# All Questions

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### Integers Positioned Around a Circle [duplicate]

Nine distinct positive integers are arranged in a circle such that the product of any two non-adjacent numbers in the circle is a multiple of n and the product of any two adjacent numbers in the ...
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### When does the two cars meet

At 10:30 am car $A$ starts from point $A$ towards point $B$ at the speed of $65$ km/hr, at the same time another car left from point $B$ towards point $A$ at the speed of $70$ km/hr, the total ...
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### Counting distinct positive valued k-tuples that sum to n where each entry can be no greater than some value.

This is motivated by the desire to count the number of ways two dice can form the sums 2,3,4,...,12 respectively. We can safely use the stars and bars method for 2,3,4,...,7 where the number of ways ...
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### Area with different unit measurements

What is the area of a rectangle, in square meters, with a length of 108 meters and a width of 300 millimeters? I think it could be 324 sqm.
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### Counterexample for Maschke's lemma

I'm trying to relax conditions and come up with a counterexample to Maschke's lemma in such a case. For example, with $k = \mathbb Z/p\mathbb Z$, I'm considering the two dimensional representation of ...
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### Is this proof of a mathematical olympiad problem correct?

I'm quite sure about the exactness of my proof, but I'd like to hear (constructive) criticism about my writing. This is the problem: Every non-negative integer is coloured white or red, so that: 1) ...
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### Mathematical presentation of a problem

The issue that I am dealing with now ends up with the solution of a second order equation. The solutions are the Z positions of a point in 3D. So, basically I have two points with the Z positions of ...
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### Does $\wp(A \cap B) = \wp(A) \cap \wp(B)$ hold? How to prove it?

I'm currently working on some discrete mathematics work and I've encountered a question I'm not sure how to answer exactly. Precisely, I'm trying to prove that two power, intersected sets statements ...
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### Series does not converge [closed]

How would I go about showing that the series$$\sum_{n + m\tau \in \Lambda} {1\over{{|n + m\tau|}^2}}$$does not converge, where $\tau \in \mathbb{H}$?
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### Determining at what points multiple variable functions are continuous

With a two variable function what is the procedure to figure out at what points it is continuous? Do I basically just look at what points it would be undefined and anywhere between those points it is ...
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### Proof Using Lagrange's Theorem

I am working on a problem in Kurzweil & Stellmacher's introductory finite group theory that looks like this: Let $A, B$, and $C$ be subgroups of the finite group $G$. Prove that if $B \leq A$, ...
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### Denumerable partition of a denumerable set where each set in the partition is denumerable. [duplicate]

Suppose that a set $A$ is denumerable. Prove that there is a partition $P$ of $A$ where $P$ is denumerable and every $X \in P$ is also denumerable. I can see that this can be done but I cannot figure ...
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### Is it possible to prove dot product by the law of cosines?

It seems many people prove the geometric definition of dot product by the law of cosines. However, i think this is incomplete because the law of cosines is for a triangle, which means we can't use it ...
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### is it possible to project any triangle on a plane as a right triangle on another plane?

I scratching my head over this problem from my projective geometry book (C. R. Wylie, Jr). Given a triangle in the plane $z = 0$, is it possible to find a viewing point, $C$, from which the triangle ...
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### Problem of understanding transitive relations

I would like to understand the transitive property in relations...I just cant get it in my brain. I mean the definition is crystal clear. However I still struggle. For example: Given the set ...
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### Set of Numbers when added in any combination always produce unique result

What I'm looking for is a set of numbers that when added in any combination they always have a unique sum? Is this called something? I have searched on google for hours and I'm having a hard time ...
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### Analyze the continuity of the following function

Here in my book I have such an exercise with the explanation given below, but still there is something the authors didn't add, but simply put "...after some operations...". Here is such an exercise: ...
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### Showing that a subrepresentation is isomorphic to the trivial representation

I'm considering $V$ to be the regular representation of a group $G$ and $W$ to be the 1-dimensional subspace of $V$ generated by the element $x=\sum_{s\in G} e_s$. I'm trying to show that $W$ is a ...
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### [Proof Verification]Prove that if f is differentiable at $c \in I$ and $f'(c) = 0$, then g is not differentiable at $d:=f(c)$.

Proposition. Let I be an interval, and let $f: I \to \mathbb{R}$ be a strictly monotone and continuous on I. Let $J := f(I)$ and let $g:J \to \mathbb{R}$ be the inverse function of f. Prove that if f ...
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### integral ring extension, maximal ideals

Let $\varphi:A\rightarrow A'$ be an integral ring extension. 1) Show that for every maximal ideal $m'\subset A'$ the ideal $\varphi^{-1}(m')\subset A$ is maximal. 2) and that for every ...