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King's Maths School

Challenge 423: Even meaner

A doubly mean problem!

The arithmetic mean of two numbers m and n is given by (m + n)/2.

The geometric mean of two numbers m and n is given by √(mn).

For two positive integers m and n with m ≥ n, their arithmetic mean and geometric mean are consecutive odd integers. (For example, 7 and 9 are consecutive odd integers.)

How many possible values of m and n are there if both are less than 500? What is the largest such pair?

Submit your solution

Please do send in your solution to this problem to weeklymaths@kcl.ac.uk You can scan or photograph your written work, or type your solutions. If this is your first weekly maths challenge solution, please include your year group and the name of the school you attend. We'll be happy to provide feedback on your solution, assuming that you are in year 11 or below. If you are older than this, we hope you enjoy trying the problems and reviewing your solutions against those we publish on the website.