# All Questions

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### What are the steps to solving this average distance problem?

I know that the answer is C. But how exactly would you go about solving this problem? Is there a specific formula I should be using? Because I can't seem to find any related ones. Thanks.
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### Find surface area by calculating surface integrals

Fix a radius $r > 0$ and two angles $ϕ_1$ and $ϕ_2$, with $−π/2 < ϕ_1 < ϕ_2 < π/2$ Find the surface area of the portion of the sphere of radius r with latitudes between $ϕ_1$ and $ϕ_2$. ...
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### Converting an optimisation problem to an integer linear formulation

Is there a way to convert the following to a linear formulation? In other words, is there a workaround for the absolute value in the objective function? Minimise: ...
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### Expectation at 2nd draw from urn.

SUppose an urn contains $r$ red balls and $b$ blue balls. Each time a ball is drawn, it is returened along with $m$ additional balls of the same colour. Let $X_k$ be the number of red balls in $k$ ...
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### Logic - find isomorphism between two modules [on hold]

I have two structures: $M_1 = (\Bbb Z,+)$ - Addition $M_2 = (\Bbb Z,*)$ - Multiplication Are they isomorphic? If so\if not, how can you prove it?
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### Humor in published articles/textbooks

Even though there are plenty examples of mathematical jokes, the mathematical literature is (in many cases) pretty dull. Nevertheless, examples exist in which an article makes you smile with a nice ...
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### Find all the functions satisfying this criterion

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$\left|f(x)-f(y)\right|=2\left|x-y\right|$$
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### GRE combinations problem involving cards and repetition.

A set of cards is numbered 1 through 5. Which of the two quantities is larger? Quantity A The number of ways to pick 3 of the 5 cards such that card number 1 is included Quantity B The number of ...
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### Existence of non-extremal point

I am stuck in the following question: Let $x,y\in \mathbb{R}^n$ such that $x\ne y$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$. I would like to know whether there exists $z\in]x,y[$ such that for every ...
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### Trying to show that $\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$

I'm trying to show that for $b>1$, $x>0$ and $x$ irrational, that $$\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$$ I know this follows immediately if we define ...
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### When does a matrix game and the sign flipped matrix game have the same nash equilibria?

Given a game $G$, we can construct another $G'$, by a positive scaling i.e. $\lambda \in \mathbb{R}_{++}$, s.t. each entry of $A$ is scaled by $\lambda$ Obviously, $G$ and $G'$ have the same nash ...
Equation given is $y= sin 2x + 2sin x..$ and $cos x = \frac 12...$ using first derivative.. How to get $\frac {5pi}{3}$ ??