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MMO 248 Moscow MO 1953 5x5 linear system + generalization

Source:

August 9, 2019
algebrasystem of equations

Problem Statement

a) Solve the system {x1+2x2+2x3+2x4+2x5=1x1+3x2+4x3+4x4+4x5=2x1+3x2+5x3+6x4+6x5=3x1+3x2+5x3+7x4+8x5=4x1+3x2+5x3+7x4+9x5=5\begin{cases} x_1 + 2x_2 + 2x_3 + 2x_4 + 2x_5 = 1 \\ x_1 + 3x_2 + 4x_3 + 4x_4 + 4x_5 = 2 \\ x_1 + 3x_2 + 5x_3 + 6x_4 + 6x_5 = 3 \\ x_1 + 3x_2 + 5x_3 + 7x_4 + 8x_5 = 4 \\ x_1 + 3x_2 + 5x_3 + 7x_4 + 9x_5 = 5 \end{cases}
b) Solve the system {x1+2x2+2x3+2x4+2x5+...+2x100=1x1+3x2+4x3+4x4+4x5+...+4x100=2x1+3x2+5x3+6x4+6x5+...+6x100=3x1+3x2+5x3+7x4+8x5+...+8x100=4...x1+3x2+5x3+7x4+9x5+...+199x100=100\begin{cases} x_1 + 2x_2 + 2x_3 + 2x_4 + 2x_5 +...+ 2x_{100}= 1 \\ x_1 + 3x_2 + 4x_3 + 4x_4 + 4x_5 +...+ 4x_{100}= 2 \\ x_1 + 3x_2 + 5x_3 + 6x_4 + 6x_5 +...+ 6x_{100}= 3 \\ x_1 + 3x_2 + 5x_3 + 7x_4 + 8x_5 +...+ 8x_{100}= 4 \\ ... \\ x_1 + 3x_2 + 5x_3 + 7x_4 + 9x_5 +...+ 199x_{100}= 100 \end{cases}
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