For each positive integer $n$, decide whether it is possible to tile a square with exactly $n+1$ similar rectangles, each with a positive area and aspect ratio $1:n$.

A triple $(a,b,c)$ of positive integers is called strong if the following holds: for each integer $m>1$, the number $a+b+c$ does not divide $a^m+b^m+c^m$. The sum of a strong triple $(a,b,c)$ is defined as $a+b+c$. Prove that there exists an infinite collection of strong triples, the sums of which are all pairwise coprime.

Determine if there exist positive real numbers $x, \alpha$, so that for any non-empty finite set of positive integers $S$, the inequality \[\left|x-\sum_{s\in S}\frac{1}{s}\right|>\frac{1}{\max(S)^\alpha}\]holds, where $\max(S)$ is defined as the maximum element of the finite set $S$.

A (not necessarily regular) tetrahedron $A_1A_2A_3A_4$ is given in space. For each pair of indices $1\leq i<j\leq 4$, an ellipsoid with foci $A_i,A_j$ and string length $\ell_{ij}$, for positive numbers $\ell_{ij}$, is given (in all 6 ellipsoids were built). For each $i=1,2$, a pair of points $X_i\neq X'_i$ was chosen so that $X_i, X'_i$ both belong to all three ellipsoids with $A_i$ as one of their foci. Prove that the lines $X_1X'_1, X_2X'_2$ share a point in space if and only if \[\ell_{13}+\ell_{24}=\ell_{14}+\ell_{23}\]Remark: An ellipsoid with foci $P,Q$ and string length $\ell>|PQ|$ is defined here as the set of points $X$ in space for which $|XQ|+|XP|=\ell$.