RMM 2025

Let $n > 10$ be an integer, and let $A_{1},A_{2},\ldots ,A_{n}$ be distinct points in the plane such that the distances between the points are pairwise different. Define $f_{10}(j,k)$ to be the $10^{\mathrm{th}}$ smallest of the distances from $A_{j}$ to $A_{1},A_{2},\ldots ,A_{k}$ , 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 $a_{1},a_{2},a_{3},\ldots$ 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 $v_{1},v_{2},\ldots ,v_{n}$ and any pairwise distinct complex numbers $z_{1},z_{2},\ldots ,z_{n}$ , 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

$$9X^{3}Y^{4} + XY^{5} + X^{6} - 2$$

is 7 because $7 = 3 + 4$ .

Each problem is worth 7 marks. Time allowed: $4\frac{1}{2}$ hours.

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

Let $A B C$ be an acute triangle with $A B< A C$ , and let $H$ and $o$ be its orthocentre and circumcentre, respectively. Let $\Gamma$ be the circumcircle of triangle $B O C$ . Circle $\Gamma$ intersects line $A O$ at points $o$ and $A^{\prime}$ , and $\Gamma$ intersects the circle of radius $A O$ with centre $A$ at points $o$ and $F$ . Prove that the circle which has diameter $A A^{\prime}$ , the circumcircle of triangle $A F H$ and $\Gamma$ pass through a common point.

Let $k$ and $m$ be integers greater than 1. Consider $k$ pairwise disjoint sets $S_{1}, S_{2}, \ldots , S_{k}$ , each of which has exactly $m + 1$ elements: one red and $m$ blue. Let $\mathcal{F}$ be the family of all subsets $T$ of $S_{1} \cup S_{2} \cup \dots \cup S_{k}$ such that, for every $i$ , the intersection $T \cap S_{i}$ is monochromatic. Determine the largest possible number of sets in a subfamily $\mathcal{G} \subseteq \mathcal{F}$ such that no two sets in $\mathcal{G}$ are disjoint.

A set is monochromatic if all of its elements have the same colour. In particular, the empty set is monochromatic.

Each problem is worth 7 marks. Time allowed: $4\frac{1}{2}$ hours.