2016/032) if $a,b,c$ are positive real numbers show that $(1+a)^7(1+b)^7(1+c)^7 > 7^7 a^4b^4c^4$

we have
$(1+a)(1+b)(1+c) = 1 + a + b + c + ab + bc + ca + abc $
$>     a + b+ c+ ab+bc+ca + abc\cdots(1)$
By am gm inequality we have $\frac{a + b+ c+ ab+bc+ca + abc}{7} >= \sqrt[7]{a^4b^4 c^4}$
or  $a + b+ c+ ab+bc+ca + abc >= 7 \sqrt[7]{a^4b^4 c^4}$
or $ 1  + a + b+ c+ ab+bc+ca + abc > 7 \sqrt[7]{a^4b^4 c^4}$
or $ (1  + a)( 1+ b)(1+ c) > 7 \sqrt[7]{a^4b^4 c^4}$ from (1)
hence  $(1  + a)^7( 1+ b)^7(1+ c)^7 > 7^7 a^4b^4 c^4$