 #1
fresh_42
Mentor
 12,338
 8,727
Summary:

Useful Equations And Inequalities
Calculus
Functional Analysis
Small Groups
Main Question or Discussion Point
Questions
1. (solved by @PeroK ) Let ##f\, : \,\mathbb{R}\longrightarrow \mathbb{R}## be a smooth, ##2\pi##periodic function with square integrable derivative, and ##\displaystyle{\int_0^{2\pi}}f(x)\,dx = 0\,.## Prove
$$
\int_0^{2\pi} \left[f(x)\right]^2\,dx \leq \int_0^{2\pi} \left[f\,'(x)\right]^2\,dx
$$
For which functions does equality hold?
2. (solved by @Antarres ) ##M## be the set of all nonnegative, convex functions ##f\, : \,[0,1]\longrightarrow \mathbb{R}## with ##f(0)=0\,.## Prove
$$
\int_0^1 \prod_{k=1}^n f_k(x)\,dx \geq \dfrac{2^n}{n+1}\prod_{k=1}^n \int_0^1 f_k(x)\,dx\quad \forall \;f_1,\ldots,f_n \in M
$$
Hint: Define and use ##\hat{f}(x)=2x\displaystyle{\int_0^1 f(x)\,dx}\,.##
3. (solved by @Antarres ) Consider the following differential operators on the space of smooth functions ##C^\infty(\mathbb{R})##
$$
A=2x\cdot \dfrac{d}{dx}\, , \,B=x^2\cdot \dfrac{d}{dx}\, , \,C=\dfrac{d}{dx}
$$
Determine the eigenvectors and a (multiplicative) structure on .
4. (solved by @Antarres ) Prove
$$
\sum_{k=0}^n \dfrac{1}{\binom{n}{k}}=\dfrac{n+1}{2^{n+1}}\sum_{k=1}^{n+1}\dfrac{2^k}{k}
$$
5. (solved by @Antarres ) Calculate
$$\displaystyle{\sum_{k=1}^\infty} \dfrac{1}{\binom{2k}{2}}$$
6. (solved by @Antarres ) Prove for ##b>0##
$$
\int_{\infty}^{\infty} f\left(x\dfrac{b}{x}\right)\,dx=\int_{\infty}^{\infty}f(x)\,dx
$$
7. (solved by @Antarres ) Let be a periodic function. Prove that
$$
\int_{\infty}^{\infty} f(x)\,\dfrac{\sin x}{x}\,dx = \int_{0}^{\pi} f(x)\,dx \text{ and }\int_{\infty}^{\infty} f(x)\,\dfrac{\tan x}{x}\,dx =\int_{0}^{\pi} f(x)\,dx
$$
assuming the integrals exist, i.e. are finite.
8. (a) (solved by @fishturtle1 ) If ##\varphi\, : \,G\longrightarrow H## is a homomorphism of finite groups, then ##\operatorname{ord}(\varphi(g))\,\,\operatorname{ord}(g)## for all elements ##g\in G\,.##
8. (b) (solved by @fishturtle1 ) Determine all group homomorphisms ##\varphi\, : \,\mathbb{Z}_4 \longrightarrow \operatorname{Sym}(3)## and ##\psi\, : \,\operatorname{Sym}(3)\longrightarrow \mathbb{Z}_4\,.##
9. Let ##\mathbf{a}=(a_1,\ldots,a_n) \in \mathbb{R}_{\geq 0}## be nonnegative real numbers. The elementary symmetric polynomials are
$$
\sigma_k(\mathbf{a}) =\sum_{1\leq j_1<\ldots <j_k\leq n}a_{j_1}a_{j_2}\ldots a_{j_k}
$$
and
$$
S_k(\mathbf{a}) =\dfrac{1}{\binom{n}{k}}\cdot \sigma_k(\mathbf{a})
$$
the corresponding elementary symmetric mean value. Prove
(a) ##S_1(\mathbf{a})\geq \sqrt{S_2(\mathbf{a})}\geq \sqrt[3]{S_3(\mathbf{a})}\geq \ldots \geq \sqrt[n]{S_n(\mathbf{a})}##
(b) ##S_m(\mathbf{a})^2\geq S_{m+1}(\mathbf{a})\cdot S_{m1}(\mathbf{a})## for ##m=1,\ldots,n1##
10. Let ##(a_n)_{n\in \mathbb{N}}## be a sequence of nonnegative real numbers, not all zero. Prove
$$
\left(\sum_{n\in \mathbb{N}}a_n\right)^4 < \pi^2 \sum_{n\in \mathbb{N}}a_n^2 \cdot \sum_{n\in \mathbb{N}}n^2a_n^2
$$
High Schoolers only
11. (solved by @etotheipi ) On how many ways can ##2020## be written as a sum of consecutive natural numbers (greater than zero)?
12. (solved by @Not anonymous ) A binary operation on a set is a mapping, which maps a pair from to . E.g. addition is a binary operation on integers. Find two different (i.e. not achievable be renaming the elements) binary operations for which have a neutral element, , and can be inverted: for all there is a with , and are associative: A binary operation on a set ##S## is a mapping, which maps a pair from ##S\times S## to ##S##. E.g. addition is a binary operation on integers. Find two different binary operations for ##S=\{\,A,B,C,D\,\}## which have a neutral element, ##A\circ X = X##, and can be inverted: for all ##X\in S## there is a ##Y\in S## with ##X\circ Y=A##, and are associative: ##X\circ (Y\circ Z)=(X\circ Y)\circ Z\,.##
13. (solved by @Not anonymous ) Find all six digit numbers with the following property: If we move the first (highest) digit at the end, we will get three times the original number.
14. (solved by @etotheipi ) The Pell sequence named after the English mathematician John Pell is defined by
$$
P(n)= \begin{cases} 0\, , \,n=0\\1 \, , \,n=1\\ P(n2)+2P(n1)\, , \,n>1 \end{cases}
$$
Calculate the limit ##\delta_s:=\lim_{n\to \infty}\dfrac{P(n)}{P(n1)}\,.##
15. (solved by @Not anonymous ) Consider the graph of ##f(x)=1/x## with ##x\geq 1## and let it rotate around the ##x##axis. This solid of revolution looks like an infinitely long trumpet. Calculate its volume ##V## and its surface ##A##.
If we fill it with paint, pour it out again, then we have painted it from inside. Explain this apparent contradiction to the surface you computed.
1. (solved by @PeroK ) Let ##f\, : \,\mathbb{R}\longrightarrow \mathbb{R}## be a smooth, ##2\pi##periodic function with square integrable derivative, and ##\displaystyle{\int_0^{2\pi}}f(x)\,dx = 0\,.## Prove
$$
\int_0^{2\pi} \left[f(x)\right]^2\,dx \leq \int_0^{2\pi} \left[f\,'(x)\right]^2\,dx
$$
For which functions does equality hold?
2. (solved by @Antarres ) ##M## be the set of all nonnegative, convex functions ##f\, : \,[0,1]\longrightarrow \mathbb{R}## with ##f(0)=0\,.## Prove
$$
\int_0^1 \prod_{k=1}^n f_k(x)\,dx \geq \dfrac{2^n}{n+1}\prod_{k=1}^n \int_0^1 f_k(x)\,dx\quad \forall \;f_1,\ldots,f_n \in M
$$
Hint: Define and use ##\hat{f}(x)=2x\displaystyle{\int_0^1 f(x)\,dx}\,.##
3. (solved by @Antarres ) Consider the following differential operators on the space of smooth functions ##C^\infty(\mathbb{R})##
$$
A=2x\cdot \dfrac{d}{dx}\, , \,B=x^2\cdot \dfrac{d}{dx}\, , \,C=\dfrac{d}{dx}
$$
Determine the eigenvectors and a (multiplicative) structure on .
4. (solved by @Antarres ) Prove
$$
\sum_{k=0}^n \dfrac{1}{\binom{n}{k}}=\dfrac{n+1}{2^{n+1}}\sum_{k=1}^{n+1}\dfrac{2^k}{k}
$$
5. (solved by @Antarres ) Calculate
$$\displaystyle{\sum_{k=1}^\infty} \dfrac{1}{\binom{2k}{2}}$$
6. (solved by @Antarres ) Prove for ##b>0##
$$
\int_{\infty}^{\infty} f\left(x\dfrac{b}{x}\right)\,dx=\int_{\infty}^{\infty}f(x)\,dx
$$
7. (solved by @Antarres ) Let be a periodic function. Prove that
$$
\int_{\infty}^{\infty} f(x)\,\dfrac{\sin x}{x}\,dx = \int_{0}^{\pi} f(x)\,dx \text{ and }\int_{\infty}^{\infty} f(x)\,\dfrac{\tan x}{x}\,dx =\int_{0}^{\pi} f(x)\,dx
$$
assuming the integrals exist, i.e. are finite.
8. (a) (solved by @fishturtle1 ) If ##\varphi\, : \,G\longrightarrow H## is a homomorphism of finite groups, then ##\operatorname{ord}(\varphi(g))\,\,\operatorname{ord}(g)## for all elements ##g\in G\,.##
8. (b) (solved by @fishturtle1 ) Determine all group homomorphisms ##\varphi\, : \,\mathbb{Z}_4 \longrightarrow \operatorname{Sym}(3)## and ##\psi\, : \,\operatorname{Sym}(3)\longrightarrow \mathbb{Z}_4\,.##
9. Let ##\mathbf{a}=(a_1,\ldots,a_n) \in \mathbb{R}_{\geq 0}## be nonnegative real numbers. The elementary symmetric polynomials are
$$
\sigma_k(\mathbf{a}) =\sum_{1\leq j_1<\ldots <j_k\leq n}a_{j_1}a_{j_2}\ldots a_{j_k}
$$
and
$$
S_k(\mathbf{a}) =\dfrac{1}{\binom{n}{k}}\cdot \sigma_k(\mathbf{a})
$$
the corresponding elementary symmetric mean value. Prove
(a) ##S_1(\mathbf{a})\geq \sqrt{S_2(\mathbf{a})}\geq \sqrt[3]{S_3(\mathbf{a})}\geq \ldots \geq \sqrt[n]{S_n(\mathbf{a})}##
(b) ##S_m(\mathbf{a})^2\geq S_{m+1}(\mathbf{a})\cdot S_{m1}(\mathbf{a})## for ##m=1,\ldots,n1##
10. Let ##(a_n)_{n\in \mathbb{N}}## be a sequence of nonnegative real numbers, not all zero. Prove
$$
\left(\sum_{n\in \mathbb{N}}a_n\right)^4 < \pi^2 \sum_{n\in \mathbb{N}}a_n^2 \cdot \sum_{n\in \mathbb{N}}n^2a_n^2
$$
High Schoolers only
11. (solved by @etotheipi ) On how many ways can ##2020## be written as a sum of consecutive natural numbers (greater than zero)?
12. (solved by @Not anonymous ) A binary operation on a set is a mapping, which maps a pair from to . E.g. addition is a binary operation on integers. Find two different (i.e. not achievable be renaming the elements) binary operations for which have a neutral element, , and can be inverted: for all there is a with , and are associative: A binary operation on a set ##S## is a mapping, which maps a pair from ##S\times S## to ##S##. E.g. addition is a binary operation on integers. Find two different binary operations for ##S=\{\,A,B,C,D\,\}## which have a neutral element, ##A\circ X = X##, and can be inverted: for all ##X\in S## there is a ##Y\in S## with ##X\circ Y=A##, and are associative: ##X\circ (Y\circ Z)=(X\circ Y)\circ Z\,.##
13. (solved by @Not anonymous ) Find all six digit numbers with the following property: If we move the first (highest) digit at the end, we will get three times the original number.
14. (solved by @etotheipi ) The Pell sequence named after the English mathematician John Pell is defined by
$$
P(n)= \begin{cases} 0\, , \,n=0\\1 \, , \,n=1\\ P(n2)+2P(n1)\, , \,n>1 \end{cases}
$$
Calculate the limit ##\delta_s:=\lim_{n\to \infty}\dfrac{P(n)}{P(n1)}\,.##
15. (solved by @Not anonymous ) Consider the graph of ##f(x)=1/x## with ##x\geq 1## and let it rotate around the ##x##axis. This solid of revolution looks like an infinitely long trumpet. Calculate its volume ##V## and its surface ##A##.
If we fill it with paint, pour it out again, then we have painted it from inside. Explain this apparent contradiction to the surface you computed.
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