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Problems
Contests
National and Regional Contests
Greece Contests
Greece JBMO TST
2004 Greece JBMO TST
2
2
Part of
2004 Greece JBMO TST
Problems
(1)
sum S=...+\frac{1}{1-x_{2004}^3}+...+\frac{1}{1-x_{2004}^{-3}}
Source: Greece JBMO TST 2004 p2
6/18/2019
Real numbers
x
1
,
x
2
,
.
.
.
x
2004
,
y
1
,
y
2
,
.
.
.
y
2004
x_1,x_2,...x_{2004},y_1,y_2,...y_{2004}
x
1
,
x
2
,
...
x
2004
,
y
1
,
y
2
,
...
y
2004
differ from
1
1
1
and are such that
x
k
y
k
=
1
x_ky_k=1
x
k
y
k
=
1
for every
k
=
1
,
2
,
.
.
.
,
2004
k=1,2,...,2004
k
=
1
,
2
,
...
,
2004
. Calculate the sum
S
=
1
1
−
x
1
3
+
1
1
−
x
2
3
+
.
.
.
+
1
1
−
x
2004
3
+
1
1
−
y
1
3
+
1
1
−
y
2
3
+
.
.
.
+
1
1
−
y
2004
3
S=\frac{1}{1-x_1^3}+\frac{1}{1-x_2^3}+...+\frac{1}{1-x_{2004}^3}+\frac{1}{1-y_1^3}+\frac{1}{1-y_2^3}+...+\frac{1}{1-y_{2004}^3}
S
=
1
−
x
1
3
1
+
1
−
x
2
3
1
+
...
+
1
−
x
2004
3
1
+
1
−
y
1
3
1
+
1
−
y
2
3
1
+
...
+
1
−
y
2004
3
1
algebra
Sum