1963 IMO Problems/Problem 4

Problem

Find all solutions of the system

$\begin{eqnarray}

x_5+x_2&=&yx_1\\ x_1+x_3&=&yx_2\\ x_2+x_4&=&yx_3\\ x_3+x_5&=&yx_4\\

x_4+x_1&=&yx_5,\end{eqnarray}$ (Error compiling LaTeX. Unknown error_msg)

where is a parameter.

Solution

Notice: The following words are Chinese. 首先,我们可以将以上5个方程相加,得到: 2(x_1+x_2+x_3+x_4+x_5)&=&y(x_1+x_2+x_3+x_4+x_5) 当$x_1+x_2+x_3+x_4+x_5&=&0$ (Error compiling LaTeX. Unknown error_msg)时,因为x_1,x_2,x_3,x_4,x_5关于原方程组轮换对称,所以 \\ 若反之,则方程两边同除以,得到,显然解为 综上所述,最终答案为

See Also

1963 IMO (Problems) • Resources
Preceded by Problem 3	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 5
All IMO Problems and Solutions