# The \(16^{\mathrm{th}}\) Romanian Master of Mathematics Competition  

Day 1: 12 February, 2025, Bucharest  

Language: English  

Problem 1. 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 \(1\leq j\leq n - 1\) .  

Problem 2. 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?  

Problem 3. 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.