MathDB

2016 preRMO p16, average of arithmetic and geometric mean, is perfect square

Source:

August 9, 2019

algebrageometric meanarithmetic meanPerfect Squarenumber theory

Problem Statement

For positive real numbers

x

and

y

, define their special mean to be average of their arithmetic and geometric means. Find the total number of pairs of integers

(x, y)

, with

x \le y

, from the set of numbers

\{1,2,...,2016\}

, such that the special mean of

x

and

y

is a perfect square.