BALKAN 2025

An integer $n > 1$ is called good if there exists a permutation $a_{1},a_{2},a_{3},\ldots ,a_{n}$ of the numbers $1,2,3,\ldots ,n$ , such that:

• $a_{i}$ and $a_{i + 1}$ have different parities for every $1 \leqslant i \leqslant n - 1$ ;

• the sum $a_{1} + a_{2} + \dots + a_{k}$ is a quadratic residue modulo $n$ for every $1 \leqslant k \leqslant n$ .

Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.

Remark: Here an integer $x$ is considered a quadratic residue modulo $n$ if there exists an integer $y$ such that $x \equiv y^{2} \pmod {n}$ .

Let $\triangle ABC$ be an acute- angled triangle with orthocentre $H$ and let $D$ be an arbitrary interior point on side $BC$ . Suppose $E$ and $F$ are points on the segments $AB$ and $AC$ respectively such that the quadrilaterals $ABDF$ and $ACDE$ are cyclic, and let $BF$ and $CE$ intersect at $P$ . Let $L$ be the point of line $HA$ such that $LC$ is tangent to the circumcircle of triangle $\triangle PBC$ at point $C$ . Let lines $BH$ and $CP$ intersect at $X$ . Prove that $D, L$ and $X$ are collinear.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that, for all real numbers $x$ and $y$,

$$f(x + y f(x)) + y = x y + f(x + y).$$

There are $n$ cities in a country, where $n \geqslant 100$ is an integer. Some pairs of cities are connected by direct (two- way) flights. For two cities $A$ and $B$ we define:

• a path between $A$ and $B$ as a sequence of distinct cities $A = C_{0}, C_{1}, \ldots , C_{k}, C_{k + 1} = B$ , $k \geqslant 0$ , such that there are direct flights between $C_{i}$ and $C_{i + 1}$ for every $0 \leqslant i \leqslant k$ ;- a long path between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has more cities;- a short path between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has fewer cities.

Assume that for any pair of cities $A$ and $B$ in the country, there exist a long path and a short path between them that have no cities in common (except $A$ and $B$ ). Let $F$ be the total number of pairs of cities in the country that are connected by direct flights. In terms of $n$ , find all possible values of $F$ .