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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 ...

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 ...

As something adjacent to the birthday problem, I found that $\frac1{\ln{365}-\ln{364}}$ is very nearly $364$ or $365$, but much closer to $364.5$. This leads to $$\ln{\frac{x+0.5}{x-0.5}}\approx \...

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 ...

I thought about the problem of finding an x such that $$ (x-6)^3 = x^{1/3} + 6 $$ for a secondary-school class, in a context where students were studying functions and their inverses. They eventually ...

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} \...

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 ...

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},\...

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 $\...

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)^{...

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 ...

I tried to come up with an integral equation for fun, and made this creature:$\def\d{\mathrm d}$ $$ f(x)-\int_x^{2x}f(t)\,\d t=0, $$ so I followed these steps: $$ \begin{aligned} &f(x)+\int_x^{2x}...

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 ...

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-...

Question: 4 points are given inside or on the boundary of a unit square. I have a conjecture that there must be 2 points at a distance $\leq 1$. Progress: I’ve found that this question is a corollary ...