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International Contests
IMO Longlists
1983 IMO Longlists
24
24
Part of
1983 IMO Longlists
Problems
(1)
show that 0 < f(x) -x < c for every 0 < x < 1
Source:
10/5/2010
Every
x
,
0
≤
x
≤
1
x, 0 \leq x \leq 1
x
,
0
≤
x
≤
1
, admits a unique representation
x
=
∑
j
=
0
∞
a
j
2
−
j
x = \sum_{j=0}^{\infty} a_j 2^{-j}
x
=
∑
j
=
0
∞
a
j
2
−
j
, where all the
a
j
a_j
a
j
belong to
{
0
,
1
}
\{0, 1\}
{
0
,
1
}
and infinitely many of them are
0
0
0
. If
b
(
0
)
=
1
+
c
2
+
c
,
b
(
1
)
=
1
2
+
c
,
c
>
0
b(0) = \frac{1+c}{2+c}, b(1) =\frac{1}{2+c},c > 0
b
(
0
)
=
2
+
c
1
+
c
,
b
(
1
)
=
2
+
c
1
,
c
>
0
, and
f
(
x
)
=
a
0
+
∑
j
=
0
∞
b
(
a
0
)
⋯
b
(
a
j
)
a
j
+
1
f(x)=a_0 + \sum_{j=0}^{\infty}b(a_0) \cdots b(a_j) a_{j+1}
f
(
x
)
=
a
0
+
j
=
0
∑
∞
b
(
a
0
)
⋯
b
(
a
j
)
a
j
+
1
show that
0
<
f
(
x
)
−
x
<
c
0 < f(x) -x < c
0
<
f
(
x
)
−
x
<
c
for every
x
,
0
<
x
<
1.
x, 0 < x < 1.
x
,
0
<
x
<
1.
algebra unsolved
algebra