Question

# If a,b and c are three positive real numbers, which one of the following are true?

A
a2+b2+c2bc+ca+ab
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B
a3+b3+c33abc
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C
(b+c)(c+a)(a+b)8abc
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D
bca+cab+abca+b+c
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Solution

## The correct options are A a2+b2+c2≥bc+ca+ab B a3+b3+c3≥3abc C (b+c)(c+a)(a+b)≥8abc D bca+cab+abc≥a+b+cWe have a2+b22≥√a2b2=ab,b2+c22≥√b2c2=bc and c2+a22≥√c2a2=ca Adding these inequalities, we get a2+b2+c2≥bc+ca+abAlso a3+b3+c33≥(a3b3c3)1/3⇒a3+b3+c3≥3abc Next, since (b+c)/2≥√bc, (c+a)/2≥√ca and (a+b)/2≥√ab, we get(b+c2)(c+a2)(a+b2)≥√a2b2c2⇒(b+c)(c+a)(a+b)≥8abcLastly, we have12(bca+cab)≥√bca.cab=c,12(cab+abc)≥√cab.abc=b,and 12(abc+bca)≥√abc.bca=b Adding the above inequalities we get bca+cab+abc≥a+b+c

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