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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1960 Kurschak Competition
2
2
Part of
1960 Kurschak Competition
Problems
(1)
every n = sum a_ if a_k < 1 + a_1 + a2_ +... + a_{k-1
Source: 1960 Hungary - Kürschák Competition p2
10/10/2022
Let
a
1
=
1
,
a
2
,
a
3
,
.
.
.
a_1 = 1, a_2, a_3,...
a
1
=
1
,
a
2
,
a
3
,
...
: be a sequence of positive integers such that
a
k
<
1
+
a
1
+
a
2
+
.
.
.
+
a
k
−
1
a_k < 1 + a_1 + a_2 +... + a_{k-1}
a
k
<
1
+
a
1
+
a
2
+
...
+
a
k
−
1
for all
k
>
1
k > 1
k
>
1
. Prove that every positive integer can be expressed as a sum of
a
i
a_i
a
i
s.
number theory
Integer
Sum