Home Competitions SE-SKOLORNAS 1979 Problem 1
SE-SKOLORNAS 1979 Problem 1
Solve the equations:
{ x 1 + 2 x 2 + 3 x 3 + ⋯ + ( n − 1 ) x n − 1 + n x n = n 2 x 1 + 3 x 2 + 4 x 3 + ⋯ + n x n − 1 + x n = n − 13 x 1 + 4 x 2 + 5 x 3 + ⋯ + x n − 1 + 2 x n = n − 2 ⋯ ⋯ ⋯ ⋯ ⋅ ⋯ ⋯ ⋯ ⋯ ⋅ ⋯ ⋯ ⋯ ⋯ ⋅ ( n − 1 ) x 1 + n x 2 + x 3 + ⋯ + ( n − 3 ) x n − 1 + ( n − 2 ) x n = 2 n x 1 + x 2 + 2 x 3 + ⋯ + ( n − 2 ) x n − 1 + ( n − 1 ) x n = 1 \left\{ \begin{array}{l}
x_1 + 2 x_2 + 3 x_3 + \cdots + (n-1) x_{n-1} + n x_n = n
2 x_1 + 3 x_2 + 4 x_3 + \cdots + n x_{n-1} + x_n = n-1
3 x_1 + 4 x_2 + 5 x_3 + \cdots + x_{n-1} + 2 x_n = n-2
\cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot
(n-1) x_1 + n x_2 + x_3 + \cdots + (n-3) x_{n-1} + (n-2) x_n = 2
n x_1 + x_2 + 2 x_3 + \cdots + (n-2) x_{n-1} + (n-1) x_n = 1
\end{array} \right. { x 1 + 2 x 2 + 3 x 3 + ⋯ + ( n − 1 ) x n − 1 + n x n = n 2 x 1 + 3 x 2 + 4 x 3 + ⋯ + n x n − 1 + x n = n − 13 x 1 + 4 x 2 + 5 x 3 + ⋯ + x n − 1 + 2 x n = n − 2 ⋯⋯⋯⋯ ⋅ ⋯⋯⋯⋯ ⋅ ⋯⋯⋯⋯ ⋅ ( n − 1 ) x 1 + n x 2 + x 3 + ⋯ + ( n − 3 ) x n − 1 + ( n − 2 ) x n = 2 n x 1 + x 2 + 2 x 3 + ⋯ + ( n − 2 ) x n − 1 + ( n − 1 ) x n = 1
Dataset text AI-MO AoPS . Quoted with source.
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