Prove that for all $a,b,c \in \mathbb{R}^+_0$ we have \[\frac{a}{bc}+\frac{b}{ac}+\frac{c}{ab} \ge \frac2a+\frac2b-\frac2c\] and determine when equality occurs.

Prove that there are no perfect squares in the array below: \[\begin{array}{cccc}11&111&1111&...\\22&222&2222&...\\33&333&3333&...\\44&444&4444&...\\55&555&5555&... \\66&666&6666&...\\77&777&7777&...\\88&888&8888&...\\99&999&9999&...\end{array}\]

Is it possible to number the $8$ vertices of a cube from $1$ to $8$ in such a way that the value of the sum on every edge is different?

Two congruent right-angled isosceles triangles (with baselength 1) slide on a line as on the picture. What is the maximal area of overlap?