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11
Find all solutions (x1, x2, . . . , xn) of the equation
Find all solutions (x1, x2, . . . , xn) of the equation
Source:
September 23, 2010
algebra
equation
roots
IMO Shortlist
Problem Statement
Find all solutions
(
x
1
,
x
2
,
.
.
.
,
x
n
)
(x_1, x_2, . . . , x_n)
(
x
1
,
x
2
,
...
,
x
n
)
of the equation
1
+
1
x
1
+
x
1
+
1
x
1
x
2
+
(
x
1
+
1
)
(
x
2
+
1
)
x
1
2
x
3
+
⋯
+
(
x
1
+
1
)
(
x
2
+
1
)
⋯
(
x
n
−
1
+
1
)
x
1
x
2
⋯
x
n
=
0
1 +\frac{1}{x_1} + \frac{x_1+1}{x{}_1x{}_2}+\frac{(x_1+1)(x_2+1)}{x{}_1{}_2x{}_3} +\cdots + \frac{(x_1+1)(x_2+1) \cdots (x_{n-1}+1)}{x{}_1x{}_2\cdots x_n} =0
1
+
x
1
1
+
x
1
x
2
x
1
+
1
+
x
1
2
x
3
(
x
1
+
1
)
(
x
2
+
1
)
+
⋯
+
x
1
x
2
⋯
x
n
(
x
1
+
1
)
(
x
2
+
1
)
⋯
(
x
n
−
1
+
1
)
=
0
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