# All Questions

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### Finding solutions to $M^3+2M^2-5M-6I=0$ for a matrix $M$

Find all matrices of the form $\left(\begin{array}{} a & c\\0&b \end{array}\right)$ which satisfy the equation: $$M^3+2M^2-5M-6I=0$$ Before this part of the question we needed to show ...
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### Absolute value of a polynomial as the Euclidean distance from its root

Say that you have a polynomial $f(x)$ of degree 2 in one real variable. Then, if the polynomial has only one unique root $r \in \mathbb{R}$, it factorizes as $f(x) = (x - r)^2$, which expresses the ...
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### Function with a set of partitions in the domain

Say I have some set $X$ and I know I'm going to have a 2-block partition $P$ over $X$. I want a function $f$ that maps from that set and any possible 2-block partition of it into $\mathbb{R}$. I think ...
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### Is it possible for a group to be a finite union of subgroups of infinite index?

Just restating the title: Does there exist a group $G$ and subgroups $H_1, \ldots, H_k$ so that $[G:H_i]$ is infinite for each $i = 1, 2, \ldots, k$, and $G = H_1 \cup \cdots \cup H_k$? If $G$ is a ...
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### What is the probability that a rod can be cut with the length of cut being 5 units?

A rod of length 10 units and breadth 3 units is cut as shown in figure. Assuming that the longest cut can be from A to C and B to D. What is the probability that the cut made is of length 5 units. The ...
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### What is the easiest way to find the radius and center of the circle of intersection between two spheres?

If given two spheres $S_1$ and $S_2$, of radius $r_1$ and $r_2$, centered at 3-space points $P_1$ and $P_2$, respectively. What is the easiest way to find the radius and center of the circle of ...
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### Geometrical meaning of orientation on vector space

Can any one explain, the geometrical meaning of orientation in a vector space?
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### Distribuion of two random Variables

I have two Bernoulli random variables, X and Y, with parameters rispectively (1/n) and (1/2n^2). Do they have the same distribution?
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### Help Proof $\underset{x\rightarrow\left(\frac{7}{4}\right)^{+}}{lim}\cfrac{3x}{4x-7}=\infty$ by definition

$$\underset{x\rightarrow\left(\frac{7}{4}\right)^{+}}{lim}\cfrac{3x}{4x-7}=\infty$$ I want to proof that for every $M>0$ exists $\delta$ for which $0<x-\cfrac{7}{4}<\delta$ such ...
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### integate $\int\sqrt{x^2+3x+3} dx$

Is there any universal method how to solve integrals like this? $$\int\sqrt{x^2+3x+3} dx$$ Or this? $$\int\sqrt{-x^2+3x+3} dx$$ I tried use first Euler subs, but it was not good idea. ...
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### Quick method to find Laurant Series? [duplicate]

Is there a quick way to find the Laurant Series of a function without doing a contour integral. Here is an example of the type of problem I am working on Find the Laurant series of the function ...
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### diophantine equation: $x^2 +y^2 = z^n$

Prove that $x^2+y^2=z^n$ has a solution $(x, y, z)$ in $\mathbb{N}$ for all $n\in\mathbb{N}$ I tried to prove this by induction, but couldn't. ( This was probably because the solution for some $n$ ...
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### How can I find a function that is in $L^1(\mathbb{R})$ with its derivative also but its limit tends to zero?

I am trying to find for a function that would full-fill these conditions below: $$f \in L^1(\mathbb{R})$$ $$f' \in L^1(\mathbb{R})$$ but its $\lim_{t \to \infty}=0$. I've tried to find a function ...
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### IMO 1997 problem 6

For each positive integer $n$ , let $f (n)$ denote the number of ways of representing $n$ as a sum of powers of $2$ with non-negative integer exponents. Representations which differ only in the ...
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An X-ray test is used to detect a disease that occurs, initially without any obvious symptoms, in 3% of the population. The test has the following error rates: 7% of people who are disease free have a ...
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### How to find $\frac{x}{y}$?

Let $x=\displaystyle\prod_{n=1}^{180}\left(\cos{\left(\dfrac{n\pi}{180}\right)}+2\right)$ and $y=\displaystyle\sum_{n=0}^{89}\binom{180}{2n+1}\left(\dfrac{3}{4}\right)^n$. How to find $\frac{x}{y}$ ? ...
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### People arranged in a queue probability of one standing in front of each other

People $O_1,O_2,\ldots,O_{10}$ were arranged in a queue. Decide whether the following events are independent: 1)$O_1$ is in front of $O_2$ and $0_3$ 2)$O_2$ is not in the end I know how to ...
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### Multiplying matrices / corresponding systems of equations

I'm having some trouble with a problem in linear algebra: Let $A$ be a matrix with dimensions $m \times n$ and $B$ also a matrix but with dimensions $n \times m$ which is not a null matrix. (That's ...
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### Given $\Phi(t)\in M_{n}(\mathcal{C}^1)$, non-singular for each $t\in\Bbb{R}$, exists only one $A(t)$ s.t. $\Phi$ is fundamental to $x'=Ax$.

Here, $M_n(\mathcal{C}^1)$ is the space of the $n\times n$ matrix of differentiable functions. To prove uniqueness I did: Suppose that $\Phi(t)$ is the fundamental matrix of $x'=A(t)x$ and of ...
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### Detailed study of the continuity of $f(x)=\sum_{n=1}^\infty (-i)^n \frac{f_1(nx)}{n}$

Let $f_1(x) = \vert x \vert$ for $\vert x \vert \le \frac{1}{2}$ and let $f_1$ be defined for other values of $x$ by periodic continuation with period $1$. Then $f$ is defined on $\mathbb R$ by ...
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### How to decide if iterations like x(n+1)=rationalFunctionOf(x(n)) have a solution as closed form?

Inspired by If $x_{n+1}= \frac{x_n^2+x_n+1}{x_n+1}$ find$\sum_{n=1}^{p}\frac{1}{1+x_n}$ I was wondering if there is a general method to decide if (and when) iterations can be solved by a closed form. ...
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### Show $-a_1a_2=b_1b_2$ for two perpendicular lines

So I was reading through a book I got at the library and it had a question in it that asks readers to prove $a_1a_2=-b_1b_2$ for two lines, $L_1: a_1x+b_1y+c_1 = 0$ and $L_2: a_2x+b_2y+c_2 = 0$. I ...

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