It can be also proved by cauchy-schwartz for the pairs of (): 1,6 2,5 3,4

$\text{Solutions use the Hölder inequalities:}$ $LHS=(x^2+y+1)(x^2+1+z)(1+y^2+z)(x+y^2+1)(x+1+z^2)(1+y+z^2)$ $\geq (\sqrt[6]{x^2.x^2.x.x}+\sqrt[6]{y.y^2.y^2.y}+\sqrt[6]{z.z.z^2.z^2})^6$ $=(x+y+z)^6$

I am just completing den_thewhitelion's proof $(x^2+y+1)(z^2+y+1^2)=((x)^2+(\sqrt y)^2+(1)^2)((1)^2+((\sqrt y)^2+(z)^2) \geq (x+y+z)^2$ by the Cauchy-Schwarz inequality. Similarly, $(x^2+z^2+1)(y^2+z^2+1) \geq (x + y + z)^2$ $(y^2+x+1)(z^2+x+1) \geq (x+y+z)^2$ Multiplying the above results we get the desired conclusion, $(x^2+y+1)(x^2+z+1)(y^2+x+1)(y^2+z+1)(z^2+x+1)(z^2+y+1)\geq (x+y+z)^6$

Is there any link to the BMO/JBMO Shortlist?

@User335559 here is your link for JBMO https://artofproblemsolving.com/community/c3240_jbmo_shortlists . BTW can anybody post the Holder inequality???

sttsmet wrote: @User335559 here is your link for JBMO https://artofproblemsolving.com/community/c3240_jbmo_shortlists . BTW can anybody post the Holder inequality??? $(x_1^n+y_1^n+z_1^n+w_1^n+\cdots)(x_2^n+y_2^n+z_2^n+w_2^n+\cdots)\cdots (x_n^n+y_n^n+z_n^n+w_n^n+\cdots) \geq (x_1x_2\cdots x_n+y_1y_2\cdots y_n+z_1z_2\cdots z_n+w_1w_2\cdots w_n+\cdots)^n$

easy-proof wrote: sttsmet wrote: @User335559 here is your link for JBMO https://artofproblemsolving.com/community/c3240_jbmo_shortlists . BTW can anybody post the Holder inequality??? $(x_1^n+y_1^n+z_1^n+w_1^n+\cdots)(x_2^n+y_2^n+z_2^n+w_2^n+\cdots)\cdots (x_n^n+y_n^n+z_n^n+w_n^n+\cdots) \geq (x_1x_2\cdots x_n+y_1y_2\cdots y_n+z_1z_2\cdots z_n+w_1w_2\cdots w_n+\cdots)^n$ basically CS but generalized

easy-proof wrote: sttsmet wrote: @User335559 here is your link for JBMO https://artofproblemsolving.com/community/c3240_jbmo_shortlists . BTW can anybody post the Holder inequality??? $(x_1^n+y_1^n+z_1^n+w_1^n+\cdots)(x_2^n+y_2^n+z_2^n+w_2^n+\cdots)\cdots (x_n^n+y_n^n+z_n^n+w_n^n+\cdots) \geq (x_1x_2\cdots x_n+y_1y_2\cdots y_n+z_1z_2\cdots z_n+w_1w_2\cdots w_n+\cdots)^n$ Really helpful! Thanks! I suspected that this exersice would be solved that way, but I didnt knew that we could generalize C-S