MathDB

1

Vector space of matrices, determiant preserving transform

M_n(\mathbb{C})

is the vector space of all complex

n\times n

matrices. Given a linear map

T:M_n(\mathbb{C})\to M_n(\mathbb{C})

s.t.

\det (A)=\det(T(A))

for every

A\in M_n(\mathbb{C})

. (1) If

T(A)

is the zero matrix, then show that

A

is also the zero matrix. (2) Prove that

\text{rank} (A)=\text{rank} (T(A))

for any

A\in M_n(\mathbb{C})

.

7

1

Condition for a series to converge

For a real number

a

and an integer

n(\geq 2)

, define

S_n (a) = n^a \sum_{k=1}^{n-1} \frac{1}{k^{2019} (n-k)^{2019}}

Find every value of

a

s.t. sequence

\{S_n(a)\}_{n\geq 2}

converges to a positive real.

6

1

Holder like inequality

A function

f:[0,\infty)\to[0,\infty)

is integrable and \int_0^\infty f(x)^2 dx<\infty, \int_0^\infty xf(x) dx <\infty Prove the following inequality.

\left(\int_0^\infty f(x) dx \right)^3 \leq 8\left(\int_0^\infty f(x)^2 dx \right) \left(\int_0^\infty xf(x) dx \right)

5

1

evaluate sum of partial products

A sequence

\{a_n\}_{n\geq 1}

is defined by a recurrence relation a_1 = 1, a_{n+1} = \log \frac{e^{a_n}-1}{a_n} And a sequence

\{b_n\}_{n\geq 1}

is defined as

b_n = \prod\limits_{i=1}^n a_i

. Evaluate an infinite series

\sum\limits_{n=1}^\infty b_n

.

4

1

unitary matrices, determinant

For any

n\times n

unitary matrices

A,B

, prove that

|\det (A+2B)|\leq 3^n

.

3

1

Multiple integral, curl, vector fields

Two vector fields

\mathbf{F},\mathbf{G}

are defined on a three dimensional region

W=\{(x,y,z)\in\mathbb{R}^3 : x^2+y^2\leq 1, |z|\leq 1\}

. \mathbf{F}(x,y,z) = (\sin xy, \sin yz, 0), \mathbf{G} (x,y,z) = (e^{x^2+y^2+z^2}, \cos xz, 0) Evaluate the following integral.

\iiint_{W} (\mathbf{G}\cdot \text{curl}(\mathbf{F}) - \mathbf{F}\cdot \text{curl}(\mathbf{G})) dV

2

1

Volume of Ellipsoid

Matrices

A

,

B

are given as follows. A=\begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}, B = \begin{pmatrix} 4 & 2 & 0 \\ 2 & 4 & 0 \\ 0 & 0 & 12\end{pmatrix} Find volume of

V=\{\mathbf{x}\in\mathbb{R}^3 : \mathbf{x}\cdot A\mathbf{x} \leq 1 < \mathbf{x}\cdot B\mathbf{x} \}

.

1

Find rank of 3by3 matrix

A = \begin{pmatrix} 2019 & 2020 & 2021 \\ 2020 & 2021 & 2022 \\ 2021 & 2022 & 2023 \end{pmatrix}

. Find

\text{rank}(A)

.

2019 Korea USCM

Subcontests

Vector space of matrices, determiant preserving transform

Condition for a series to converge

Holder like inequality

evaluate sum of partial products

unitary matrices, determinant

Multiple integral, curl, vector fields

Volume of Ellipsoid

Find rank of 3by3 matrix