Romanian Master of Mathematics (RMM) 2025 - THE PROBLEMS

Problem 1

Let \(n\) be an integer, and let \(P_1, P_2, \dots, P_n\) be distinct points in the plane such that the distances between the points are pairwise different. Define \(d(P_i, P_j)\) to be the distance between points \(P_i\) and \(P_j\). For each point \(P_i\), define \(r(P_i)\) to be the 10th smallest of the distances from \(P_i\) to \(P_j\), excluding \(j=i\). Suppose that for all \(i\) and \(j\) satisfying \(r(P_i) < r(P_j)\), we have \(i < j\). Prove that \(i \le 10\) for all \(i\) in the range \(1 \le i \le n\).

Proposed by Morteza Saghafian, Iran

Problem 2

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

Proposed by Pavel Kozlov, Russia

Problem 3

Fix an integer \(k \ge 2\). Determine the smallest positive integer \(m\) satisfying the following condition:

For any tree \(T\) with \(k\) vertices \(v_1, v_2, \dots, v_k\) and any pairwise distinct complex numbers \(z_1, z_2, \dots, z_k\), there is a polynomial \(P(x_1, x_2, \dots, x_k)\) with complex coefficients of total degree at most \(m\) such that for all \(i, j\) satisfying \(1 \le i < j \le k\), we have \(P(z_1, z_2, \dots, z_k) = 0\) if and only if there is an edge in \(T\) joining \(v_i\) to \(v_j\).

Proposed by Andrei Chiriță, Romania

Day 2 - February 13, 2025

Problem 4

Let \(\mathbb{Z}\) denote the set of integers and \(\mathbb{Z}_{\ge -1}\) be the set of integers that are at least \(-1\). Fix a positive integer \(k\). Determine all functions \(f: \mathbb{Z}_{\ge -1} \to \mathbb{Z}\) satisfying \(f(n) f(n+2) = f(n+1)^2 - 1\), for all \(n \in \mathbb{Z}_{\ge -1}\).

Problem 5

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 centered at \(O\) with radius \(OA\). The line \(AH\) and the circle of radius \(AH\) centered at \(A\) cross \(\Gamma\) at \(D\) and \(E\), respectively. Prove that the circle on diameter \(DE\) and circle \(\Gamma\) are concurrent.

Proposed by Romania, Radu-Andrew Lecoiu

Problem 6

Let \(n\) and \(k\) be integers greater than 1. Consider \(nk\) pairwise disjoint sets \(A_{i,j}\), where \(1 \le i \le n\) and \(1 \le j \le k\); each of these sets has exactly 2 elements, one of which is red and the other is blue. Let \(\mathcal{F}\) be the family of all subsets \(S\) of \(\bigcup_{i=1}^n \bigcup_{j=1}^k A_{i,j}\) such that, for every \(i \in \{1, \dots, n\}\) and every \(j \in \{1, \dots, k\}\), the intersection \(S \cap A_{i,j}\) is monochromatic; the empty set is also monochromatic. Determine the largest cardinality of a subfamily \(\mathcal{F}' \subseteq \mathcal{F}\), no two sets of which are disjoint.

Proposed by Russia, Andrew Kupavskii and Maksim Turevskii