# All Questions

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### Prove that a perfect square is either a multiple of $4$ or of the form $4q+1$.

Prove that a perfect square is either a multiple of $4$ or of the form $4q+1$ for some $q\in \mathbb{Z}$. Any ideas on how to start? Do I use a proof by contraposition? Also what's the definition of ...
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### Visualizing Markov and Chebyshev inequalities

I am helping a class on introductory probability covering Markov and Chebyshev's inequalities. I would like to give the students a nice visualization for why they are true or at least to show what ...
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### Finding a sufficient condition for a set to be finitely decomposable into open sets..

Let us call a set in a topological space finitely decomposable set (FDS) iff it can be rewritten using the standard set operations $\cup$ and $\sim$ and only finitely many open sets. I'm looking for ...
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### inclusion exclusion principle basic question

Hello I have found a question about exclusion principle and I have love that you will help me with that question. Prove that for each 201 number from[1,300] we can find that there is always two ...
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### Distance from point to vertices of convex hull

let $P = \{p_1, \ldots, p_k\}$ be any $k$ points on the unit $n$-sphere in $\mathbb{R}^{n+1}$ and $p_0 = 0$ the origin. Furthermore, let $CH(P\cup \{p_0\})$ denote the (possibly degenerate) convex ...
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### How to prove that $x/y$ is continuous in R

$f:R^2$ \{y=0} $\Rightarrow R$ , $f:(x,y)\Rightarrow x/y$. Prove (formally) that $f$ is continuous. I think what I should show is that any point that belongs to an open ball of radius $\epsilon$ of ...
### If $A$ Hurwitz, $(A+A^*)$ is Hurwitz?
If I have $A$ Hurwitz matrix, is $(A+A^*)$, with $A*$ the transpose of $A$, still Hurwitz? Any reference or proof? Because if $(A+A^*)$ is still Hurwitz I can say that it is even negative definite ...