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fresh_42

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Summer is coming and brings .... Oops, time for a change!

Fall (Spring) is here and what's better than to solve some tricky problems on a long dark evening (with the power of returning vitality all around).

a) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions will be posted around 15th of the following month.

b) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common knowledge to all mathematicians". Whether the latter is satisfied will be decided on a case-by-case basis.

c) If you have seen the problem before and remember the solution, you cannot participate in the solution to that problem.

d) You are allowed to use google, wolframalpha or any other resource. However, you are not allowed to search the question directly. So if the question was to solve an integral, you are allowed to obtain numerical answers from software, you are allowed to search for useful integration techniques, but you cannot type in the integral in wolframalpha to see its solution.

e) Mentors, advisors and homework helpers are kindly requested not to post solutions, not even in spoiler tags, for the challenge problems, until 16th of each month. This gives the opportunity to other people including but not limited to students to feel more comfortable in dealing with / solving the challenge problems. In case of an inadvertent posting of a solution the post will be deleted.

We have quite a couple of old problems, which are still open. Seven of Ten in each of the September threads to be exact:

https://www.physicsforums.com/threads/basic-math-challenge-september-2018.954490/

https://www.physicsforums.com/threads/intermediate-math-challenge-september-2018.954495/

so you can still try to solve them. Answers will be given on demand if someone is interested in it, otherwise we will leave them as is. Maybe someone will come one day and tackle them.

As it seems more and more unnecessary to split threads between "Basic" and "Intermediate" as apperently "Calculus" and "other" were more significant, we decided to post all questions with a varying number of them in one thread instead, so that everyone can pick the problems he likes best, regardless of what we think is difficult or not. I also want to thank @wrobel for the problem about Noether's theorem which he provided.

Some of the questions are on a level, which could be solved by High Schoolers. I will mark them with an "H" and hope those of you which aren't in school anymore will leave them for actual high school students - maybe at least for two week. Thanks.

$$L(t,x,\dot x)=\frac{1}{2}\dot x^2-\frac{t}{x^4}.$$(by @wrobel )

(by @fresh_42 )

$$

f(x,y) =

\begin{cases}

1 & \text{if } x \geq 0 \text{ and }x \leq y < x+1\\

-1 & \text{if } x \geq 0 \text{ and }x+1 \leq y < x+2\\

0 & \text{elsewhere }

\end{cases}

$$

Now calculate ##\int_\mathbb{R}\left[\int_\mathbb{R}f(x,y)\,d\lambda(x) \right] \,d\lambda(y)## and ##\int_\mathbb{R}\left[\int_\mathbb{R}f(x,y)\,d\lambda(y) \right]\,d\lambda(x)\,,## and why isn't it a contradiction to Fubini's theorem.

$$

\int_A \dfrac{1}{x^2+y}\,d\lambda(x,y)

$$

with ##A=(0,1)\times (0,1)\subseteq \mathbb{R}^2## is finite. (by @fresh_42 )

(by @fresh_42 )

$$

\lim_{r \to 0}\int_\mathbb{R}f_r(x)\,dx \neq \int_\mathbb{R} \lim_{r \to 0} f_r(x) \,dx

$$

$$

\dfrac{d}{dx}\int_\mathbb{R}f(x,y)\,dy \neq \int_\mathbb{R}\dfrac{\partial}{\partial x}f(x,y)\,dy

$$

(by @fresh_42 )

$$

\hat{\mathcal{T}}=\{\,U\subseteq \hat{\mathbb{C}}\,|\,\infty \notin U \,\wedge \,U\subseteq \mathbb{C}\text{ open}\,\}\,\cup \,\{\,U\subseteq \hat{\mathbb{C}}\,|\,\infty \in U \,\wedge \,U^C\subseteq \mathbb{C}\text{ compact}\,\}

$$

(by @fresh_42 )

(by @fresh_42 )

\int_{-\infty}^{+\infty}\dfrac{4}{x^2-x+1}\,dx

$$

(by @fresh_42 )

Did the beetle eventually reach the top of the tree? And if so, how many nights will he need at least? (by @fresh_42 )

(by @QuantumQuest )

Fall (Spring) is here and what's better than to solve some tricky problems on a long dark evening (with the power of returning vitality all around).

__RULES:__a) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions will be posted around 15th of the following month.

b) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common knowledge to all mathematicians". Whether the latter is satisfied will be decided on a case-by-case basis.

c) If you have seen the problem before and remember the solution, you cannot participate in the solution to that problem.

d) You are allowed to use google, wolframalpha or any other resource. However, you are not allowed to search the question directly. So if the question was to solve an integral, you are allowed to obtain numerical answers from software, you are allowed to search for useful integration techniques, but you cannot type in the integral in wolframalpha to see its solution.

e) Mentors, advisors and homework helpers are kindly requested not to post solutions, not even in spoiler tags, for the challenge problems, until 16th of each month. This gives the opportunity to other people including but not limited to students to feel more comfortable in dealing with / solving the challenge problems. In case of an inadvertent posting of a solution the post will be deleted.

We have quite a couple of old problems, which are still open. Seven of Ten in each of the September threads to be exact:

https://www.physicsforums.com/threads/basic-math-challenge-september-2018.954490/

https://www.physicsforums.com/threads/intermediate-math-challenge-september-2018.954495/

so you can still try to solve them. Answers will be given on demand if someone is interested in it, otherwise we will leave them as is. Maybe someone will come one day and tackle them.

As it seems more and more unnecessary to split threads between "Basic" and "Intermediate" as apperently "Calculus" and "other" were more significant, we decided to post all questions with a varying number of them in one thread instead, so that everyone can pick the problems he likes best, regardless of what we think is difficult or not. I also want to thank @wrobel for the problem about Noether's theorem which he provided.

Some of the questions are on a level, which could be solved by High Schoolers. I will mark them with an "H" and hope those of you which aren't in school anymore will leave them for actual high school students - maybe at least for two week. Thanks.

**1.**Solve and describe the solution step by step in quadrature a Lagrangian differential equation with Lagrangian$$L(t,x,\dot x)=\frac{1}{2}\dot x^2-\frac{t}{x^4}.$$(by @wrobel )

**2.**(solved by @Math_QED )**a)**Let ##X## be a set and ##\mathcal{F}=\{\,\{\,x\,\}\,|\,x\in X\,\}##. Determine the ##\sigma-##algebra ##\sigma(\mathcal{F})##.**b)**Let ##X## be a set and ##S\subseteq \mathcal{P}(X)##. Show that for ##A\in \sigma(S)## there is an ##S_0\subseteq S## such that ##S_0## is countable and ##A \in \sigma(S_0)##.(by @fresh_42 )

**3.**Let [itex]f(x)\in\mathbb{Q}[x][/itex] be a degree 5 polynomial with splitting field [itex]K[/itex]. Suppose that there is a unique extension [itex]F/\mathbb{Q}[/itex] such that [itex]F\subset K[/itex] and [itex][K:F]=3[/itex]. Show that [itex]f(x)[/itex] is divisible by a degree 3 irreducible element of [itex]\mathbb{Q}[x][/itex]. (by @Infrared )**4. a)**Let ##f\, : \,\mathbb{R}^2 \longrightarrow \mathbb{R}## be defined as$$

f(x,y) =

\begin{cases}

1 & \text{if } x \geq 0 \text{ and }x \leq y < x+1\\

-1 & \text{if } x \geq 0 \text{ and }x+1 \leq y < x+2\\

0 & \text{elsewhere }

\end{cases}

$$

Now calculate ##\int_\mathbb{R}\left[\int_\mathbb{R}f(x,y)\,d\lambda(x) \right] \,d\lambda(y)## and ##\int_\mathbb{R}\left[\int_\mathbb{R}f(x,y)\,d\lambda(y) \right]\,d\lambda(x)\,,## and why isn't it a contradiction to Fubini's theorem.

**4. b)**(solved by @benorin ) Show that the integral$$

\int_A \dfrac{1}{x^2+y}\,d\lambda(x,y)

$$

with ##A=(0,1)\times (0,1)\subseteq \mathbb{R}^2## is finite. (by @fresh_42 )

**5. a)**Let [itex]n[/itex] be a positive integer. Let [itex]a_1,\ldots,a_k[/itex] be (positive) factors of [itex]n[/itex] such that [itex]\gcd(a_1,\ldots,a_k,n)=1[/itex]. How many solutions [itex](x_1,\ldots,x_k)[/itex] does the equation [itex]a_1x_1+\ldots+a_kx_k\equiv 0\mod n[/itex] have subject to the restriction that [itex]0\leq x_i<n/a_i[/itex] for each [itex]i[/itex]?**5. b)**How does the solution change if [itex]\gcd(a_1,\ldots,a_k,n)=d>1[/itex]? (by @Infrared)**6.**(solved by @lpetrich ) Calculate**a)**$$\sum_{n=0}^\infty \left(\dfrac{2}{2+3i} \right)^n$$**b)**$$\sum_{n=0}^\infty \left(2\sqrt{n}-4\sqrt{n+1}+2\sqrt{n+2} \right)$$**c)**$$\sum_{n=3}^\infty \dfrac{8n}{(n^2-1)^2}$$**d)**$$\lim_{x \to 0}\dfrac{\cos(x^2)-\sqrt{1+x^3}}{x^3}$$(by @fresh_42 )

**7.**(solved by @benorin )**7. a)**Determine ##\int_1^\infty \frac{\log(x)}{x^3}\,dx\,.##**7. b)**Determine for which ##\alpha## the integral ##\int_0^\infty x^2\exp(-\alpha x)\,dx## converges.**7. c)**Find a sequence of functions ##f_n\, : \,\mathbb{R}\longrightarrow \mathbb{R}\, , \,n\in \mathbb{N}## such that $$\sum_\mathbb{N}\int_\mathbb{R}f_n(x)\,dx \neq \int_\mathbb{R}\left(\sum_\mathbb{N}f_n(x) \right) \,dx$$**7. d)**Find a family of functions ##f_r\, : \,\mathbb{R}^+\longrightarrow \mathbb{R}\, , \,r\in \mathbb{R}## such that$$

\lim_{r \to 0}\int_\mathbb{R}f_r(x)\,dx \neq \int_\mathbb{R} \lim_{r \to 0} f_r(x) \,dx

$$

**7. e)**Find an example for which$$

\dfrac{d}{dx}\int_\mathbb{R}f(x,y)\,dy \neq \int_\mathbb{R}\dfrac{\partial}{\partial x}f(x,y)\,dy

$$

(by @fresh_42 )

**8.**Let ##f##, ##g##: ##\mathbb{R} \rightarrow \mathbb{R}## be two functions with ##f\,''(x) + f\,'(x)g(x) - f(x) = 0##. Show that if ##f(a) = f(b) = 0## then ##f(x) = 0## for all ##x\in [a,b]##. (by @QuantumQuest )**9.**(solved by @nuuskur ) Let ##\hat{\mathbb{C}}=\mathbb{C}\cup \{\,\infty\,\}## with the usual Euclidean topology on ##\mathbb{C}## and$$

\hat{\mathcal{T}}=\{\,U\subseteq \hat{\mathbb{C}}\,|\,\infty \notin U \,\wedge \,U\subseteq \mathbb{C}\text{ open}\,\}\,\cup \,\{\,U\subseteq \hat{\mathbb{C}}\,|\,\infty \in U \,\wedge \,U^C\subseteq \mathbb{C}\text{ compact}\,\}

$$

**9. a)**##\hat{\mathcal{T}}## is a topology on ##\hat{\mathbb{C}}##.**9. b)**##(\hat{\mathbb{C}},\hat{\mathcal{T}})## is Hausdorff.**9. c)**##(\hat{\mathbb{C}},\hat{\mathcal{T}})## is compact.(by @fresh_42 )

**10.**(solved by @lpetrich )**10. a)**Show that ##D_4=\langle r,s\,|\,r^2=s^2=rsrs=1\rangle## is the smallest non-cyclic group.**10. b)**Show that the converse of Lagrange's theorem is false, i.e. that there is a finite group with ##n## elements which has no subgroup to one of the divisors of ##n\,.##**10. c)**Give an example of a non-Abelian finite and a non-Abelian infinite group.**10. d)**Show that ##A_5## is simple, i.e. has only trivial normal subgroups.(by @fresh_42 )

**11.**(solved by @lpetrich ; @Math_QED ) Calculate $$\int_{-\infty}^{+\infty}\dfrac{4}{x^2-x+1}\,dx

$$

(by @fresh_42 )

**H 12.**Let [itex](x_n)[/itex] be a sequence of positive real numbers such that [itex]x_{n+1}\geq\dfrac{x_n+x_{n+2}}{2}[/itex] for each [itex]n[/itex]. Show that the sequence is (weakly) increasing, i.e. [itex]x_n\leq x_{n+1}[/itex] for each [itex]n[/itex]. (by @Infrared )**H 13.**Let [itex]p(x)[/itex] be a non-constant real polynomial. Suppose that there exists a real number [itex]a[/itex] such that [itex]p(a)\neq 0[/itex] and [itex]p'(a)=p''(a)=0\,.[/itex] Show that not all of the roots of [itex]p[/itex] are real. (by @Infrared )**H 14.**Find the area enclosed by the curve ##r^2 = a^2\cos 2\theta\,##. (by @QuantumQuest )**H 15.**(You may use wolframalpha.com for calculations) One tiny night-active and long-living beetle decided one night to climb a sequoia. The tree was exactly ##100\, m## high at this time. Every night the beetle made a distance of ##10\, cm##. The tree grew every day evenly ##20\, cm## along its entire length.Did the beetle eventually reach the top of the tree? And if so, how many nights will he need at least? (by @fresh_42 )

**H 16.**Show that ##\lim_{n \to \infty} \sqrt[n]{p_1a_1^n + p_2a_2^n + \cdots + p_ka_k^n} = max \{a_1, a_2, \cdots ,a_k\}## where ##p_1, p_2, \cdots, p_k \gt 0## and ##a_1, a_2, \cdots, a_k \geq 0##.(by @QuantumQuest )

**H 17.**Show that the sequence ##(a_n)## with ##a_1 = \sqrt[]{b}##, ##a_{n+1} = \sqrt[]{a_n + b}## for ##n = 1, 2, 3, \ldots## converges to the positive root of the equation ##x^2 - x - b = 0##. (by @QuantumQuest )
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