# Tagged Questions

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72 views

### Solve the equation $2x^2+5y^2+6xy-2x-4y+1=0$

The problem does not say it but I think solutions should be from $\mathbb{R}$. I tried to express the left sum as a sum of squares but that does not work out. Any suggestions?
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### $n^2 + 7n + 1$ is odd

Prove that for any integer $n$, the integer $n^2 + 7n + 1$ is odd. I have $n=2k+1$ for some $k\in Z$ i really do not how to do this problem. any help in understanding would be greatly appreciated.
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### Proving facts regarging a graph with a degree of $n$ and no cycles of length more than 3.

Let H be a simple graph that has no cycles of length more than 3. Each vertex has degree of n $n$. Is it possible to prove H has at least $2n$ vertices?
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### Does this proof account for all integers? I am new to proofs.

I am learning proofs and, have the following statement: Let $x$ belong to the set of integers. If $x$ has the property that for each integer $m$, $m+x=m$, then $x=0$. Here is my strategy: ...
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### Proofs by Contradiction (cont.) [closed]

This question is somewhat of a continuation of the very interesting question and its responses: Can every proof by contradiction also be shown without contradiction? I did a rough count of proofs ...
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### Discrete Structures Math

Let $x > 0$ be a real number. Prove that $x + \dfrac1{4x} \ge 1$. I don't know where to begin with this question, I was hoping someone could help me out with this.
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Define the function $g:\mathbb N \rightarrow \mathbb N$ with $g(d)= d^2 + d + 1$ I started out by assuming that if two arbitrary elements of $\mathbb N$, $x$ and $y$,where $x>y$ without loss of ...
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### Proving square of nonzero integer is natural number

I am learning proofs with $\mathbb N$ and have this proposition: Let $m \in\mathbb Z$. If $m \ne 0$, then $m^2 \in\mathbb N$. Previously, I have proven: For $m \in\mathbb Z$, one and only one of the ...
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### $[x]_0,[x]_1\ldots [x]_n$ is a basis for vector space $V$.

here is a lemma which requires the use of falling factorials which are written as $[x]_n=x(x-1)\ldots(x-(n-1))$ : Lemma:Let $V$ be a vector space of polynomials over $\mathbb C$ , then ...
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### An interval with width greater than one contains an integer.

If I have an interval $(a, b)$ such that $b - a > 1$, how can I prove that this contains an integer? It seems 'obvious', but a formal proof eludes me.
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### Defining prime numbers for proofs

In my discrete mathematics book under existence proofs it has Prove that there exists a prime $p$ such that $2^p -1$ is composite. It then goes on to say by trial and error we find $2^{11}-1$ ...
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### Proof that bernstein-coefficients of $p(x)=x$ are $b_i=a+i\frac{b-a}{n},\ i=0,…,n$

I want to proof that the bernstein-coefficients for $p(x)=x$ on $[a,b]$ are described by $$b_i=a+i\frac{b-a}{n},\ i=0,...,n$$ Where the Bernstein polynomials on $[a,b]$ are defined by ...