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(a -1)(b -1)+(a -1)(c -1)+(b -1)(c -1) >= 6 2011 Romania NMO VIII p2

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August 15, 2024
algebrainequalities

Problem Statement

Let a,b,ca, b, c be distinct positive integers.
a) Prove that a2b2+a2c2+b2c2≥9a^2b^2 + a^2c^2 + b^2c^2 \ge 9.
b) if, moreover, ab+ac+bc+3=abc>0,ab + ac + bc +3 = abc > 0, show that (a−1)(b−1)+(a−1)(c−1)+(b−1)(c−1)≥6.(a -1)(b -1)+(a -1)(c -1)+(b -1)(c -1) \ge 6.
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