Each year since 2008 I have been a tutor at the UKMT National Mathematics Summer School. This is a five-day residential summer school for students aged 15 and 16. It is a nice opportunity to introduce able students to mathematics beyond the school curriculum, and especially how to think like a mathematician. Topics I have taught include inequalities, cyclic quadrilaterals, the pigeonhole principle, and graph theory.
During my PhD in Orsay, I participated as a jury member and team leader in the Tournoi Français de Jeunes Mathématicien(ne)s (2012 and 2013) and in the International Tournament of Young Mathematicians (2012). In these competitions, students in their final year of secondary school work in teams of five over a period of several months to solve open-ended problems set by the organisers, and then present their work at a regional, national or international event.