MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
IMOC Shortlist
2023-IMOC
N6
N6
Part of
2023-IMOC
Problems
(1)
Number of solution of a system of equations
Source: IMOC 2023 N6
9/9/2023
Let
S
(
b
)
S(b)
S
(
b
)
be the number of nonuples of positive integers
(
a
1
,
a
2
,
…
,
a
9
)
(a_1, a_2, \ldots , a_9)
(
a
1
,
a
2
,
…
,
a
9
)
satisfying
3
b
−
1
=
a
1
+
a
2
+
…
+
a
9
3b-1=a_1+a_2+\ldots+a_9
3
b
−
1
=
a
1
+
a
2
+
…
+
a
9
and
b
2
+
1
=
a
1
2
+
…
+
a
9
2
b^2+1=a_1^2+\ldots+a_9^2
b
2
+
1
=
a
1
2
+
…
+
a
9
2
. Prove that for all
ϵ
>
0
\epsilon>0
ϵ
>
0
, there exists
C
ϵ
>
0
C_{\epsilon}>0
C
ϵ
>
0
such that
S
(
b
)
≤
C
ϵ
b
3
+
ϵ
S(b)\leq C_{\epsilon}b^{3+\epsilon}
S
(
b
)
≤
C
ϵ
b
3
+
ϵ
.
algebra
system of equations