Let $n > 10$ be an integer, and let $A_1, A_2, \dots, A_n$ be distinct points in the plane such that the distances between the points are pairwise different. Define $f_{10}(j, k)$ to be the 10th smallest of the distances from $A_j$ to $A_1, A_2, \dots, A_k$, excluding $A_j$ if $k \geq j$. Suppose that for all $j$ and $k$ satisfying $11 \leq j \leq k \leq n$, we have $f_{10}(j, j - 1) \geq f_{10}(k, j - 1)$. Prove that $f_{10}(j, n) \geq \frac{1}{2} f_{10}(n, n)$ for all $j$ in the range $1 \leq j \leq n - 1$.

Consider an infinite sequence of positive integers $a_1, a_2, a_3, \dots$ such that $a_1 > 1$ and $(2^{a_n} - 1)a_{n+1}$ is a square for all positive integers $n$. Is it possible for two terms of such a sequence to be equal?

Fix an integer $n \geq 3$. Determine the smallest positive integer $k$ satisfying the following condition: For any tree $T$ with vertices $v_1, v_2, \dots, v_n$ and any pairwise distinct complex numbers $z_1, z_2, \dots, z_n$, there is a polynomial $P(X, Y)$ with complex coefficients of total degree at most $k$ such that for all $i \neq j$ satisfying $1 \leq i, j \leq n$, we have $P(z_i, z_j) = 0$ if and only if there is an edge in $T$ joining $v_i$ to $v_j$. Note, for example, that the total degree of the polynomial $$ 9X^3Y^4 + XY^5 + X^6 - 2 $$is 7 because $7 = 3 + 4$. Proposed by Andrei Chiriță, Romania

Let $\mathbb{Z}$ denote the set of integers and $S \subset \mathbb{Z} $ be the set of integers that are at least $10^{100}$. Fix a positive integer $c$. Determine all functions $f: S \rightarrow \mathbb{Z} $ satisfying $f(xy+c)=f(x)+f(y)$, for all $x,y \in S$

Let triangle $ABC$ be an acute triangle with $AB<AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. Let $\Gamma$ be the circle $BOC$. The line $AO$ and the circle of radius $AO$ centered at $A$ cross $\Gamma$ at $A’$ and $F$, respectively. Prove that $\Gamma$ , the circle on diameter $AA’$ and circle $AFH$ are concurrent. Proposed by Romania, Radu-Andrew Lecoiu

Let $k$ and $m$ be integers greater than $1$. Consider $k$ pairwise disjoint sets $S_1,S_2, \cdots S_k$; each of these sets has exactly $m+1$ elements, one of which is red and the other $m$ are all blue. Let $\mathcal{F}$ be the family of all subsets $F$ of $S_1 \bigcup S_2\bigcup \cdots S_k$ such that, for every $i$ , the intersection $F \bigcap S_i$ is monochromatic; the empty set is also monochromatic. Determine the largest cardinality of a subfamily $\mathcal{G} \subseteq \mathcal{F}$, no two sets of which are disjoint. Proposed by Russia, Andrew Kupavskii and Maksim Turevskii