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Summer is coming and brings a new basic math challenge! Enjoy! For more advanced problems you can check our other intermediate level math challenge thread!

1) 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.

2) 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.

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

4) 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.

5) 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.

$$

\sum_{k=0}^\infty \sum_{m=0}^{2k+1} \dfrac{\sqrt{5}^m}{m!}\cdot \left(\dfrac{(2k)!}{k!}\right)^2 \dfrac{2^{-6k-2}}{(2k-m+1)!}

$$

$$

v(x,y)=\begin{bmatrix}

y \\ x-y

\end{bmatrix}\; , \;w(x,y)=\begin{bmatrix}

y-x \\ -y

\end{bmatrix}

$$

and two curves in ##\mathbb{R}^2## given as:

##\gamma_1## is the half circle from ##(0,-1)## to ##(0,1)## with radius ##1## and origin ##(0,0)##, run anti-clockwise from bottom to top.

##\gamma_2## is the straight line segments from~~(-1,0)~~ correction: ##(0,-1)## to ##(1,0)## and from ##(1,0)## to ##(0,1)##, also run through from bottom to top.

$$

\int_0^{\pi} \dfrac{\sin(\varphi)}{3\cos^2(\varphi)+2\cos(\varphi)+3}\,d\varphi

$$

$$

F(x) =

\begin{cases}

0 & \text{if } x < 0 \\

a-be^{-\lambda x}1 & \text{if } x \geq 0

\end{cases}

$$

for some parameters ##a, b, \lambda \in \mathbb{R}## with ##\lambda > 0\,.## We further assume that ##P(X=0)=0.5##, i.e. there is a ##50\,\%## chance not to wait at all, and ##P(X>1[min])=0.25##.

__RULES:__1) 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.

2) 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.

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

4) 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.

5) 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.

__QUESTIONS:__**1.**Given a non-negative, monotone decreasing sequence ##(a_n)_{n \in \mathbb{N}}\subseteq \mathbb{R}\,.## Prove that ##\sum_{n \in \mathbb{N}}a_n## converges if and only if ##\sum_{n \in \mathbb{N}_0}2^na_{2^n}## converges.**2.**(solved by @lpetrich ) Calculate$$

\sum_{k=0}^\infty \sum_{m=0}^{2k+1} \dfrac{\sqrt{5}^m}{m!}\cdot \left(\dfrac{(2k)!}{k!}\right)^2 \dfrac{2^{-6k-2}}{(2k-m+1)!}

$$

**3.**(solved by @nuuskur , @lpetrich ) Show that the product ##P=xyz## of a Pythagorean triple ##x^2+y^2=z^2## is always divisible by ##60\,|\,P##. Since this is an easy problem, please make sure you won't forget to name an argument!**4.**(solved by @lpetrich, resp. resolved ) Given two vector fields ##v,w\, : \,\mathbb{R}^2 \longrightarrow \mathbb{R}^2## by$$

v(x,y)=\begin{bmatrix}

y \\ x-y

\end{bmatrix}\; , \;w(x,y)=\begin{bmatrix}

y-x \\ -y

\end{bmatrix}

$$

and two curves in ##\mathbb{R}^2## given as:

##\gamma_1## is the half circle from ##(0,-1)## to ##(0,1)## with radius ##1## and origin ##(0,0)##, run anti-clockwise from bottom to top.

##\gamma_2## is the straight line segments from

- Compute all ##4## path integrals of ##v## and ##w## with both paths ##\gamma_1,\gamma_2\,.##
- Determine whether ##v## or ##w## have potentials.

**5.**(solved by @nuuskur ) Show that ##\mathbb{Z}[x]/\langle x^2+2x+4\; , \;5 \rangle \cong \mathbb{Z}_5[4+\sqrt{2}]## are isomorphic rings.**6.**(solved by @lpetrich ) Calculate$$

\int_0^{\pi} \dfrac{\sin(\varphi)}{3\cos^2(\varphi)+2\cos(\varphi)+3}\,d\varphi

$$

**7.**(solved by @lpetrich ) Integrate $$\int_1^5 \dfrac{dx}{\sqrt{x^2+3x-4}}$$**8.**(solved by @Math_QED ) The random waiting time ##X## on a telephone hotline is characterized by the distribution function ##F## with$$

F(x) =

\begin{cases}

0 & \text{if } x < 0 \\

a-be^{-\lambda x}1 & \text{if } x \geq 0

\end{cases}

$$

for some parameters ##a, b, \lambda \in \mathbb{R}## with ##\lambda > 0\,.## We further assume that ##P(X=0)=0.5##, i.e. there is a ##50\,\%## chance not to wait at all, and ##P(X>1[min])=0.25##.

- Determine the parameters with the given information, such that ##F## is actually a distribution function.

- Can the distribution be described by a density function? Why? If yes, calculate the density function.

**9.**(solved by @Danny Sleator ) A princess decided one day to go swimming in the circular lake far from the castle of her father. As soon as she got into the water, suddenly a witch appeared, who wanted to kidnap the girl. The princess swam quickly into the middle of the lake to think of an escape plan. She noticed three things:- The witch can run four times as fast as I can swim.
- The witch always tries to stay closest to me.
- On land, I'm faster than the witch.

**10.**(solved by @Dewgale ) Let ##f\, : \,\mathbb{R}^2 \longrightarrow \mathbb{R}## be defined as ##f(x,y)=x(x-1)^2-2y^2\,.## Determine all critical points of ##f##, decide whether there are extrema, and which, and at last consider, whether ##f## has global extrema or not.
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