(184, 49, 47)]solved by[/COLOR][/URL] [USER=691758]@ergospherical[/USER] ) Let ##A## by a cylindric surface (without base or cover) that rotates around the ##z##-axis and stands on the plane ##\{z=0\}##, with radius ##R>0## and height ##h>0##. Give a parameterization and calculate the surface integral $$ \int_A \langle F,n \rangle\,d^2r $$ for the vector field ##F\, : \,\mathbb{R}^3\rightarrow \mathbb{R}^3## defined by ##F(x,y,z)=(xz,yz,123).## [B]31.[/B] Let ##\mathcal{P}## be a finite set of points in a plane, that are not all collinear. Then there is a straight, that contains exactly two points. . [/QUOTE]
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Math Challenge - December 2021
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