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**Summary:**Gamma function. Combinatorics. Stochastic. Semisimple Modules. Topological Groups. Metric spaces. Logarithmic inequality. Stochastic. Primes. Approximation theory.

**1.**(solved by @julian and @benorin ) Let ##f## be a function defined on ##(0,\infty)## such that ##f(x)>0## for all ##x>0.## Suppose that ##f## has the following properties:

- ##\log f(x)## is a convex function.
- ##f(x+1)=x\cdot f(x)## for all ##x>0.##
- ##f(1)=1.##

**2.**(solved by @TeethWhitener ) Let ##T = (x_1, x_2, \ldots , x_m)## be a sequence of not necessarily distinct reals. For any positive ##b##, define

$$

T_b := \{(x_i, x_j ) \,|\, 1 \leq i, j \leq m, |x_i-x_j | \leq b\}.

$$

Show that for any sequence ##T## and for every integer ##r>1,##

$$|T_r| < (2r-1)|T_1|.$$

**3.**(solved by @Office_Shredder ) Let ##X,Y## be two independent identically distributed real random variables. For a positive ##b,## define ##p_b:=\operatorname{prob}\left(|X-Y|\leq b\right).## Then for every integer ##r,## ##p_r\leq (2r-1)p_1.## Thus

$$

\operatorname{prob}\left(|X-Y|\leq 2\right) \leq 3\cdot \operatorname{prob}\left(|X-Y|\leq 1\right)

$$

**4.**Let ##\mathbb{F}## be a field and ##G## a finite group, such that ##\operatorname{char}\mathbb{F}\,\nmid\,|G|.## Prove that ##\mathbb{F}G## is semisimple, and show that this is not true if ##\operatorname{char}\mathbb{F}\,\mid\,|G|.##

**5.**A group ##G## together with a topology, such that the mapping on ##G\times G## (equipped with the product topology) to ##G## given by ##(x,y)\longmapsto xy^{-1}## is continuous, is called a topological group (e.g. a Lie group). Prove

- (solved by @jbergman ) ##G## is a topological group if and only if inversion and multiplication are continuous.
- (solved by @jbergman ) The mappings ##x\longmapsto xg## and ##x\longmapsto gx## are homeomorphisms for each ##g\in G.##
- (solved by @jbergman ) Each open subgroup ##U\leq G## is closed, and each closed subgroup ##U\leq G## of finite index is open. If ##G## is compact, then each open subgroup is of finite index.
- (solved by @jbergman ) Let ##H \leq G## be a subgroup equipped with the subspace topology, ##K\trianglelefteq G## a normal subgroup, and ##G/K## equipped with the quotient space topology. Then ##H## and ##G/K## are again topological groups and the projection ##\pi\, : \,G \twoheadrightarrow G/K## is open.
- ##G## is Hausdorff if and only if ##\{1\}## is a closed set in ##G.## ##G/K## is Hausdorff for a normal subgroup ##K\trianglelefteq G,## if and only if ##K## is closed in ##G.## If ##G## is totally disconnected, then ##G## is Hausdorff.
- Let ##G## be a compact topological group and ##\{X_j \;(j\in I)\}\subseteq G## a family of closed subsets such that for all ## i ,j\in I## there is a ##k\in I## with ##X_k\subseteq X_i\cap X_j.## Then we have for any closed subset ##Y\subseteq G##

$$Y\cdot \left(\bigcap_{i\in I}X_i\right) = \bigcap_{i\in I} YX_i $$

**6.**Let ##(X_n,d_n)_{n\in \mathbb{N}_0}## be a sequence of complete metric spaces, and let ##(f_n)_{n\in \mathbb{N}_0}## be a sequence of continuous functions ##f_n:X_{n+1}\longrightarrow X_n## such that the image ##f_n(X_{n+1}) \subseteq (X_n,d_n)## is dense for all ##n\in \mathbb{N}_0##. Then

$$

M_0:= \left\{v_0\in X_0\,|\,\exists\,(v_n)_{n\in \mathbb{N}} \,\forall\,n\in \mathbb{N}\, : \,v_n\in X_n\,\wedge \,f_{n-1}(v_n)=v_{n-1} \right\}

$$

and

$$

M_0\subseteq M:= \bigcap_{n=0}^\infty(f_0\circ f_1\circ\ldots\circ f_n)(X_{n+1})

$$

are dense in ##(X_0,d_0).## In particular ##M\neq \emptyset## in case ##X_0\neq \emptyset.##

**7.**(solved by @julian ) Prove for all ##x>-1##

$$

x-(1+x)\log (1+x)\leq -\dfrac{3x^2}{2(x+3)}

$$

**8.**(solved by @julian ) Let ##(\Omega, \mathcal{A},\mathcal{P})## be a probability space, ##B,T,\sigma ## positive real numbers, and ##n\in \mathbb{N}.## For independently distributed random variables ##X_1,\ldots,X_n\, : \,\Omega \longrightarrow \mathbb{R}## with expectation values ##E(X_k)=0## and ##E(X_k^2)\leq \sigma^2##, and boundary ##\|X_k\|_\infty \leq B## for all ##k=1,\ldots,n ## prove

$$

P\left(\dfrac{1}{n}\sum_{k=1}^nX_k \geq \sqrt{\dfrac{2\sigma^2 T}{n}}+\dfrac{2BT}{3n}\right) \leq e^{-T}.

$$

**9.**Let ##\mathbb{P}## be the set of all primes, ##p\in \mathbb{P},## and ##n\in \mathbb{N}## a positive integer. ##\operatorname{ord}_p(N)## denotes the number of primes ##p## which occur as divisor in ##\{1,2,\ldots,N\}## counted by multiplicity. E.g. ##N=24=4!## and ##p=3## yields in ##\{3=3^1,6=3^1\cdot 2,9=3^2,12=3^1\cdot 4,15=3^1\cdot 5,18=3^2\cdot 2,21=3^1\cdot 7,24=3^1\cdot 8\}##

$$

\operatorname{ord}_3(24) = 1+1+2+1+1+2+1+1=10

$$

Prove

- (solved by @TeethWhitener ) ##\displaystyle{\operatorname{ord}_p(n)=\sum_{k\geq 1}\left\lfloor \dfrac{n}{p^k}\right\rfloor}##
- (solved by @TeethWhitener )##2\left.\,\right|\binom{2n}{n}## and ##p\left.\,\right|\binom{2n}{n}## for all ##n<p\leq 2n##
- (solved by @TeethWhitener ) ##p\geq 3\,\wedge \,2n/3 <p\leq n \,\Longrightarrow \,p\,\nmid\,\binom{2n}{n}##
- (solved by @julian ) ##p^r\left.\,\right|\binom{2n}{n}\,\Longrightarrow \,p^r\leq 2n##
- (solved by @julian ) ##\dfrac{2^{2n-1}}{n}\leq \binom{2n}{n}\leq 2^{2n-1}##
- (solved by @julian ) ##\displaystyle{\prod_{p\leq n}p<4^n}##

**10.**Let ##K## be compact and ##C(K):=\{f:K\to \mathbb{R} \text{ or }\mathbb{C}\,|\,f \text{ is continuous}.## A ##n-##dimensional subspace ##M\subseteq C(K)## is called Haar space, if all ##f\in M-\{0\}## have at most ##n-1## zeros. Linear independent functions ##S:=\{\varphi_1,\ldots,\varphi_n \}\subseteq C(K)## are called a Chebyshev- or Haar-system, if ##\operatorname{span}(S)## is a Haar space. We denote the (compact) unit circle ##\mathbb{T} := \{e^{2\pi it} \,|\, t\in [0,1)\}.## Let ##K\subseteq \mathbb{R}## be compact or ##K=\mathbb{T}\, , \,f\in C(K).##

We call a point ##\xi \in K## with ##f(\xi)=0## a simple zero of ##f## if ##\xi## is either on the boundary of ##K## or ##f## changes sign in ##\xi.## If ##f(\xi -t)f(\xi +t)>0## in a neighborhood of ##\xi,## then we speak of a double zero.

- A subspace ##M\subseteq C(K)## with ##\dim M=n## is a Haar space if and only if each ##f\in M-\{0\}## that has ##j\in \mathbb{N}_0## simple zeros and ##k\in \mathbb{N}_0## double zeros holdsSo each element $f\in M-\{0\}$ has at most $n-1$ different zeros.

$$j+2k<n. $$ So each element ##f\in M-\{0\}## has at most ##n-1## different zeros. - The space of all real-valued trigonometric polynomials on ##[0,1)## of degree at most ##n## is a Haar space of dimension ##2n+1.##
- Let ##n\in \mathbb{N}_0,\;p\in T_n,## and ##x\in \mathbb{T}.## Then

$$

\left|p'(x)\right|\leq 2\pi n\sqrt{\|p\|^2_\infty -|p(x)|^2}.

$$

**Remark:**Consider the linear differential operator ##D(p)=p'## on ##T_n.## From ##|D(\sin(2\pi nx))|=|2\pi n\cos(2\pi nx)|## we conclude that

$$

2\pi n \leq \|D\|=\sup_{\|p\|_\infty \leq 1}\|p'\|_\infty=\sup_{\|p\|_\infty = 1}\sup_{x\in \mathbb{T}}|p'(x)| <\infty

$$

because ##T_n## is finite-dimensional.

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

11.(solved by @Not anonymous ) Let ##S## be a set of real numbers that is closed under multiplication (that is, if ##a## and ##b## are in ##S##, then so is ##ab##). Let ##T## and ##U## be disjoint subsets of ##S## whose union is ##S##. Given that the product of any three (not necessarily distinct) elements of ##T## is in ##T## and that the product of any three elements of ##U## is in ##U##, show that at least one of the two subsets ##T,U## is closed under multiplication.

11.

**12.**(solved by @Not anonymous ) Suppose we have a necklace of ##n## beads. Each bead is labeled with an integer and the sum of all these labels is ##n-1.## Prove that we can cut the necklace to form a string whose consecutive labels ##x_1, x_2,\ldots, x_n## satisfy

$$

\sum_{i=1}^k x_i \leq k-1 \quad (k=1,\ldots,n).

$$

**13.**(solved by @Not anonymous ) Let ##d:=d_1d_2\ldots d_9## be a number with not necessarily distinct nine decimal digits. A number ##e:=e_1e_2\ldots e_9## is such that each of the nine digit numbers formed by replacing just one of the digits ##d_j## by the corresponding digit ##e_j## is divisible by ##7## for all ##1\leq j\leq 9.## A number ##f:=f_1f_2\ldots f_9## is formed the same way by starting with ##e,## i.e. each of the nine numbers formed by replacing a ##e_k## by ##f_k## is divisible by ##7.## Example: If ##d=20210901## then ##e_6\in \{0,7\}## since ##7\,|\,20210001## and ##7\,|\,20210701 \,.##

Show that, for each ##j,## ##d_j-f_j## is divisible by ##7.##

**14.**An ellipse, whose semi-axes have lengths ##a## and ##b##, rolls without slipping on the curve ##y=c\sin(x/a).## How are ##a,b,c## related, given that the ellipse completes one revolution when it traverses one period of the curve?

**15.**(solved by @Not anonymous ) For a partition ##\pi## of ##N:=\{1,2,\ldots\, , \,9\}##, let ##\pi(x)## be the number of elements in the part containing ##x.## Prove that for any two partitions ##\pi_1## and ##\pi_2,## there are two distinct numbers ##x## and ##y## in ##N## such that ##\pi_j(x)=\pi_j(y)## for ##j=1,2.##

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