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Moscow Mathematical Olympiad
1957 Moscow Mathematical Olympiad
368
MMO 368 Moscow MO 1957 system 99x99 ... 1-x_1^2=x_2
MMO 368 Moscow MO 1957 system 99x99 ... 1-x_1^2=x_2
Source:
March 20, 2021
algebra
system of equations
Problem Statement
Find all real solutions of the system : (a)
{
1
−
x
1
2
=
x
2
1
−
x
2
2
=
x
3
.
.
.
1
−
x
98
2
=
x
99
1
−
x
99
2
=
x
1
\begin{cases}1-x_1^2=x_2 \\ 1-x_2^2=x_3\\ ...\\ 1-x_{98}^2=x_{99}\\ 1-x_{99}^2=x_1\end{cases}
⎩
⎨
⎧
1
−
x
1
2
=
x
2
1
−
x
2
2
=
x
3
...
1
−
x
98
2
=
x
99
1
−
x
99
2
=
x
1
(b)*
{
1
−
x
1
2
=
x
2
1
−
x
2
2
=
x
3
.
.
.
1
−
x
98
2
=
x
n
1
−
x
n
2
=
x
1
\begin{cases} 1-x_1^2=x_2\\ 1-x_2^2=x_3\\ ...\\1-x_{98}^2=x_{n}\\ 1-x_{n}^2=x_1\end{cases}
⎩
⎨
⎧
1
−
x
1
2
=
x
2
1
−
x
2
2
=
x
3
...
1
−
x
98
2
=
x
n
1
−
x
n
2
=
x
1
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