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As to why the question is hard - one could reformulate it as $$\frac{x+y+z+t}{4}\stackrel{?}{\ge} \sqrt[4]{\frac{(x^8 y^4 z^2 t)^{4/15}+(y^8 z^4 t^2 x)^{4/15}+(z^8 t^4 x^2 y)^{4/15}+(t^8 x^4 y^2 z)^{4 …

The "implausible" inequality is true for any $\epsilon$, for some $C$ and $K$, if $\space a>K\log(d)+K$. Taking logarithms and then derivatives with respect to $t$ one can see that the unique maximum …

This answer implements a very head on, non creative approach to the problem, as mentioned in David Speyer's comment to Gjergji Zaimi's answer. I thought it's worth having it here, but I regret unneces …

Equivalently, we want to know if $\mathrm{sqceiling}(n^2/3) > n^4/(3n^2-5) = n^2/3 + 5/9 + 25/(27n^2) + ...$ where $\mathrm{sqceiling}()$ is the function taking a real to the next exact square. Thi …

Puzzled by this still open question, I tried comparing the arithmetic mean $A(x,y)=(x+y)/2$ with a mean intermediate between a geometric-type mean $G(X)=(x^a y^{1-a}+x^{1-a} y^a)/2\;$ for $0\le a \le …

This is a pair of long comments, not an answer. Hopefully it can help towards a full answer. First, I intend to show that the separation condition on the $x_i$ vectors formulated in the question may …

For $p, x_i\in \mathbb{R}^n$, with $1\le i\le n$, define $$\theta_i=\angle(x_i,\text{span}(\{x_j \;|\; j\ne i\})),\quad \theta=\min(\{\theta_i\})$$ $$\beta_i=\frac{\pi}{2}-\angle(p,x_i),\quad\beta=\ma …

Define $P(x)$ to be positive if $P(x)>0$ for $x>0$. I can prove that a quadratic positive polynomial is the ratio of 2 polynomials with non negative coefficients, for example $\displaystyle x^2-x+1/3= …

Do there exist functions $F,G$ on $[0,1]$ with $0\le F,G< 1$, such that for all $x, y\in [0,1]$ with $x+y\le 1$, the following hold? 1) $G(x)\le x$, 2) $G(1)<1$, 3) $F(x)>0$ if $x>0$, 4) $\min(y,F …

It seems that $$\lim_{x\to\infty}\sum_{n=-\infty}^{\infty}\frac{x}{n^2+x^2}=\pi.$$ But I can't prove it. I cannot prove that the function is decreasing in $x$ either.

This is only a comment to @DimaPasechnik, but I cannot put the picture in a comment. The surface to the right of the $x=y$ is a plot of Dima's function (barring mistakes); clearly not convex.

Consider an $n\times n$ real matrix $A=(a_{ij})$ with non negative entries. Assume that - the sum of the three largest entries in each row is a constant $R$ (the same for all rows), - the sum of the …