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International Contests
KoMaL A Problems
KoMaL A Problems 2018/2019
A. 741
A. 741
Part of
KoMaL A Problems 2018/2019
Problems
(1)
Existence of real sequence with specific properties
Source: KöMaL A. 741
2/13/2019
Let
f
f
f
be a function defined on the positive integers with
f
(
n
)
≥
0
f(n) \ge 0
f
(
n
)
≥
0
and
f
(
n
)
≤
f
(
n
+
1
)
f(n) \le f(n+1)
f
(
n
)
≤
f
(
n
+
1
)
for all
n
n
n
. Prove that if
∑
n
=
1
∞
f
(
n
)
n
2
\sum_{n = 1}^{\infty} \frac{f(n)}{n^2}
n
=
1
∑
∞
n
2
f
(
n
)
diverges, there exists a sequence
a
1
,
a
2
,
…
a_1, a_2, \dots
a
1
,
a
2
,
…
such that the sequence
a
n
n
\tfrac{a_n}{n}
n
a
n
hits every natural number, while
a
n
+
m
≤
a
n
+
a
m
+
f
(
n
+
m
)
a_{n+m} \le a_n + a_m + f(n+m)
a
n
+
m
≤
a
n
+
a
m
+
f
(
n
+
m
)
holds for every pair
n
n
n
,
m
m
m
.
algebra
combinatorics