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Welcome to the reinstatement of the monthly math challenge threads!

Rules:

1. You may use google to look for anything except the actual problems themselves (or very close relatives).

2. Do not cite theorems that trivialize the problem you're solving.

3. Have fun!

1) ##A## and ##B## are both connected.

2) ##A## and ##B## are both path-connected.

3) ##A## is path-connected and ##B## is connected.[/COLOR]

Rules:

1. You may use google to look for anything except the actual problems themselves (or very close relatives).

2. Do not cite theorems that trivialize the problem you're solving.

3. Have fun!

**1.**(solved by @Throwaway_for_June ) Given a rectangle and a triangle in the plane, describe how to construct a line that cuts both in half by area using a compass and straightedge. [Edit: This is a lot more annoying than I intended. Feel free to ignore this problem, but if you do solve of course post it!]**2.**(Main problem solved by @fresh_42 and @mathwonk . Counterexample given by @pasmith ) For a function ##f:\mathbb{R}\to\mathbb{R}## define ##P(f)=\{T\in\mathbb{R}: f(x+T)=f(x) \text{ for all } x\in\mathbb{R}\}##. Note that ##f## is periodic if and only if ##P(f)## contains a nonzero element. If ##f## is continuous and not constant, show that ##P(f)=\{nT:n\in\mathbb{Z}\}## for some real number ##T.## Give a counterexample when ##f## is not continuous.**3.**(solved by @julian) Show that ##\sqrt{5}+\sqrt[3]{7}+\sqrt[5]{3}## is irrational.**4.**(solved by @projective) Suppose that ##G## is a group of order ##63## with normal subgroups of order ##7## and ##9##. Show that ##G## is abelian.**5.**(solved by @martinbn) Let ##\zeta=e^{2\pi i/n}=\cos(2\pi/n)+i\sin(2\pi/n).## Evaluate the product ##(1-\zeta)(1-\zeta^2)...(1-\zeta^{n-1}).##**6.**(solved by @PeroK) A rook starts on the square a1 on an otherwise empty chessboard. Every turn, the rook makes a legal move uniformly at random. What is the expected number of turns it will take it for it to reach the opposite corner h8? Staying on the same square does not count as a turn.**7.**(solved by @fresh_42) Let ##GL_n^+(\mathbb{R})## be the set of invertible ##n\times n## real matrices with positive determinant. Suppose that ##A## and ##B## are elements of ##GL_n^+(\mathbb{R})## and similar in the sense that ##A=PBP^{-1}## for some ##P\in GL_n(\mathbb{R})##. Can you necessarily find a matrix ##Q\in GL_n^+(\mathbb{R})## such that ##A=QBQ^{-1}?## In other words, if two matrices in ##GL_n^+(\mathbb{R})## are conjugate as elements of ##GL_n(\mathbb{R})##, are they also conjugate as elements of ##GL_n^+(\mathbb{R})?##**8.**(solved by @nuuskur) Let ##\mathcal{F}## be a collection of subsets of ##\mathbb{N}## which is totally ordered by inclusion, i.e. for any ##A,B\in\mathcal{F}##, either ##A\subseteq B## or ##B\subseteq A.## Is it possible for ##\mathcal{F}## to be uncountable?**9.**(solved by @mathwonk) Identify ##S^3## with ##\{(z,w)\in\mathbb{C}^2: |z|^2+|w|^2=1\}.## Consider the projection map ##\pi:S^3\to\mathbb{C}P^1, \pi(z,w)=[z:w]##. This is called the Hopf map. Compute explicitly the preimages of ##[1:0]## and ##[0:1]## as subsets of ##S^3## and verify that they are linked circles.**10.**(solved by @mathwonk) Suppose ##A## and ##B## are disjoint (thanks @mfb !) subsets of ##[0,1]^2## such that ##(0,0)## and ##(1,1)## are elements of ##A## and ##(0,1)## and ##(1,0)## are elements of ##B.## Which of the following are possible?1) ##A## and ##B## are both connected.

2) ##A## and ##B## are both path-connected.

3) ##A## is path-connected and ##B## is connected.[/COLOR]

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