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For $5\le n\le 19$, $a_1, a_2,\cdots, a_n\ge 0$ and $\displaystyle\sum\limits_{k=1}^na_k^2=1$, prove that:

$$\sqrt{1-a_1a_2}+\sqrt{1-a_2a_3}+\sqrt{1-a_3a_4}+\cdots+\sqrt{1-a_na_1}\ge\sqrt{n(n-1)}$$

Update(counterexamples to $n$ beyond the scope)

I originally expected that the global minimum of lhs is exactly rhs. However, I check by numerical optimization in Mathematica by Differential Evolution algorithm that, for $n>20$, smaller than rhs local minimizer can be numerically achieved. The codes are:

ClearAll["Global`*"];
n=22;
Print["sqrt of n(n-1):"]
Sqrt[n(n-1)]//N[#,40]&
var=Table[Subscript[a,k],{k,n}];
Print["When all variables are equal:"]
{Sqrt[1-Subscript[a,1] Subscript[a,n]]+Sum[Sqrt[1-Subscript[a,k] Subscript[a,k+1]],{k,n-1}]}/.Thread[var->1/Sqrt[n]]//N[#,40]&
Print["Random seed:"]
rseeds=RandomInteger[99]
Print["NMinimize Solution"]
sol=NMinimize[{Sqrt[1-Subscript[a,1] Subscript[a,n]]+Sum[Sqrt[1-Subscript[a,k] Subscript[a,k+1]],{k,n-1}]}~Join~Thread[1>=var>=0]~Join~{Total@(var^2)==1},var,Method->{"DifferentialEvolution","RandomSeed"->rseeds},WorkingPrecision->50]
{Sqrt[1-Subscript[a,1] Subscript[a,n]]+Sum[Sqrt[1-Subscript[a,k] Subscript[a,k+1]],{k,n-1}]}/.Last@sol

Results:

sqrt of n(n-1):

21.49418526020467703924401057548171299596

When all variables are equal:

21.49418526020467703924401057548171299596

Random seed:

97

NMinimize Solution

{21.493958088361718757508756191304024461882549266188, $$\begin{array}{rl} a_1\to&0.27551074101367342222780049146552671553141115720781\\ a_2\to&0.24122101942591859516896035008786114084222080634210\\ a_3\to&0.20686031925778095999625358660583743030950178927919\\ a_4\to&0.17577853782894785232564071925930202897837797620107\\ a_5\to&0.14967776232864740441800976481499462604156652376732\\ a_6\to&0.12919237259005274526751896356388549836657598687909\\ a_7\to&0.11439952750687089613978518627483806246516035842270\\ a_8\to&0.10507233393317968236379206706172547805340715644522\\ a_9\to&0.10101354732229610358754955223851289466594306555616\\ a_{10}\to&0.10215969656778251068580715934518992632384781887479\\ a_{11}\to&0.10851219137727185962358704265924136455115927676940\\ a_{12}\to&0.12020966542805764693753571335235270582328771351713\\ a_{13}\to&0.13749524018489170480661186121587722167726253796253\\ a_{14}\to&0.16047577556457054303088930254045653682972975693984\\ a_{15}\to&0.18889853876426131135197586875434934304601287018713\\ a_{16}\to&0.22171666754106704676947535912078689424836159977455\\ a_{17}\to&0.25655322766782152064878861383995289461345663919666\\ a_{18}\to&0.28932738110835094055030451354655289547312222971662\\ a_{19}\to&0.31452382324755012672904620877170420894722451824547\\ a_{20}\to&0.32680622312811119929265771484173921350691759958035\\ a_{21}\to&0.32326625988701552021297482437342530033165491570368\\ a_{22}\to&0.30475064029155834486956174539864169866495135183824 \end{array}$$}

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    $\begingroup$ Is it given any counterexample for $n = 20$? $\endgroup$ – PenasRaul Oct 26 '16 at 12:34
  • $\begingroup$ See my updates. $\endgroup$ – user6043040 Oct 26 '16 at 18:51
  • $\begingroup$ The inequality does not hold for $n\ge 20$. The idea of a proof is to construct $x_1, \ x_2, \ \cdots, \ x_n$ which are functions of a parameter $t$ and when $t=0$ it holds that $x_1 = x_2 = \cdots = x_n = \frac{1}{n}$, then let $f(t) = LHS - RHS$ (so $f(0)=0$) and prove that $f'(0) = 0$ and $f''(0) < 0$. $\endgroup$ – River Li Jul 19 '19 at 14:22

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