Fie \( n \in \mathbb{N} \), \( n \ge 2 \). Aratati ca exista o infinitate de \( (x_1,...,x_n) \in \mathbb{N}^n \) cu \( 1 < x_1 < \cdots < x_n \) si astfel incat urmatorul numar este patrat perfect:
\( 1 + x_1^2 + \cdots + x_n^2 \).
Concursul "La scoala cu ceas", 2008, Problema 1
Ecuatie polinomiala cu patrate
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Filip Chindea
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Ecuatie polinomiala cu patrate
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Laurian Filip
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Fie \( x_i=2y_i \) pentru \( i \in \lbrace 1,2,...,n-1 } \) si
\( x_n=2y_1^2+2y_2^2+...+2y_{n-1}^2 \).
De unde avem
\( 1+x_1^2+...+x_n^2=(1+(2y_1^2+2y_2^2+...+2y_{n-1}^2))^2 \)
\( x_n=2y_1^2+2y_2^2+...+2y_{n-1}^2 \).
De unde avem
\( 1+x_1^2+...+x_n^2=(1+(2y_1^2+2y_2^2+...+2y_{n-1}^2))^2 \)
Last edited by Laurian Filip on Sun May 11, 2008 10:26 pm, edited 1 time in total.