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infinite triples of positive rational, 16xyz = (x + y)^2(x + z)^2, x+y+z =M, max

Source: 48th Austrian Mathematical Olympiad National Competition (Final Round, part 2) 25th May 2017 p4

May 25, 2019

maximuminequalitiesthree variable inequalityrationalalgebra

Problem Statement

(a) Determine the maximum

M

x+y +z

where

x, y

and

z

are positive real numbers with

16xyz = (x + y)^2(x + z)^2

. (b) Prove the existence of infinitely many triples

(x, y, z)

of positive rational numbers that satisfy

16xyz = (x + y)^2(x + z)^2

and

x + y + z = M

.
Proposed by Karl Czakler