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fresh_42

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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!

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. As we think that most of them aren't too difficult, we want to give them another try. One reason is, we want to find out why they have been untouched. So in case you need a hint or two in order to tackle them, don't hesitate to ask! We've also some new questions. Our special thanks go to @Infrared who contributed some new questions.

$$

\mathcal{I}=\{\, z\stackrel{p}{\mapsto} z\; , \; z\stackrel{q}{\mapsto} -z\; , \;z\stackrel{r}{\mapsto} z^{-1}\; , \;z\stackrel{s}{\mapsto}-z^{-1}\,\}

$$

We also consider the two functions $$\mathcal{J}=\{\,z\stackrel{u}{\longmapsto}\frac{1}{2}(-1+i \sqrt{3})z\; , \;z\stackrel{v}{\longmapsto}-\frac{1}{2}(1+i \sqrt{3})z\,\}$$

and the set ##\mathcal{F}## of functions which we get, if we combine any of them: ##\mathcal{F}=\langle\mathcal{I},\mathcal{J} \rangle## by consecutive applications. We now define for ##\mathcal{K}\in \{\mathcal{I},\mathcal{J}\}## a relation on ##\mathcal{F}## by

$$

f(z) \sim_\mathcal{K} g(z)\, :\Longleftrightarrow \, (\forall \,h_1\in \mathcal{K})\,(\exists\,h_2\in \mathcal{K})\,: f(h_1(z))=g(h_2(z))

$$

With these rules in place, is it possible to form more than ##r## teams?

Larry, and Moe have chances of ##0.75## and ##0.70##, respectively, of winning when they serve. Since Moe doesn't serve in the very first game, he gets to choose: should the servers alternate game by game or should the winner of a given game serve in the next game. What's the best choice for Moe?

__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. As we think that most of them aren't too difficult, we want to give them another try. One reason is, we want to find out why they have been untouched. So in case you need a hint or two in order to tackle them, don't hesitate to ask! We've also some new questions. Our special thanks go to @Infrared who contributed some new questions.

__QUESTIONS:__

**1.**Let ##f## be a differentiable function in ##\mathbb{R}##. If ##f'## is invertible and ##(f')^{-1}## is differentiable in ##\mathbb{R}##, show that ##[I_A (f')^{-1} - f \circ [(f')^{-1} ]]' = (f')^{-1}## where ##I_A## with ##I_A(x) = x## is the identity function ##I_A : A \to A##**Hint:**Let ##y = f'(x)## then ##(f')^{-1}(y) = x##. By differentiating, we can get to a useful result including the second derivative of ##f(x)##. Next, we can utilize ##[f\circ (f')^{-1}]'## and incorporate the identity function ##I_A##.**2.**(solved by @nuuskur ) Given a nonnegative, 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.**Let's consider complex functions in one variable and especially the involutions****3.**$$

\mathcal{I}=\{\, z\stackrel{p}{\mapsto} z\; , \; z\stackrel{q}{\mapsto} -z\; , \;z\stackrel{r}{\mapsto} z^{-1}\; , \;z\stackrel{s}{\mapsto}-z^{-1}\,\}

$$

We also consider the two functions $$\mathcal{J}=\{\,z\stackrel{u}{\longmapsto}\frac{1}{2}(-1+i \sqrt{3})z\; , \;z\stackrel{v}{\longmapsto}-\frac{1}{2}(1+i \sqrt{3})z\,\}$$

and the set ##\mathcal{F}## of functions which we get, if we combine any of them: ##\mathcal{F}=\langle\mathcal{I},\mathcal{J} \rangle## by consecutive applications. We now define for ##\mathcal{K}\in \{\mathcal{I},\mathcal{J}\}## a relation on ##\mathcal{F}## by

$$

f(z) \sim_\mathcal{K} g(z)\, :\Longleftrightarrow \, (\forall \,h_1\in \mathcal{K})\,(\exists\,h_2\in \mathcal{K})\,: f(h_1(z))=g(h_2(z))

$$

- Show that ##\sim_\mathcal{K}## defines an equivalence relation.
- Show that ##\mathcal{F}/\sim_\mathcal{I}## admits a group structure on its equivalence classes by consecutive application.
- Show that ##\mathcal{F}/\sim_\mathcal{J}## does not admit a group structure on its equivalence classes by consecutive applications.

**Hint:**Determine the groups ##F=\langle \mathcal{F}\rangle\, , \,I=\langle \mathcal{I} \rangle\, , \,J=\langle \mathcal{J} \rangle## and what distinguishes ##F/I## from ##F/J\,.##**4.**There are ##r## sports 'enthusiasts' in a certain city. They are forming various teams to bet on upcoming events. A pair of people dominated last year, so there are new rules in place this year. The peculiar rules are:- each team must have an odd number of members
- each and every 2 teams must have an even number of members in common.

With these rules in place, is it possible to form more than ##r## teams?

**Hint:**Model these rules with matrix multiplication and select a suitable field.**5.**We define an equivalence relation on the topological two-dimensional unit sphere ##\mathbb{S}^2\subseteq \mathbb{R}^3## by ##x \sim y \Longleftrightarrow x \in \{\,\pm y\,\}## and the projection ##q\, : \,\mathbb{S}^2 \longrightarrow \mathbb{S}^2/\sim \,.## Furthermore we consider the homeomorphism ##\tau \, : \,\mathbb{S}^2 \longrightarrow \mathbb{S}^2## defined by ##\tau (x)=-x\,.## Note that for ##A \subseteq \mathbb{S}^2## we have ##q^{-1}(q(A))= A \cup \tau(A)\,.## Show that- ##q## is open and closed.
- ##\mathbb{S}^2/\sim ## is compact, i.e. Hausdorff and covering compact.
- Let ##U_x=\{\,y\in \mathbb{S}^2\,:\,||y-x||<1\,\}## be an open neighborhood of ##x \in \mathbb{S}^2\,.## Show that ##U_x \cap U_{-x} = \emptyset \; , \;U_{-x}=\tau(U_x)\; , \;q(U_x)=q(U_{-x})## and ##q|_{U_{x}}## is injective. Conclude that ##q## is a covering.

**Hint:**For (a) consider ##O\cup \tau(O)## for an open set ##O\,.## For (b) use part (a) and that ##\mathbb{S}^2## is Hausdorff. For (c) a covering map is a local homeomorphism, so we need continuity, openess, closedness and bijectivity. Again use the previous parts.**(solved by @PeroK ) Two Tennis players have a match that consists of just one set. Larry will serve in the first game and the first player to get ##12## games wins the match (whether ahead by only ##1## game or ##2+## games - either way wins the match).**

6.6.

Larry, and Moe have chances of ##0.75## and ##0.70##, respectively, of winning when they serve. Since Moe doesn't serve in the very first game, he gets to choose: should the servers alternate game by game or should the winner of a given game serve in the next game. What's the best choice for Moe?

**7.**How many times a day is it impossible to determine what time it is, if you have a clock with same length (identical looking) hour and minutes hands, supposing that we always know if it's morning or evening (i.e. we know whether it's am or pm).**For the class of ##n \times n## matrices whose entries, (if flattened and sorted) would be ##1, 2, 3, ..., n^2 -1 ,n^2## prove that there always exists two neighboring entries (in same row or same column) that must differ by at least ##n##.****8.****(solved by @Math_QED ) A fixed point of a function ##f:X\to X## is an element ##x\in X## satisfying ##f(x)=x##. Let ##f:\mathbb{R}\to\mathbb{R}## be a continuous function with no fixed points. Show that ##f\circ f## also has no fixed points.****9.****(solved by @nuuskur ) Let ##f:\mathbb{R}\to\mathbb{R}## be differentiable and let ##a,b## be real numbers with ##a<b##. Show that if ##f'(a)<m<f'(b)##, then there exists some ##c\in (a,b)## such that ##f'(c)=m.##****10.**
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