Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. Proposed by Vidan Govedarica, Serbia

A rectangle is partitioned into finitely many small rectangles. We call a point a cross point if it belongs to four different small rectangles. We call a segment on the obtained diagram maximal if there is no other segment containing it. Show that the number of maximal segments plus the number of cross points is $3$ more than the number of small rectangles.

In a convex quadrilateral $ABCD$ with $\angle ABC = \angle ADC = 135^\circ$, points $M$ and $N$ are taken on the rays $AB$ and $AD$ respectively such that $\angle MCD = \angle NCB = 90^\circ$. The circumcircles of triangles $AMN$ and $ABD$ intersect at $A$ and $K$. Prove that $AK \perp KC.$

Let $A$ and $B$ be two fixed points in the plane. Consider all possible convex quadrilaterals $ABCD$ with $AB = BC, AD = DC$, and $\angle ADC = 90^\circ$. Prove that there is a fixed point $P$ such that, for every such quadrilateral $ABCD$ on the same side of $AB$, the line $DC$ passes through $P.$

Let $\delta$ be a symbol such that $\delta \neq 0$ and $\delta^2 = 0$. Define $\mathbb R[\delta] = \{a + b \delta | a, b \in \mathbb R\}$, where $a+ b \delta = c+ d \delta$ if and only if $a = c$ and $b = d$, and define \[(a + b \delta) + (c + d \delta) = (a + c) + (b + d) \delta,\]\[(a + b \delta) \cdot (c + d \delta) = ac + (ad + bc) \delta.\] Let $P(x)$ be a polynomial with real coefficients. Show that $P(x)$ has a multiple real root if and only if $P(x)$ has a non-real root in $\mathbb R[\delta].$

Let $G$ be a simple graph with $100$ edges on $20$ vertices. Suppose that we can choose a pair of disjoint edges in $4050$ ways. Prove that $G$ is regular.