MathDB
Problems
Contests
National and Regional Contests
Romania Contests
District Olympiad
2022 District Olympiad
P4
Romania District MO 2022 Grade 11 P4
Romania District MO 2022 Grade 11 P4
Source: Romania District MO 2022 Grade 11
March 27, 2022
matrix
rank
college contests
romania
Problem Statement
Let
A
∈
M
n
(
C
)
A\in\mathcal{M}_n(\mathbb{C})
A
∈
M
n
(
C
)
where
n
≥
2.
n\geq 2.
n
≥
2.
Prove that if
m
=
∣
{
rank
(
A
k
)
−
rank
(
A
k
+
1
)
"
:
k
∈
N
∗
}
∣
m=|\{\text{rank}(A^k)-\text{rank}(A^{k+1})":k\in\mathbb{N}^*\}|
m
=
∣
{
rank
(
A
k
)
−
rank
(
A
k
+
1
)
"
:
k
∈
N
∗
}
∣
then
n
+
1
≥
m
(
m
+
1
)
/
2.
n+1\geq m(m+1)/2.
n
+
1
≥
m
(
m
+
1
)
/2.
Back to Problems
View on AoPS