Question Number 7507 by Master Moon last updated on 01/Sep/16
![Given r, t, j >0 and n≥1; Prove that (([Σ_(p_1 =1) ^n (p_1 ^r +r)+Σ_(p_2 =1) ^n (p_2 ^t +t)]^2 )/(n^2 [(n!)^((r+t)/n) +r(n!)^(t/n) +t(n!)^(r/n) +rt])) +(([Σ_(p_2 =1) ^n (p_2 ^t +t)+Σ_(p_3 =1) ^n (p_3 ^j +j)]^2 )/(n^2 [(n!)^((t+j)/n) +t(n!)^(j/n) +j(n!)^(t/n) +tj])) ≥ 8 By: Mr. Chheang Chantria](Q7507.png)
$$\boldsymbol{{Given}}\:\boldsymbol{{r}},\:\boldsymbol{{t}},\:\boldsymbol{{j}}\:>\mathrm{0}\:\boldsymbol{{and}}\:\boldsymbol{{n}}\geqslant\mathrm{1};\:\boldsymbol{{Prove}}\:\boldsymbol{{that}} \\ $$ $$\frac{\left[\underset{\boldsymbol{{p}}_{\mathrm{1}} =\mathrm{1}} {\overset{\boldsymbol{{n}}} {\sum}}\left(\boldsymbol{{p}}_{\mathrm{1}} ^{\boldsymbol{{r}}} +\boldsymbol{{r}}\right)+\underset{\boldsymbol{{p}}_{\mathrm{2}} =\mathrm{1}} {\overset{\boldsymbol{{n}}} {\sum}}\left(\boldsymbol{{p}}_{\mathrm{2}} ^{\boldsymbol{{t}}} +\boldsymbol{{t}}\right)\right]^{\mathrm{2}} }{\boldsymbol{{n}}^{\mathrm{2}} \left[\left(\boldsymbol{{n}}!\right)^{\frac{\boldsymbol{{r}}+\boldsymbol{{t}}}{\boldsymbol{{n}}}} +\boldsymbol{{r}}\left(\boldsymbol{{n}}!\right)^{\frac{\boldsymbol{{t}}}{\boldsymbol{{n}}}} +\boldsymbol{{t}}\left(\boldsymbol{{n}}!\right)^{\frac{\boldsymbol{{r}}}{\boldsymbol{{n}}}} +\boldsymbol{{rt}}\right]}\:+\frac{\left[\underset{\boldsymbol{{p}}_{\mathrm{2}} =\mathrm{1}} {\overset{\boldsymbol{{n}}} {\sum}}\left(\boldsymbol{{p}}_{\mathrm{2}} ^{\boldsymbol{{t}}} +\boldsymbol{{t}}\right)+\underset{\boldsymbol{{p}}_{\mathrm{3}} =\mathrm{1}} {\overset{\boldsymbol{{n}}} {\sum}}\left(\boldsymbol{{p}}_{\mathrm{3}} ^{\boldsymbol{{j}}} +\boldsymbol{{j}}\right)\right]^{\mathrm{2}} }{\boldsymbol{{n}}^{\mathrm{2}} \left[\left(\boldsymbol{{n}}!\right)^{\frac{\boldsymbol{{t}}+\boldsymbol{{j}}}{\boldsymbol{{n}}}} +\boldsymbol{{t}}\left(\boldsymbol{{n}}!\right)^{\frac{\boldsymbol{{j}}}{\boldsymbol{{n}}}} +\boldsymbol{{j}}\left(\boldsymbol{{n}}!\right)^{\frac{\boldsymbol{{t}}}{\boldsymbol{{n}}}} +\boldsymbol{{tj}}\right]}\:\geqslant\:\mathrm{8}\: \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{By}}:\:\boldsymbol{{Mr}}.\:\boldsymbol{{Chheang}}\:\boldsymbol{{Chantria}} \\ $$