The Hardest Question in the BMO1. (2020)
Accessible for all HS grades
The British Mathematics Olympiad 1 is a competition only accessible to the top 1000 scorers of the Senior Mathematics Challenge. It is of a great difficulty which is evident by the fact that you are given 3.5 hours to solve a mere 6 number of questions. These six questions grow in difficulty as you complete one after another and this article focuses on the hardest question of the paper.
The first step to solving this behemoth of a question is to write this equation into L (Left-Hand-Side), M (Middle) and R (Right-Hand-Side). This will make it easier when managing the expressions given in the question. Afterwards, simplify the equation by taking g(m) for f(m,-m). This will leave you with equation for L = R as:
You can simplify the equation to the one below by first expanding, moving the m on the Left hand side to the right, then crossing out values. This is shown in the image below:
The equation we obtain at the end contains no variables with exponents, creating a linear equation. Thus we know that g(0)=0 and g(m) = am where a is an integer. We can then rule out that a is an odd number after taking m=1 and n=0. Substituting as 2b + 1 to represent that a will always be an odd number, you will get the equation below:
This equation shows all the possible “good functions” and that it can work for any value of b. The difficulty of this problem is as so due to two things: that it is hard to see the recurrence relation that can be used to rule things about f(m,n); and that the substitutions needed to solve the problem are as equally hard to see.
