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Let $n$-regular polygon $X_1X_2\cdots X_n$ with the circumcribed circle $(O)$. Let $n$ points $A_1, A_2,\cdots,A_n$ lie on the circle $(O)$. Let $x_{ij}=X_iX_j$ (for $1 \le i<j \le n) $. Let $a_{ij}=A_iA_j$ (for $1 \le i<j \le n)$.

Now we denote:

$\{ x_1, x_2, \cdots ,x_{\frac{n(n+1)}{2}} \}=\{x_{ij}, 1 \le i < j \le n \},$

$\{ a_1, a_2, \cdots ,a_{\frac{n(n+1)}{2}} \}=\{a_{ij}, 1 \le i < j \le n \}.$

Conjecture that:

$p_k(x_1,x_2, \cdots , x_{\frac{n(n+1)}{2}}) \ge p_k(a_1,a_2, \cdots , a_{\frac{n(n+1)}{2}})$ for all $k=1,2,...,n-1$

$e_k(x_1,x_2, \cdots , x_{\frac{n(n+1)}{2}}) \ge e_k(a_1,a_2, \cdots , a_{\frac{n(n+1)}{2}})$ for all $k=1,2,...,n$

Equality holds iff $A_1A_2 \cdots A_n$ is n-regular polygon.

Here $p_k$ and $e_k$ are the pơer sum and elementary symmetric functions, respectively see Newton's identities.

See also: My previous question

My question: Could you give a proof, or a reference, or a counterexamples?

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See if your claims are contained in http://igm.univ-mlv.fr/~fpsac/FPSAC07/SITE07/PDF-Proceedings/Posters/82.pdf

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  • $\begingroup$ I am not sure, but I think it is not proof. It is a reference. $\endgroup$ – Oai Thanh Đào Sep 10 '16 at 5:23
  • $\begingroup$ Because your reference both sides are the same $(x)$, but this question left side is $(x)$ and right side is $(a)$ $\endgroup$ – Oai Thanh Đào Sep 11 '16 at 15:38

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