I argue for the inclusion of topics in high school mathematics curricula that are traditionally reserved for high achieving students preparing for mathematical contests. These include the arithmetic mean - geometric mean inequality which has many practical applications in mathematical modelling. The problem of extremalising functions of more than one independent variable usually falls outside the secondary school curriculum and is generally discussed in topics on single and multivariable calculus in undergraduate mathematics. In particular the method of Lagrange multipliers is efficient in solving a wide class of problems where a function of many variables is to be maximised or minimised subject to a number of constraints on the variables. We explore some non-calculus approaches to solving some classes of problems. The well known arithmetic mean - geometric mean inequality is discussed and is shown to be useful in solving types of problems involving more than two variables but in a manner accessible to high school students. The method involves writing the constraint function as a sum of quantities with a finite sum and whose product must be maximised or minimised in the target function. This is a powerful idea that should be be exploited in high school mathematics.