Squares $ABB_{1}A_{2}$ and $BCC_{1}B_{2}$ are externally drawn on the hypotenuse $AB$ and on the leg $BC$ of a right triangle $ABC$ . Show that the lines $CA_{2}$ and $AB_{2}$ meet on the perimeter of a square with the vertices on the perimeter of triangle $ABC .$

Let $f$ be an increasing mapping from the family of subsets of a given finite set $H$ into itself, i.e. such that for every $X \subseteq Y\subseteq H$ we have $f (X )\subseteq f (Y )\subseteq H .$ Prove that there exists a subset $H_{0}$ of $H$ such that $f (H_{0}) = H_{0}.$

Find all sequences $x_{1},x_{2},...,x_{n}$ of distinct positive integers such that $\frac{1}{2}=\sum_{i=1}^{n}\frac{1}{x_{i}^{2}}$.

Does there exist a sequence of $2005$ consecutive positive integers that contains exactly $25$ prime numbers?

Let $F_{n}$ be the $n-$ th Fibonacci number (where $F_{1}= F_{2}= 1$). Consider the functions $f_{n}(x)=\parallel . . . \parallel |x|-F_{n}|-F_{n-1}|-...-F_{2}|-F_{1}|, g_{n}(x)=| . . . \parallel x-1|-1|-...-1|$ ($F_{1}+...+F_{n}$ one’s). Show that $f_{n}(x) = g_{n}(x)$ for every real number $x.$

There are seven rods erected at the vertices of a regular heptagonal area. The top of each rod is connected to the top of its second neighbor by a straight piece of wire so that, looking from above, one sees each wire crossing exactly two others. Is it possible to set the respective heights of the rods in such a way that no four tops of the rods are coplanar and each wire passes one of the crossings from above and the other one from below?