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4

Part of 2017 Federal Competition For Advanced Students, P2

Problems(1)

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

5/25/2019
(a) Determine the maximum MM of x+y+zx+y +z where x,yx, y and zz are positive real numbers with 16xyz=(x+y)2(x+z)216xyz = (x + y)^2(x + z)^2. (b) Prove the existence of infinitely many triples (x,y,z)(x, y, z) of positive rational numbers that satisfy 16xyz=(x+y)2(x+z)216xyz = (x + y)^2(x + z)^2 and x+y+z=Mx + y + z = M.
Proposed by Karl Czakler
maximuminequalitiesthree variable inequalityrationalalgebra