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.