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Welcome to this month's math challenge thread!

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 Do not cite theorems that trivialize the problem you're solving.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

$$\lim_{x\to\infty} e^{-\sqrt{x}}\left(1+\frac{1}{\sqrt{x}}\right)^x.$$

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 Do not cite theorems that trivialize the problem you're solving.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 @AndreasC) I start watching a stream from my favorite youtuber 10 minutes after the stream starts. To catch up to the stream, I make the playback speed 1.5x. How long will it take until I am caught up with the stream?**2.**(solved by @julian) Evaluate the following limit:$$\lim_{x\to\infty} e^{-\sqrt{x}}\left(1+\frac{1}{\sqrt{x}}\right)^x.$$

**3.**(solved by @julian) Evaluate ##\int_0^1\frac{1-x^2}{\ln(x)} dx.##**4.**(solved by @AndreasC) If ##T:V\to V## and ##S:W\to W## are linear maps with eigenvalues ##\alpha## and ##\beta##, respectively, check that ##\alpha\beta## is an eigenvalue of ##T\otimes S:V\otimes W\to V\otimes W.## Since every polynomial is the characterstic polynomial of a linear map (up to scalar multiplication), conclude that the product of two algebraic numbers is algebraic. Construct a linear transformation on ##V\otimes W## with eigenvalue ##\alpha+\beta## to show that the sum of two algebraic numbers is algebraic.**5.**(solved by @mathwonk) Let ##A## be an ##n\times m## matrix with rank ##r## and let ##B## be a ##p\times q## matrix with rank ##s.## Consider the set of all ##m\times p## matrices ##X## satisfying ##AXB=0.## This set is a vector space. What is its dimension?**6.**What is the smallest positive integer ##m## such that there exist integer polynomials ##p_1(x),\ldots,p_n(x)## with ##mx=p_1(x)^3+\ldots+p_n(x)^3.##**7.**(solved by @mathwonk)Give examples of groups ##G## and ##H## such that ##G## contains a subgroup isomorphic to ##H## and ##H## contains a subgroup isomorphic to ##G##, but ##G## is not isomorphic to ##H.##**8.**(solved by @AndreasC) Let ##A## be an ##n\times n## matrix all of whose entries are non-negative real numbers. Show that ##A## has a non-negative real eigenvalue.**9.**(solved for ##n=2## by @AndreasC and @mathwonk. (Solved for ##n>2## by @mathwonk) Let ##f:\mathbb{R}^n\to\mathbb{R}^n## be a continuous function such that ##f(f(x))=x## for all ##x\in\mathbb{R}^n.## Does ##f## necessarily have a fixed point?
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