Most active questions
1,912 questions from the last 30 days
26
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6
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Why is this angle always less than $49°$?
I was drawing this configuration in GeoGebra, repeating it dozens of times, always considering any triangle $ABC$ with the centroid $G$, while maintaining the $25°$ angle.
An observation then occurred ...
- 1,748
40
votes
7
answers
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How far can an infinite number of unit length planks bridge a right-angled gap?
Problem: You are standing at the origin of an infinite flat earth. One quadrant of the earth is gone, leaving an infinite abyss. You have an infinite supply of unit-length zero-width planks. How far ...
17
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6
answers
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What is the correct definition of a limit point in real analysis?
This question relates two (seemingly) conflicting definitions of Limit Points in real analysis.
The definition of limit points and closed sets from my notes are written as:
A much more general ...
- 561
5
votes
8
answers
949
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How to find the angle created when joining two vertices of four adjacent squares
The text of the problem, taken from the picture:
Optional challenge (required to understand it once we go over it):
4 squares are drawn and two segments are drawn between vertices. Find the value of ...
- 1,026
11
votes
5
answers
2k
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Why Is the Dual Basis Mathematically Unavoidable?
In linear algebra and differential geometry, we always introduce the dual space and the dual basis, defined as linear functionals that extract the components of vectors. But I still do not understand ...
- 1,955
2
votes
6
answers
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Example that the union of an infinite number of closed sets is not necessarily closed.
Example. Let $I_n = [1/n, 1]$, which is clearly closed, and consider
$$S=\bigcup_{n=2}^{\infty}I_n=[1/2,1]\cup[1/3,1]\cup[1/4,1]\cdots\tag{1}$$
This is the set
$$S=\bigg\{x\bigg\lvert x\in \mathbb{R},\...
- 561
12
votes
5
answers
553
views
Non-recursive, explicit rational sequence that converges to $\sqrt{2}$
Is there a non-recursive, explicit sequence of rational numbers that has $\sqrt{2}$ as a limit?
I know of rational sequences such as $x_{n+1}=(x_n+2/x_{n})/2$ and $q_n=[10^n\sqrt{2}]/10^n$ that have $\...
16
votes
3
answers
2k
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Why does the leading coefficient of a quadratic trinomial resemble some sort of a slope?
It's known that a unique parabola of the form $y=ax^{2}+bx+c$ exists for any three distinct points, provided that the points are non-collinear and their $x$ coordinates are distinct.
Consider the ...
- 183
8
votes
9
answers
568
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Evaluate $\int_0^{\frac\pi2}\frac{\mathrm dx}{a\sin ^2x+b\cos ^2x}$ without using its antiderivative
Find the value of
$$\int_0^{\frac\pi2} \frac{1}{a \sin ^2x+b \cos ^2x} \, \mathrm dx,$$
where $a$, $b>0$.
The corresponding indefinite integral evaluates to
$$\int \frac{1}{a \sin ^2x+b \cos ^2x} \...
- 5,070
14
votes
2
answers
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Is this formula for cosine correct? (And is it significant) [closed]
I have recently found this formula for cosine that seems to work, yet I am unable to prove that it works. The formula in question is
$$\lim _{n\rightarrow \infty }\sum _{k=0}^{2n}\sum _{j=0}^{k}( -1)^{...
- 167
23
votes
1
answer
2k
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Continuous extensions of GCD to $\Bbb R^{+}\!\!\times \Bbb R^{+}\!$ still commutative and distributive
Question. Is there an extension of the GCD function? Since the concept of divisibility breaks down in $\mathbb{R}$, is there an established analytic interpretation of $\gcd(m, n)$ for non-integer ...
- 5,309
9
votes
3
answers
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Are all natural numbers (except 1 and 2) part of at least one primitive pythagorean triplet?
I'm referring to triplets that are primitive and have only natural numbers as their terms.
What I've found-
$k^2 - (k - 1)^2 = 2k - 1$, and
$(2k - 1)^{1/2}$ generates all odd natural numbers.
Hence, ...
- 93
13
votes
2
answers
2k
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Is the transfer principle useless?
In this MathOverflow thread, a comment states:
Any proof using a transfer principle can be rewritten without it, so in some sense it can't play an “essential” role in a proof.
Is there a known ...
- 413
7
votes
5
answers
388
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Prove $\int_0^af(x)\mathrm dx\ge a\int_0^1f(x)\mathrm dx$ for decreasing function
Let $f$ be a decreasing function on $[0,1]$ and $a\in(0,1)$. Prove that
$$\int_0^af(x)\mathrm dx\ge a\int_0^1f(x)\mathrm dx.$$
This would be quite obvious if $f$ were continuous. But for non-...
- 5,070
11
votes
3
answers
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Can there be a continuous function with infinite derivative everywhere?
Let $f:\mathbb{R}\to\mathbb{R}$ be continuous. For all nowhere-differentiable examples that I know of, for each $a\in\mathbb{R}$ there exist sequences $x_n\to a$ and $y_n\to a$ such that
$$\frac{f(...
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