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Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2016 China Second Round Olympiad
Q10
China Second Round Olympiad 2016 Test 1 Q10
China Second Round Olympiad 2016 Test 1 Q10
Source: Sep 11, 2016
September 11, 2016
algebra
function
Problem Statement
Let
f
(
x
)
f(x)
f
(
x
)
is an odd function on
R
R
R
,
f
(
1
)
=
1
f(1)=1
f
(
1
)
=
1
and
f
(
x
x
−
1
)
=
x
f
(
x
)
f(\frac{x}{x-1})=xf(x)
f
(
x
−
1
x
)
=
x
f
(
x
)
(
∀
x
<
0
)
(\forall x<0)
(
∀
x
<
0
)
. Find the value of
f
(
1
)
f
(
1
100
)
+
f
(
1
2
)
f
(
1
99
)
+
f
(
1
3
)
f
(
1
98
)
+
⋯
+
f
(
1
50
)
f
(
1
51
)
.
f(1)f(\frac{1}{100})+f(\frac{1}{2})f(\frac{1}{99})+f(\frac{1}{3})f(\frac{1}{98})+\cdots +f(\frac{1}{50})f(\frac{1}{51}).
f
(
1
)
f
(
100
1
)
+
f
(
2
1
)
f
(
99
1
)
+
f
(
3
1
)
f
(
98
1
)
+
⋯
+
f
(
50
1
)
f
(
51
1
)
.
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