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Problems
Contests
International Contests
APMO
2018 APMO
2
2
Part of
2018 APMO
Problems
(1)
Bounding difference of sum of fractions in an interval
Source: APMO 2018 P2
6/24/2018
Let
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
be given by
f
(
x
)
=
1
x
+
1
x
−
2
+
1
x
−
4
+
⋯
+
1
x
−
2018
f(x) = \frac{1}{x} + \frac{1}{x-2} + \frac{1}{x-4} + \cdots + \frac{1}{x-2018}
f
(
x
)
=
x
1
+
x
−
2
1
+
x
−
4
1
+
⋯
+
x
−
2018
1
g
(
x
)
=
1
x
−
1
+
1
x
−
3
+
1
x
−
5
+
⋯
+
1
x
−
2017
g(x) = \frac{1}{x-1} + \frac{1}{x-3} + \frac{1}{x-5} + \cdots + \frac{1}{x-2017}
g
(
x
)
=
x
−
1
1
+
x
−
3
1
+
x
−
5
1
+
⋯
+
x
−
2017
1
.Prove that
∣
f
(
x
)
−
g
(
x
)
∣
>
2
|f(x)-g(x)| >2
∣
f
(
x
)
−
g
(
x
)
∣
>
2
for any non-integer real number
x
x
x
satisfying
0
<
x
<
2018
0 < x < 2018
0
<
x
<
2018
.
algebra
APMO