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1500 questions
132
votes
4 answers

What is the probability that a point chosen randomly from inside an equilateral triangle is closer to the center than to any of the edges?

My friend gave me this puzzle: What is the probability that a point chosen at random from the interior of an equilateral triangle is closer to the center than any of its edges? I tried to draw the picture and I drew a smaller (concentric)…
terrace
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132
votes
5 answers

Help find hard integrals that evaluate to $59$?

My father and I, on birthday cards, give mathematical equations for each others new age. This year, my father will be turning $59$. I want to try and make a definite integral that equals $59$. So far I can only think of ones that are easy to…
Argon
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132
votes
12 answers

Can you be 1/12th Cherokee?

I was watching an old Daily Show clip and someone self-identified as "one twelfth Cherokee". It sounded peculiar, as people usually state they're "$1/16$th", or generally $1/2^n, n \in \mathbb{N}$. Obviously you could also be some summation of same…
Nick T
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131
votes
4 answers

How to show that a set of discontinuous points of an increasing function is at most countable

I would like to prove the following: Let $g$ be a monotone increasing function on $[0,1]$. Then the set of points where $g$ is not continuous is at most countable. My attempt: Let $g(x^-)~,g(x^+)$ denote the left and right hand limits of $g$…
AKM
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131
votes
27 answers

Proofs of AM-GM inequality

The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual proofs on the internet but I was wondering if someone…
Michiel Van Couwenberghe
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131
votes
9 answers

Prove that $C e^x$ is the only set of functions for which $f(x) = f'(x)$

I was wondering on the following and I probably know the answer already: NO. Is there another number with similar properties as $e$? So that the derivative of $ e^x$ is the same as the function itself. I can guess that it's probably not, because…
Timo Willemsen
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131
votes
19 answers

Past open problems with sudden and easy-to-understand solutions

What are some examples of mathematical facts that had once been open problems for a significant amount of time and thought hard or unsolvable by contemporary methods, but were then unexpectedly solved thanks to some out-of-the-box flash of genius,…
Damian Reding
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131
votes
1 answer

Application of Hilbert's basis theorem in representation theory

In Smalø: Degenerations of Representations of Associative Algebras, Milan J. Math., 2008 there is an application of Hilbert's basis theorem that I don't understand: Two orders are defined on the set of $d$-dimensional modules over an algebra…
Julian Kuelshammer
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131
votes
8 answers

Are half of all numbers odd?

Plato puts the following words in Socrates' mouth in the Phaedo dialogue: I mean, for instance, the number three, and there are many other examples. Take the case of three; do you not think it may always be called by its own name and also be called…
Jon Ericson
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130
votes
1 answer

Prove that simultaneously diagonalizable matrices commute

Two $n\times n$ matrices $A, B$ are said to be simultaneously diagonalizable if there is a nonsingular matrix $S$ such that both $S^{-1}AS$ and $S^{-1}BS$ are diagonal matrices. a) Show that simultaneously diagonalizable matrices commute: $AB =…
diimension
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130
votes
26 answers

Unexpected examples of natural logarithm

Quite often, mathematics students become surprised by the fact that for a mathematician, the term “logarithm” and the expression $\log$ nearly always mean natural logarithm instead of the common logarithm. Because of that, I have been gathering…
José Carlos Santos
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130
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0 answers

If polynomials are almost surjective over a field, is the field algebraically closed?

Let $K$ be a field. Say that polynomials are almost surjective over $K$ if for any nonconstant polynomial $f(x)\in K[x]$, the image of the map $f:K\to K$ contains all but finitely many points of $K$. That is, for all but finitely many $a\in K$,…
Eric Wofsey
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130
votes
5 answers

Incremental averaging

Is there a way to incrementally calculate (or estimate) the average of a vector (a set of numbers) without knowing their count in advance? For example you have a = [4 6 3 9 4 12 4 18] and you want to get an estimate of the average but you don't…
Ali
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130
votes
7 answers

Can an irrational number raised to an irrational power be rational?

Can an irrational number raised to an irrational power be rational? If it can be rational, how can one prove it?
John Hoffman
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129
votes
7 answers

Is there a function that grows faster than exponentially but slower than a factorial?

In big-O notation the complexity class $O(2^n)$ is named "exponential". The complexity class $O(n!)$ is named "factorial". I believe that $f(n) = O(2^n)$ and $g(n) = O(n!)$ means that $\dfrac{f(n)}{g(n)}$ goes to zero in the limit as $n$ goes to…
Robert L. Read
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