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National and Regional Contests
Russia Contests
Moscow Mathematical Olympiad
1952 Moscow Mathematical Olympiad
222
222
Part of
1952 Moscow Mathematical Olympiad
Problems
(1)
MMO 222 Moscow MO 1952 15x15 system of equations (1 - x_1x_2 = 0)
Source:
8/8/2019
a) Solve the system of equations
{
1
−
x
1
x
2
=
0
1
−
x
2
x
3
=
0
.
.
.
1
−
x
14
x
15
=
0
1
−
x
15
x
1
=
0
\begin{cases} 1 - x_1x_2 = 0 \\ 1 - x_2x_3 = 0 \\ ...\\ 1 - x_{14}x_{15} = 0 \\ 1 - x_{15}x_1 = 0 \end{cases}
⎩
⎨
⎧
1
−
x
1
x
2
=
0
1
−
x
2
x
3
=
0
...
1
−
x
14
x
15
=
0
1
−
x
15
x
1
=
0
b) Solve the system of equations
{
1
−
x
1
x
2
=
0
1
−
x
2
x
3
=
0
.
.
.
1
−
x
n
−
1
x
n
=
0
1
−
x
n
x
1
=
0
\begin{cases} 1 - x_1x_2 = 0 \\ 1 - x_2x_3 = 0 \\ ...\\ 1 - x_{n-1}x_{n} = 0 \\ 1 - x_{n}x_1 = 0 \end{cases}
⎩
⎨
⎧
1
−
x
1
x
2
=
0
1
−
x
2
x
3
=
0
...
1
−
x
n
−
1
x
n
=
0
1
−
x
n
x
1
=
0
How does the solution vary for distinct values of
n
n
n
?
algebra
system of equations