[JEE Advanced 2004] if a b c 0 prove a 1 7 b 1 7 c 1
If $a,b,c>0$, prove $(a+1)^7(b+1)^7(c+1)^7>7^7\,a^4b^4c^4$.
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VAVidaara Admin
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· 4d ago
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Answer: Proved.
By AM–GM on the seven parts $\tfrac13,\tfrac13,\tfrac13,\tfrac a4,\tfrac a4,\tfrac a4,\tfrac a4$, $(1+a)^7\ge\frac{7^7a^4}{3^3\cdot4^4}$. Multiplying the three such bounds and using $\frac{7^{14}}{(27\cdot256)^3}>1$ gives the strict inequality.
JEE Advanced 2004 · Quadratic Equations and Inequations — verified solution by the Vidaara Team.
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