An arithmetic sequence is a sequence of numbers for which the difference between two consecutive numbers applies terms is constant. So this is an arithmetic sequence with difference $\frac56$: $$\frac13,\frac76, 2,\frac{17}{6},\frac{11}{3},\frac92.$$The sequence of seven natural numbers $60$, $70$, $84$, $105$, $140$, $210$, $420$ has the property that the sequence inverted numbers (i.e. the row $\frac{1}{60}$, $\frac{1}{70}$, $\frac{1}{84}$, $\frac{1}{105}$, $\frac{1}{140}$, $\frac{1}{210}$,$\frac{1}{420}$ ) is an arithmetic sequence. (a) Is there a sequence of eight different natural numbers whose inverse numbers are one form an arithmetic sequence? (b) Is there an infinite sequence of distinct natural numbers whose inverses are form an arithmetic sequence?

In the plane, the point $M$ is the midpoint of a line segment $[AB]$ and $\ell$ is an arbitrary line that has no has a common point with the line segment $[AB]$ (and is also not perpendicular to $[AB]$). The points $X$ and $Y$ are the perpendicular projections of $A$ and $B$ onto $\ell$, respectively. Show that the circumscribed circles of triangle $\vartriangle AMX$ and triangle $\vartriangle BMY$ have the same radius.

The vertices of a regular $4$-gon, $6$-gon and $12$-goncan be brought together in one point to form a complete angle of $360^o$ (see figure). Determine all triples $a, b, c \in N$ with $a < b < c$ for which the angles of a regular $a$-gon, $b$-gon and $c$-gon together also form $360^o$ .

There are $12$ mathematicians living in a village, each of whom belongs to the $\sqrt2$-clan or belong to the $\pi$-clan. Moreover every mathematician's birthday is in a different month and every mathematician has an odd number of friends among them the mathematicians. We agree that if mathematician $A$ is a friend of mathematician $B$, then so is $B$ is a friend of $A$. On his birthday, every mathematician looks at which clan the majority of his friends belong to, and decides to join that clan until his next birthday. Prove that the mathematicians no longer change clans after a certain point.