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**Summary:**Linear Programming, Trigonometry, Calculus, PDE, Differential Matrix Equation, Function Theory, Linear Algebra, Irrationality, Group Theory, Ring Theory.

**1.**(solved by @suremarc , 1 other solutions possible) Let ##A\in \mathbb{M}_{m,n}(\mathbb{R})## and ##b\in \mathbb{R}^m##. Then exactly one of the following two statements is true:

- ##Ax=b\, , \,x\geq 0 \,,## is solvable for a ##x\in \mathbb{R}^n.##
- ##A^\tau y\leq 0\, , \,b^\tau y> 0\,,## is solvable for some ##y\in \mathbb{R}^m.##

**2.**(solved by @Antarres ) Prove ##\pi =\displaystyle{\lim_{n \to \infty}2^n\underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}}_{n\text{ square roots }}}.##

**3.**(solved by @julian ) Let ##z(t)## be a non-negative continuous real function on the interval ##[a,b]## and ##t_0\in [a,b].##

Prove that if

$$

z(t) \leq C+L\left|\int_{t_0}^t z(s)\,ds\right|\quad (*)

$$

for all ##t\in [a,b]## with any constants ##C,L\geq 0,## then

$$

z(t)\leq Ce^{L|t-t_0|} \quad (**)

$$

for all ##t\in [a,b].##

**4.**(solved by @Fred Wright ) Solve the partial differential equation

\begin{align*}

u:D\longrightarrow \mathbb{R}\, &, \, D\subseteq \mathbb{R}^3\\[10pt]

xu_x+yu_y+(x^2+y^2)u_z&=0\\[10pt]

u(1,0,0)&=0\\u_x(1,0,1)&=0\\

u_y(-1,1,(\pi+2)/2)&=1\\u_z(-1,1,(\pi+2)/2)&=-1

\end{align*}

**5.**(solved by @suremarc ) Let ##A\in \mathbb{M}(n,\mathbb{R})## be a real square matrix. Show that there is a parameterized path ##x\, : \,\mathbb{R}\longrightarrow \mathbb{R}^n## as solution of the differential equation ##\dot x(t)=Ax(t)## which is unique for any given initial condition ##x(t_0)=x_0.##

**6.**(solved by @etotheipi , and @julian ) Calculate

- ##\displaystyle{\int_{-\infty}^\infty \dfrac{x^2}{x^4+2x^2+1}\,dx}##
- ##\displaystyle{\int_{0}^{\frac{\pi}{2}} \dfrac{1}{1+\sin^2 t}\,dt}##

**7.**(solved by @nuuskur and @StoneTemplePython ) Prove the following well known theorem by using topological and analytical tools only.

*For every real symmetric matrix*##A##

*there is a real orthogonal matrix*##Q##

*such that*##Q^\tau AQ##

*is diagonal.*

**Hint:**'Topological and analytical tools only' forbids the words 'characteristic' and 'eigen'. You can use Heine-Borel instead.

**8.**(solved by @julian ) We define ##e=\displaystyle{\sum_{k=0}^\infty \dfrac{1}{k!}}.## Prove that ##e^2## is irrational.

**9.**(solved by @fishturtle1 ) Let ##p<q## be two primes, ##b\in \mathbb{N},## and ##G## a group with ##p^2q^b## elements. Show that:

- If there is no normal ##q-##Sylow subgroup in ##G##, then ##(p,q)=(2,3),## and there is a non trivial homomorphism from ##G## to ##S_4.##
- ##G## is always solvable.

**10.**(solved by @suremarc ) Let ##f(x)=2x^5-6x+6\in \mathbb{Z}[x]##. In which of the following rings is ##f## irreducible and why?

**(a)**##\mathbb{Z}[x]##

**(b)**##(S^{-1}\mathbb{Z})[x]## with ##S=\{2^n\,|\,n\in \mathbb{N}_0\}##

**(c)**##\mathbb{Q}[x]##

**(d)**##\mathbb{R}[x]##

**(e)**##\mathbb{C}[x]##

An extra question from @julian as all others have been solved:

https://www.physicsforums.com/threads/math-challenge-january-2021.997970/page-4#post-6447092

**16.**Let ##r/s## be any positive or negative rational number, prove that ##e^{r/s}## is an irrational number.

Hint: make use of the polynomial:

\begin{align*}

P_n (x) = \frac{x^n (1-x)^n}{n!} .

\end{align*}

**High Schoolers only**

11.Given a set ##A## of ##32## pairwise distinct, positive integers less than ##112##. Decide right or wrong:

11.

- (solved by @Not anonymous ) There is a number that occurs at least five times among the differences between two numbers of ##A##.
- (solved by @Not anonymous ) There is a number that occurs at least six times among the differences between two numbers of ##A##.

**Hint:**A difference in this context is always positive, and only counted once between any two numbers of ##A##.

**12.**The harmonic numbers are

$$

H_n := \sum_{k=1}^n \dfrac{1}{k} = 1+ \dfrac{1}{2}+\dfrac{1}{3}+\ldots +\dfrac{1}{n}\; , \;(n\in \mathbb{N})

$$

We define

$$

T_n := \sum_{k=1}^n \dfrac{1}{k\cdot H_k^2} = \dfrac{1}{H_1^2}+ \dfrac{1}{2\cdot H_2^2}+\dfrac{1}{3\cdot H_3^2}+\ldots +\dfrac{1}{n\cdot H_n^2}\; , \;(n\in \mathbb{N})

$$

Show that ##T_n<2## for all ##n\in \mathbb{N}.##

**13.**We have a four sided pyramid with summit ##S## and a quadratic base ##A,B,C,D.## Let ##A',B',C',D'## be four points on the edges ##AS,BS,CS,DS,## resp. with positive distances ##a,b,c,d## from ##S,## resp. Show that ##A',B',C',D'## are coplanar if and only if

$$

\dfrac{1}{a}+\dfrac{1}{c}=\dfrac{1}{b}+\dfrac{1}{d}

$$

**14.**(solved by @Not anonymous ) Let ##f(n)=\left[\,2\sqrt{n}\,\right]-\left[\,\sqrt{n-1}+\sqrt{n+1}\;\right]## for ##n\in \mathbb{N}.## Determine all values of ##n## such that ##f(n)=1## and all ##n## such that ##f(n)=0.##

**Hint:**If ##r\in \mathbb{R}## with ##s\leq r < s+1## then ##\left[r\right]=\lfloor r \rfloor =s.##

**15.**Determine all pairs of non negative integers ##(m,n)## such that ##2^m-5^n=7.##

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