# All Questions

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I encountered the following question: Find a natural number x which satisfies the following: 12345 mod 54321 = 6 mod 54321 I tried using the extended Euclidean algorithm, but failed to solve the ...
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### Help understanding a particular proof of the compactness theorem for Propositional Calculus.

I've reading through this proof, I don't understand the last part: the claim $\tau \models \Sigma$. Note: I'll use $AP(\varphi)$ and $\text{Var}(\varphi)$ interchangeably, to mean the variables that ...
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### Books or website about solving IMO problems

Hey I want to solve IMO problems like the problem in the image below, but I cannot solve the problem or any of the problems in the IMO, so do you guys have some good website or books that teach how to ...
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### Does there exist a $k$ such that for all $n \ge 3$, $\text{gpf}(\lfloor n^{(\log{n})^k} \rfloor) \gt n$?

Does there exist a $k \in \mathbb{R}$ such that for all $n \in \mathbb{N}, n \ge 3$, $\text{gpf}(\lfloor n^{(\log{n})^k} \rfloor) \gt n$, where $\text{gpf}(x)$ is the greatest prime factor of $x$? I ...
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### Conditional probability,two conditions

A doctor operates on patient with a certain disease if he is 80% sure that he has it.We have a patient for whom the doctor is 60% sure that he has the disease,so he makes him do another test which is ...
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### What is the maximum of the following function?

Let $f(x,y) = \frac{xy^\alpha}{x+y},\alpha\in(0,\infty)$. How to compute $$\sup_{(x,y)\in[a,b]\times [0,c]}\frac{xy^\alpha}{x+y},$$ with $b>a>0$?
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### given $-\pi < \theta \leq \pi$ prove $f(z) = z^{1/3}$ is not entire.

I don't want the solution at all, but I'm incredibly stuck, and I really need some (hopefully not much) help. What I've considered: Liouville's Theorem Not applicable because f is not bounded. ...
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### Simplify $(k +1)! > (k + 1)^2$ to prove true for $k ≥ 4$

I am trying to prove this statement is true for $k ≥ 4$. I don't know how to work with $k + 1$ factorial, so I'm a little lost on proving this.
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### How does computing the determinant of a matrix with unit vectors give you the Cross Product?

Say you had $(a_x,a_y,a_z)\times(b_x,b_y,b_z)$, you would set up a matrix like the following: And the resulting would be your Cross Product or the coordinates of an orthogonal vector. My question ...
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### Stationary process vs stationary increments

Am I right that these are not the same, i.e. a stationary process need not have stationary increments and vice versa? example: Brownian motion is not a stationary process but it has stationary ...
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### Why is that if $z^n = |z|^2$, then $|z| = 1$?

We have $z^{n-1} = \bar{z}\ \forall\ n > 2$ which gives us $z^n = |z|^2$, but I dont see why that means $|z| = 1$?
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### Number of field homomorphisms [on hold]

Let $E$ be a finite field extension over $K$. Show there are at most $[E:K]$ $K$-homomorphisms $E \to F$, where $F$ is an algebraic closed field extension of $E$.
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### Calculate the discrete density of the variables of a Markov chain

$X$ and $Y$ are independent random variables of Bernoulli with parameter $\frac{2}{3}$. $Z=X+Y$ $\{X_n\}_{n \in \mathbb{N}}$ with values in {0,1,2} having $Z$ such as initial law and the transition ...
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### Implications of $n a_n=O((n^{\frac{1}{\alpha}}|b_n|)^\alpha)$

Consider two sequences of real numbers $\{a_n\}_n$, $\{b_n\}_n$. Suppose $n a_n=O((n^{\frac{1}{\alpha}}|b_n|)^\alpha)$ where $\alpha \in \mathbb{R}$, $na_n\geq 0$ and big $O$ notation is explained ...
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### what happens to the wave equation?

What happens to the wave equation when the solutions are $$v(x,0)=f(x)=x$$$$v_t(x,0)=g(x)=0.$$ I have a homework question that asks to discuss the solutions of wave equations and this is the first ...

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