Problem

Source: 2nd National Women's Contest of Mexican Mathematics Olympiad 2023 , level 1+2 p7

Tags: Mexico, algebra, inequalities



Suppose a and b are real numbers such that 0<a<b<1. Let x=1b1b+a,y=1ba1bandz=1ba1b. Show that x, y, z are always ordered from smallest to largest in the same way, regardless of the choice of a and b. Find this order among x, y, z.