Maths Olympiad Prep

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Algebra Difficulty 6.1 National olympiad Prove it

Example 1.10.11. Let a,b,c,da, b, c, d be non-negative real numbers. Prove that
4(a3+b3+c3+d3)+15(abc+bcd+cda+dab)≥(a+b+c+d)34\left(a^{3}+b^{3}+c^{3}+d^{3}\right)+15(a b c+b c d+c d a+d a b) \geq(a+b+c+d)^{3}

Solution

SOLUTION. Because this is a third-degree symmetric inequality of four variables, according to the generalization of the SD3 theorem, it suffices to check this inequality in case a=b=c=d=1a=b=c=d=1 or a=0,b=c=d=1a=0, b=c=d=1 or a=b=0,c=d=1a=b=0, c=d=1 or a=b=c=0,d=1a=b=c=0, d=1. They are all obvious, so we have the desired result. The equality holds for a=b=c=da=b=c=d or a=b=c,d=0a=b=c, d=0 up to permutation.

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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.