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The greatest value that x^2 -yz, y^2 - xz, z^2 - xy can have

Source: IMO LongList 1970 - P36

May 21, 2011
inequalities proposedinequalities

Problem Statement

Let x,y,zx, y, z be non-negative real numbers satisfying x^2 + y^2 + z^2 = 5   \text{ and }   yz + zx + xy = 2. Which values can the greatest of the numbers x2−yz,y2−xzx^2 -yz, y^2 - xz and z2−xyz^2 - xy have?
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