Maths Olympiad Prep

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Algebra Difficulty 6.2 National olympiad Prove it

Example 19 (Self-created problem, 2007.01.21) Let a,b,c,k∈Ra, b, c, k \in \mathbf{R}, then
(k2+a2)(k2+b2)(k2+c2)⩾2k3(a2b+b2c+c2a+abc)\left(k^{2}+a^{2}\right)\left(k^{2}+b^{2}\right)\left(k^{2}+c^{2}\right) \geqslant 2 k^{3}\left(a^{2} b+b^{2} c+c^{2} a+a b c\right)

Equality in (31) holds if and only if k=a=b=ck=a=b=c or k=0,abc=0k=0, a b c=0.

Solution

Prove that the left side - right side =(k3−abc)2+k2a2(k−b)2+k2b2(k−c)2+=\left(k^{3}-a b c\right)^{2}+k^{2} a^{2}(k-b)^{2}+k^{2} b^{2}(k-c)^{2}+ k2c2(k−a)2⩾0k^{2} c^{2}(k-a)^{2} \geqslant 0

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.