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Miklós Schweitzer
1994 Miklós Schweitzer
3
3
Part of
1994 Miklós Schweitzer
Problems
(1)
analysis
Source: miklos schertzer 1994 q3
10/16/2021
Let p be an odd prime, A be a non-empty subset of residue classes modulo p,
f
:
A
→
R
f:A\to\mathbb R
f
:
A
→
R
. Suppose that f is not constant and satisfies
f
(
x
)
≤
f
(
x
+
h
)
+
f
(
x
−
h
)
2
f(x) \leq \frac{f(x + h) + f(x-h)}{2}
f
(
x
)
≤
2
f
(
x
+
h
)
+
f
(
x
−
h
)
whenever
x
,
x
+
h
,
x
−
h
∈
A
x,x+h,x-h\in A
x
,
x
+
h
,
x
−
h
∈
A
. Prove that
∣
A
∣
≤
p
+
1
2
|A| \leq \frac{p + 1}{2}
∣
A
∣
≤
2
p
+
1
.
real analysis