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1500 questions
129
votes
14 answers
Olympiad Inequality $\sum\limits_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$
$x,y,z >0$, prove
$$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$
Note:
Often Stack Exchange asked to show some work before answering the question. This inequality was used as a proposal problem…
HN_NH
- 4,361
129
votes
1 answer
Is There An Injective Cubic Polynomial $\mathbb Z^2 \rightarrow \mathbb Z$?
Earlier, I was curious about whether a polynomial mapping $\mathbb Z^2\rightarrow\mathbb Z$ could be injective, and if so, what the minimum degree of such a polynomial could be. I've managed to construct such a quartic and rule out the existence of…
Milo Brandt
- 60,888
128
votes
9 answers
Continuity of the roots of a polynomial in terms of its coefficients
It's commonly stated that the roots of a polynomial are a continuous function of the coefficients. How is this statement formalized? I would assume it's by restricting to polynomials of a fixed degree n (maybe monic? seems like that shouldn't…
Harry Altman
- 4,652
128
votes
13 answers
Advantages of Mathematics competition/olympiad students in Mathematical Research
Everyone in this community I think would be familiar with International Mathematical Olympiad, which is an International Mathematics Competition held for high school students, with many countries participating from around the world.
What's…
user9413
128
votes
1 answer
Can $x^{x^{x^x}}$ be a rational number?
If $x$ is a positive rational number, but not an integer, then can $x^{x^{x^x}}$ be a rational number ?
We can prove that if $x$ is a positive rational number but not an integer, then $x^x$ can not be rational:
Denote…
lsr314
- 15,806
128
votes
8 answers
When to learn category theory?
I'm a undergraduate who wishes to learn category theory but I only have basic knowledge of linear algebra and set theory, I've also had a short course on number theory which used some basic concepts about groups and modular arithmetic. Is it too…
Vicfred
- 2,787
128
votes
9 answers
Normal subgroup of prime index
Generalizing the case $p=2$ we would like to know if the statement below is true.
Let $p$ the smallest prime dividing the order of $G$. If $H$ is a subgroup of $G$ with index $p$ then $H$ is normal.
Sigur
- 6,416
- 3
- 25
- 45
127
votes
8 answers
Why is the derivative of a circle's area its perimeter (and similarly for spheres)?
When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle.
Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is differentiated with respect to $r$, we get $4 \pi…
bryn
- 9,746
127
votes
1 answer
Motivation of irrationality measure
I have a question about the irrationality of $e$:
In proving the irrationality of $e$, one can prove the irrationality of $e^{-1}$ by using the series $$e^x = 1+x+\frac{x^2}{2!} + \cdots + \frac{x^n}{n!}+ \cdots$$ So the series for $e^{-1}$ is…
Damien
- 4,291
127
votes
3 answers
All real numbers in $[0,2]$ can be represented as $\sqrt{2 \pm \sqrt{2 \pm \sqrt{2 \pm \dots}}}$
I would like some reference about this infinitely nested radical expansion for all real numbers between $0$ and $2$.
I'll use a shorthand for this expansion, as a string of signs, $+$ or $-$, with infinite periods denoted by brackets.
$$2=\sqrt{2 +…
Yuriy S
- 31,474
126
votes
9 answers
Produce an explicit bijection between rationals and naturals
I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but can anyone produce an explicit formula for such…
Alex Basson
- 4,231
126
votes
3 answers
Are all limits solvable without L'Hôpital Rule or Series Expansion
Is it always possible to find the limit of a function without using L'Hôpital Rule or Series Expansion?
For example,
$$\lim_{x\to0}\frac{\tan x-x}{x^3}$$
$$\lim_{x\to0}\frac{\sin…
lab bhattacharjee
- 274,582
126
votes
8 answers
How to prove that eigenvectors from different eigenvalues are linearly independent
How can I prove that if I have $n$ eigenvectors from different eigenvalues, they are all linearly independent?
Corey L.
- 1,279
126
votes
3 answers
Why, historically, do we multiply matrices as we do?
Multiplication of matrices — taking the dot product of the $i$th row of the first matrix and the $j$th column of the second to yield the $ij$th entry of the product — is not a very intuitive operation: if you were to ask someone how to mutliply two…
msh210
- 3,860
126
votes
0 answers
Mondrian Art Problem Upper Bound for defect
Divide a square of side $n$ into any number of non-congruent rectangles. If all the sides are integers, what is the smallest possible difference in area between the largest and smallest rectangles?
This is known as the Mondrian Art Problem. For…
Ed Pegg
- 20,955