exam: RMM

problem: 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 \(x, y \in S\) .

problem_match: Problem 4.


problem: 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.

problem_match: Problem 5.


problem:
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.

problem_match: Problem 6.