1971 IMO P1   Prove that the following assertion is true for $$n = 3$$ and $$n = 5$$, and that it is false for every other natural number $$n\gt2$$:

If $$x_1$$, $$x_2$$, $$\dotsc$$, $$x_n$$ are arbitrary real numbers, then

$$A_n(x)= \sum_{i=1}^n\prod_{j\ne i}\Big(x_i-x_j\Big)\geqslant 0.$$

Anneli Lax  and Peter Lax 1978 年在 [1] 指出: $$A_5(x)$$ 不能表成二次型的平方和.

$$\begin{split}&\hspace3.25ex(x_1-x_2)(x_1-x_3)(x_1-x_4)(x_1-x_5)+(x_2-x_1)(x_2-x_3)(x_2-x_4)(x_2-x_5)\\&=(x_1-x_2)\Big((x_1-x_3)(x_1-x_4)(x_1-x_5)-(x_2-x_3)(x_2-x_4)(x_2-x_5)\Big)\\&\geqslant0,\end{split}$$

$$(x_3-x_1)(x_3-x_2)(x_3-x_4)(x_3-x_5)\geqslant0,$$

• 设 $$x_1$$, $$x_2$$, $$x_3$$, $$x_4$$, $$x_5$$ 是实数, 有 $$A_5(x)\geqslant0$$ 为真 ;
• $$A_5(x)$$ 不能写成多项式的平方和.

$$A_5=\sum Q_j^2,$$

$$x_1=x_2,\;\; x_3=x_4=x_5,$$

Lemma 4.2  二次型 $$Q$$, 只要条件 $$(5)$$ 或者 $$(5)$$ 的指标进行置换得到的条件任何一个为真, 总能导致 $$Q=0$$, 那么二次型 $$Q\equiv0$$.

Proof of the Lemma  记

$$Q(x)=\sum c_{jk}x_jx_k,\;\; c_{jk}=c_{kj}.$$

$$\begin{split}Q&=(c_{11}+2c_{12}+c_{22})y^2\\&+2(c_{13}+c_{14}+c_{15}+c_{23}+c_{24}+c_{25})yz\\&+(c_{33}+c_{44}+c_{55}+2c_{34}+2c_{35}+c_{45})z^2=0.\end{split}$$

$$c_{11}+2c_{12}+c_{22}=0;$$

$$c_{13}+c_{14}+c_{15}+c_{23}+c_{24}+c_{25}=0;$$

$$c_{33}+c_{44}+c_{55}+2c_{34}+2c_{35}+c_{45}=0.$$

$$c_{33}+2c_{34}+c_{44}=0.$$

$$c_{55}+2c_{35}+2c_{45}=0.$$

$$c_{j5}+c_{k5}=-\frac12 c_{55}$$

$$c_{15}=c_{25}=c_{35}=c_{45}.$$

$$c_{51}=c_{21}=c_{31}=c_{41}.$$

$$c_{12}=c_{52}=c_{32}=c_{42}.$$

$$c_{1j}=c_{2j}=c_{12}\qquad j=3, 4,5.$$

$$c_{jk}=0 \qquad j\ne k.$$

$$c_{jj}=0.$$

#### References

1. Anneli Lax, Peter Lax, On sums of squares, Linear Algebra and its applications, 20, 71-75 (1978)

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