MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2016 China Second Round Olympiad
Q10
Q10
Part of
2016 China Second Round Olympiad
Problems
(1)
China Second Round Olympiad 2016 Test 1 Q10
Source: Sep 11, 2016
9/11/2016
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
)
.
algebra
function