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IMC
1999 IMC
3
3
Part of
1999 IMC
Problems
(2)
Constant function
Source: IMC 1999 day 1 problem 3
11/19/2005
Suppose that
f
:
R
→
R
f: \mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
fulfils
∣
∑
k
=
1
n
3
k
(
f
(
x
+
k
y
)
−
f
(
x
−
k
y
)
)
∣
≤
1
\left|\sum^n_{k=1}3^k\left(f(x+ky)-f(x-ky)\right)\right|\le1
∑
k
=
1
n
3
k
(
f
(
x
+
k
y
)
−
f
(
x
−
k
y
)
)
≤
1
for all
n
∈
N
,
x
,
y
∈
R
n\in\mathbb{N},x,y\in\mathbb{R}
n
∈
N
,
x
,
y
∈
R
. Prove that
f
f
f
is a constant function.
function
search
real analysis
real analysis unsolved
very well-known
Source: IMC 1999 day 2 problem 3
11/19/2005
Let
x
i
≥
−
1
x_i\ge -1
x
i
≥
−
1
and
∑
i
=
1
n
x
i
3
=
0
\sum^n_{i=1}x_i^3=0
∑
i
=
1
n
x
i
3
=
0
. Prove
∑
i
=
1
n
x
i
≤
n
3
\sum^n_{i=1}x_i \le \frac{n}{3}
∑
i
=
1
n
x
i
≤
3
n
.
real analysis
real analysis unsolved