What is a sharp constant?

The term sharp means we can find a best bound, that cannot be improved by a better number. I.e., to say a function is bounded is to say there exists an $M$ such that $|f(x)|\le M$. To say a particular number is sharp, say $|f(x)|\le 3$, means that there is no number smaller than 3 we could put there and have it be true.

Envelope of Projectile Trajectories

is the inverse of a absolutely continuous function with almost everywhere positive derivation absolutely continuous?

A curious algebraic fraction that converges to $\frac{\sqrt{2}}{2}$

Does there exist a polynomial $p(x) \in \mathbb C[x]$ such that $p(x) \notin \mathbb R[x]$ and $p(x)p(-x)=p(x^2)$?

Irreducible polynomial over $\mathbb{Q}$ implies polynomial is irreducible over $\mathbb{Z}$

Show that:$\sum\limits_{n=1}^{\infty}{n\over (4n^2-1)(16n^2-1)}={1\over 12}(1-\ln{2})$

$x^5 + y^2 = z^3$

What is the theory behind rigorous hypothesis testing?

Simple proof that if $A^n=I$ then $\mathrm{tr}(A^{-1})=\overline{\mathrm{tr}(A)}$

Prove that $\int_0^\infty\,\frac{\sin(kx)}{x(x^2+1)}\,\text{d}x=\frac{\pi}{2}\,\left(1-\exp(-k)\right)$ for all $k\in\mathbb{R}_{\ge0}$.

Finding all possible proofs

How to evaluate $\int \frac{dx}{\sin(\ln(x))}$?