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[sum(x-y)^7-(x-y)(y-z)(z-x)((x-y)^4+(y-z)^4+(z-x)^4)/sum(x-y)^5 >=3

Source: 41st Austrian Mathematical Olympiad National Competition (Final Round, part 2) 2nd June 2010 p1

September 5, 2019
algebrainequalities

Problem Statement

Show that (x−y)7+(y−z)7+(z−x)7−(x−y)(y−z)(z−x)((x−y)4+(y−z)4+(z−x)4)(x−y)5+(y−z)5+(z−x)5≥3\frac{(x - y)^7 + (y - z)^7 + (z - x)^7 - (x - y)(y - z)(z - x) ((x - y)^4 + (y - z)^4 + (z - x)^4)} {(x - y)^5 + (y - z)^5 + (z - x)^5} \ge 3 holds for all triples of distinct integers x,y,zx, y, z. When does equality hold?
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