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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 not a full answer, but a pair of soft arguments suggesting that $A$ is dense in $[0, +\infty)$. First Argument Given any triple $(a,b,c)$, let $\displaystyle r(a,b,c)=\frac{c}{\text{rad}(abc) …

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 …

Let $A=\{mn(m+n)\mid n,m\in \mathbb{N}_0\}$. Sorted, this is OEIS sequence A088915. What is its asymptotic behavior? It seems approximately $a(n)=O(n^{1.5})$, but not quite. I'm actually even more in …

The cops should always win. Here is a sketch of a proof. One cop (for short) trying to catch the thief crossing a segment: the cop can run on the segment at top speed towards the thief (provided the …

Given a set $S$, a function $M: S\times S \rightarrow S$ is a mean if it satisfies the properties: $M(a,a)=a\qquad$ (identity) $M(a,b)=M(b,a)\qquad$ (commutativity). and possibly $M(M(a,b),M(a,c …

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.

Claim. The thief $T$ can escape if $C$ is a circle, with a simple strategy of dribbling left and right each policeman at a time in such a way that he is left out of reach of the thief no matter what t …