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**Summary:**Calculus, Measure Theory, Convergence, Infinite Series, Topology, Functional Analysis, Real Numbers, Algebras, Complex Analysis, Group Theory

**1.**(solved by @Office_Shredder ) Let ##f## be a real, differentiable function such that there is no ##x\in \mathbb{R}## with ##f(x)=0=f'(x)##.

Show that ##f## has at most finitely many zeros in the interval ##[0,1]##.

**2.**(solved by @wrobel ) Let ##(X,\Omega,\omega)## be a measure space and ##f## be a ##\omega-##integrable function.

Show that for every ##\varepsilon>0 ## there is a set ##W\in \Omega## such that ##\omega(W)< \infty ## and ##\int_{X-W}|f|\,d\omega <\varepsilon .##

**3.**(solved by @suremarc ) Prove or find a counterexample to:

**(a)**##L^2## convergence implies pointwise convergence.

**(b)**##\displaystyle{\lim_{n \to \infty}\int_{0}^{\infty}\dfrac{\sin x^n}{x^n}\,dx}=1##

**(c)**Let ##(f_n)## be a sequence of measurable functions which converge uniformly to zero on ##[0,\infty).## Then

$$

\lim_{n \to \infty}\int_{[0,\infty)} f_n(x)\,dx=\int_{[0,\infty)} \lim_{n \to \infty} f_n(x)\,dx\,.

$$

**4.**(solved by @julian ) Let ##(a_n)## be a sequence of positive real numbers such that the series ##\displaystyle{\sum_{n=1}^\infty}a_n=:C<\infty## converges. Show that ##\displaystyle{\sum_{n=1}^\infty\left(\prod_{k=1}^n a_k\right)^{1/n}}\leq e\cdot C.##

**5.**Let ##(E,\mathcal{T})## be a normal Hausdorff space, and ##U_1,\ldots,U_n## a finite open covering of ##E##. Then there are continuous functions ##g_1,\ldots,g_n\, : \,(E,\mathcal{T})\longrightarrow [0,1]## such that ##g_1+\ldots+g_n=1## on ##E## and ##g_j(E-U_j)=\{0\}## for all ##1\leq j\leq n.##

**6.**Let ##(X,\Omega,\omega)## be a measure space and ##1\leq p<\infty.## Show that

**(a)**##\tilde{L}^p:=L^p(X,\Omega,\omega)## is a Banach space with respect to ##\|\cdot\|_p\,.##

**(b)**The sequence ##(\|f_n\|_p)\subseteq \mathbb{R}## is bounded for every Cauchy sequence ##(f_n)\subseteq L^p(X,\Omega,\omega).##

**7.**(solved by @nuuskur ) We know that there are only two sets in ##(\mathbb{R},|\cdot|),## which are open and closed, the empty set and the entire topological space. Prove it.

**8.**Let ##(V,\alpha)## and ##(W,\beta)## be irreducible representations of an associative, complex algebra ##\mathcal{A}##. Assume that ##V## and ##W## are complex and of countable dimension. Then

$$

\dim \operatorname{Hom}_{\mathcal{A}}(V,W) = \begin{cases}1&\text{ if }(V,\alpha)\cong(W,\beta)\\0&\text{ otherwise }\end{cases}

$$

**9.**(solved by @StoneTemplePython , @nuuskur ) Let ##U\subseteq\mathbb{C}## be an open connected neighborhood and ##f:U\longrightarrow\mathbb{C}## a holomorphic function. If ##|f|## has a local maximum in ##z_0\in U,## i.e. there is an open neighborhood ##z_0\in U_0\subseteq U## with ##|f(z_0)|\geq |f(z)|## for all ##z\in U_0,## then ##f## is constant.

**10.**(solved by @fishturtle1 ) Show that all groups ##G## of order ##pqr## with pairwise distinct primes ##p<q<r## are solvable.

**(solved by @Not anonymous ) Let ##z## be a natural number with ##1995## decimal digits and ##1\leq n\leq 1994.## Then we note the number, which we get by cutting off the first ##n## digits and append them in the same order at the end of ##z## by ##z^{[n]}.## Show that if ##z## is divisible by ##27,## then all ##z^{[n]}## are divisible by ##27,## too.
**

High Schoolers only

11.

High Schoolers only

11.

**12.**(solved by @Not anonymous ) Let ##a,b,c,d## be positive real numbers. Prove (in the logically correct order)

$$

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

$$

**13.**(solved by @Not anonymous ) Let ##m\geq 2## be a given natural number. We define a sequence ##(x_0,x_1,x_2,\ldots)## of numbers by ##x_0=0,x_1=1,## and for ##n\geq 0## we set ##x_{n+2}## to be the remainder of ##x_{n+1}+x_n## by division by ##m,## chosen such that ##0\leq x_{n+2} < m.## Decide whether for every ##m\geq 2## there exists a natural number ##k\geq 1,## such that ##x_{k+2}=1\, , \,x_{k+1}=1\, , \,x_k=0.##

**14.**We define real functions

$$

f_n(x):=x^3+(n+3)\cdot x^2+2n\cdot x -\dfrac{n}{n+1}

$$

for every non-negative integer ##n\geq 0.## Determine all values of ##n,## such that all zeros of ##f_n(x)## are contained in an interval of length ##3.##

**15. (a)**(solved by @Not anonymous ) Determine the number of all pairs of integers ##(x,y) \in \mathbb{N}_0^2## with ##\sqrt{x}+\sqrt{y}=1993.##

**15. (b)**Determine for every ##n\in \mathbb{N}## the greatest power of ##2## which divides ##[( 4+\sqrt{18} )^n ].##

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