AoPS - Problem

Source: 2019 Belarusian National Olympiad 10.3

Tags: algebra, polynomial

Vlados021

01.09.2019 10:49

The polynomial of seven variables $$ Q(x_1,x_2,\ldots,x_7)=(x_1+x_2+\ldots+x_7)^2+2(x_1^2+x_2^2+\ldots+x_7^2) $$is represented as the sum of seven squares of the polynomials with nonnegative integer coefficients: $$ Q(x_1,\ldots,x_7)=P_1(x_1,\ldots,x_7)^2+P_2(x_1,\ldots,x_7)^2+\ldots+P_7(x_1,\ldots,x_7)^2. $$Find all possible values of $P_1(1,1,\ldots,1)$. (A. Yuran)

Problem is not hard. As $x_1^2$ has coefficient $3$ then $x_1$ in all $P_i$ has coefficient $0$ or $1$ and there are exactly $3$ polynomials where coefficient is $1$ As for every $i,j$ pair $x_ix_j$ has coefficient $2$ so exactly one polynomial has sum $x_i+x_j$ Let $a_i$ - is number of coefficients $1$ in $P_i$ Then there are $\frac{a_i(a_i-1)^2}{2}$ pairs $(x_i,x_j)$ in $P_i^2$ $\sum a_i=3*7=21$ $\sum \frac{a_i(a_i-1)}{2}=\frac{7*6}{2}=21$ So $ \sum a_i^2= 63$ but $\sum a_i^2 \geq \frac{(\sum a_i)^2}{7}=63$ so $a_i=3$ for every $i$ and so $P_1(1,1,1,1,1,1,1)=3$