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Problem 3 of First round - inequalities

Source: VII International Festival of Young Mathematicians Sozopol, 2016 Theme for 10-12 grade

April 21, 2022
algebrainequalities

Problem Statement

Let x≤y≤zx\leq y\leq z be real numbers such that x+y+z=12x+y+z=12, x2+y2+z2=54x^2+y^2+z^2=54. Prove that: a) x≤3x\leq 3 and z≥5z\geq 5 b) xyxy, yzyz, zx∈[9,25]zx\in [9,25]
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