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National and Regional Contests
Estonia Contests
Estonia National Olympiad
1999 Estonia National Olympiad
2
sum f(i/2000)+ f(2000/i)+ f(2000/2000) , when i=1,...,1999
sum f(i/2000)+ f(2000/i)+ f(2000/2000) , when i=1,...,1999
Source: 1999 Estonia National Olympiad Final Round grade 11 p2
March 11, 2020
Sum
algebra
Problem Statement
Find the value of the expression
f
(
1
2000
)
+
f
(
2
2000
)
+
.
.
.
+
f
(
1999
2000
)
+
f
(
2000
2000
)
+
f
(
2000
1999
)
+
.
.
.
+
f
(
2000
1
)
f\left( \frac{1}{2000} \right)+f\left( \frac{2}{2000} \right)+...+ f\left( \frac{1999}{2000} \right)+f\left( \frac{2000}{2000} \right)+f\left( \frac{2000}{1999} \right)+...+f\left( \frac{2000}{1} \right)
f
(
2000
1
)
+
f
(
2000
2
)
+
...
+
f
(
2000
1999
)
+
f
(
2000
2000
)
+
f
(
1999
2000
)
+
...
+
f
(
1
2000
)
assuming
f
(
x
)
=
x
2
1
+
x
2
f(x) =\frac{x^2}{1 + x^2}
f
(
x
)
=
1
+
x
2
x
2
.
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