Answer: Option A : A, B, C, and D (a), (b), (c), (d) (a) a4+b4+c43>(a+b+c3)4 or (a+b+c3)(a+b+c3)3>a+b+c3[(abc)13]3; ∵A.M.>G.M. or >(a+b+c3) ∴a4+b4+c4>abc(a+b+c) (b) As above (c) (a5+b5+c53)>(a+b+c3)5 or >(a+b+c3)3(a+b+c3)2 or >[(abc)13]3(a2+b2+c2+2ab+2bc+2ca9) But we know that a2+b2+c2>ab+bc+ca by result of two by two rule. ∴a5+b5+c53>abc3(ab+bc+ca)9 etc. (d) a8+b8+c8>a2b2c2(bc+ca+ab) Now a8+b8+c83>(a+b+c3)8 or >(a+b+c3)6(a+b+c3)2>[(abc)13]6[a2+b2+c2+2ab+2bc+2ca9] ∵A.M.>G.M But by two by two rule a2+b2+c2>ab+bc+ca ∴a8+b8+c83>a2b2c2(3ab+3bc+3ca)9 ∴a8+b8+c8>a2b2c2(ab+bc+ca) or a8b8c8a3b3c3>ab+bc+caabc or >1a+1b+1c (e) b2+c22>(b+c2)2∴b2+c2b+c>b+c2 Write similar inequalities and add.
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