# Tagged Questions

For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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### Let p be prime. Prove that:

Let p be prime: $p^2\choose p$ is congruent to p (mod $p^2$) and $2p\choose p$ is congruent to 2(mod$p^2)$ I know that when p is prime p|$p\choose k$ where $p\choose k$ can be defined as ...
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### Formal logic behind defining variables in a proof (not E.I. or U.G, etc.)

This is something I have been curious about and hopefully has a simple answer. Often when looking at proofs I will come upon a step that goes along the lines of "define y = ..." and then proceed to ...
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### Numbers in the form $10^n + 1$ with square divisors

Basically, describe every number in the form $10^n + 1$ with square divisors meaning at least one of it's divisors is a square. Of course, there's infinite, but give a general algorithm for finding ...
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### Suppose $f:\mathbb{R} \to \mathbb{R}$ is differentiable and $f'(x) \geq c > 0, \forall x.$ Then $f(\mathbb{R}) =\mathbb{R} \$

Suppose $f:\mathbb{R} \to \mathbb{R}$ is differentiable and there exists $c>0$ such that $f'(x) \geq c, \forall x.$ Could anyone advise me how to prove $f(\mathbb{R}) =\mathbb{R} \$ I have ...
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### Squaring any element of the empty set.

I am asked to prove that when squaring any element of the empty set, one should always get zero. Of course the empty set is the set which contains no elements. If you square nothing then you should ...
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### Generalising mean value theorem for integrals

I want to generalize mean value theorem for Riemann integrals to $\Bbb R^n$, but I do not know how to formulate it. Can someone please help me with formulating the theorem? I think I can prove it ...
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### Matrix Invertibility: AB invertible implies A,B invertible

AB invertible $\implies$ AB is the product of elementary matrices $\implies$ A, B are the product of elementary matrices $\implies$ A,B are invertible, since the products of elementary matrices are ...
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### Need help with starting a proof of linear independence [closed]

Let $\mathbb{R}^m$, $\mathbb{R}^n$ be Euclidean spaces. Let $T: \mathbb{R}^m → \mathbb{R}^n$ be a linear transformation with corresponding $n × m$ matrix A. Let $x_1, x_2, ..., x_k \in \mathbb{R}^m$ ...
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### Proof : Do 4 days fall on the same day?

I was working my way through some discrete math proof examples from Discrete Math by Rosen and being a newbie am stuck on this problem : Show that at least four of any 22 days must fall on the ...
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### Prove that for every rational number z and every irrational number x, there exists a unique irrational number such that x+y=z

This is a homework assignment, please tell me if my proof is correct! Prove that for every rational number z and every irrational number x, there exists a unique irrational number such that x + y = ...
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### if $2a+3b \geq 12m+1$, then either $a \geq 3m+1$ or $b \geq 2m+1$

Not sure how to go about proving this. So far I've declared the contrapositive but can't seem to get further... Let $a,\ b$ and $m$ be integers. Prove that if $2a+3b \geq 12m+1$, then $a \geq 3m+1$ ...
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### Prove that the solution of the differential equation exists for $0\le x\le min(a,{b\over a^2+b^2})$

Consider the initial value problem $y'=x^2+y^2, y(0)=0$ and let $R$ the rectangle $0\le x\le a$, $-b\le y \le b$. Prove that: The solution $y(x)$ exists for $0\le x\le min(a,{b\over a^2+b^2})$ I ...
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### The number of regions into which a plane is divided by n lines in generic position [duplicate]

Suppose that $n$ lines are drawn on a plane in such a way that no lines are parallel and no three of them intersect at a point. Let $r(n)$ be the number of regions the plane is divided into after ...
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### Prove that det(AB) = det(A) det(B) in AB ∈ $GL_2(\mathbb{R} \!\,)$

Prove that det(AB) = det(A) det(B) in AB ∈ $GL_2(\mathbb{R} \!\,)$. Use this result to show that the binary operation in the group AB ∈ $GL_2(\mathbb{R} \!\,)$ is closed; that is, if A and B are in ...
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### Prove that there exists irrational numbers $x$ and $y$ such that $x + y$ is rational, without using subtraction

My homework has this problem: Prove that there exist irrational numbers $x$ and $y$ such that $x + y$ is rational. There is an easy solution that I found on mathbitsnotebook.com: ...
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### Uniform Convergence Analysis

Uniform Convergence of $n^{2}(x)^{3}e^{-nx^{2}}$ on $[0,1]$ My attempt: criterion: suppose $f_n:I\to\ J$ is a sequence of functions which converges point wise to a function $f$, then the ...
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### Is this (sketch of a) proof sound? Uniform cont.

I want to show that a specific trigonometric function is not uniform continuous, as far away from 0, it oscillates like crazy. What I (think I) want to show: I can find an $\epsilon$ such that ...
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### Linear Transformation between Isomorphic Vector Spaces

Suppose $f:V\to W$ is a linear transformation, and that there exists a basis $\{v_1,\ldots,v_n\}$ for $V$ such that $\{f(v_1),\ldots,f(v_n)\}$ is a basis for $W$. Prove that $f$ is an ...
### Let $x_{0}=1$ and $x_{1}=-1$ For $n\geq0$ inductively define $x_{n+2}=x_{n+1}+6x_{n}$
I need to prove that if $a$ and $b$ are negative then $gcd(a,b) = 2gcd(\frac{a}{2},\frac{b}{2})$ I feel like it should be more complicated than $2\cdot \frac{a}{2} = a$, so it's the same GCD.