MathDB

1

Sequence with strange inequality relation

\{a_n\}_{n\geq 0}

satisfies

a_0 = 0

,

\frac{100}{101} <a_{100}<1

, and

2a_n - a_{n-1} -a_{n+1} \leq 2 (1-a_n )^3

for every

n\geq 1

. (1) Define a sequence

b_n = a_n - \frac{n}{n+1}

. Prove that

b_n\leq b_{n+1}

for any

n\geq 100

. (2) Determine whether infinite series

\sum_{n=1}^\infty \frac{a_n}{n^2}

converges or diverges.

7

1

Non-symmetric but positive definite

Suppose a

3\times 3

matrix

A

satisfies

\mathbf{v}^t A \mathbf{v} > 0

for any vector

\mathbf{v} \in\mathbb{R}^3 -\{0\}

. (Note that

A

may not be a symmetric matrix.) (1) Prove that

\det(A)>0

. (2) Consider diagonal matrix

D=\text{diag}(-1,1,1)

. Prove that there's exactly one negative real among eigenvalues of

AD

.

6

1

Easy gronwall type inequality

Suppose a continuous function

f:[0,1]\to\mathbb{R}

is differentiable on

(0,1)

and

f(0)=1

,

f(1)=0

. Then, there exists

0<x_0<1

such that

|f'(x_0)| \geq 2018 f(x_0)^{2018}

5

1

Range of spectral radius

A real symmetric

2018\times 2018

matrix

A=(a_{ij})

satisfies

|a_{ij}-2018|\leq 1

for every

1\leq i,j\leq 2018

. Denote the largest eigenvalue of

A

by

\lambda(A)

. Find maximum and minumum value of

\lambda(A)

.

4

1

Find determinant of matrix obtained by two permuations

n\geq 2

is a given integer. For two permuations

(\alpha_1,\cdots,\alpha_n)

and

(\beta_1,\cdots,\beta_n)

of

1,\cdots,n

, consider

n\times n

matrix

A= \left(a_{ij} \right)_{1\leq i,j\leq n}

defined by

a_{ij} = (1+\alpha_i \beta_j )^{n-1}

. Find every possible value of

\det(A)

.

3

1

Maximum value of Integral on subset of plane

\Phi

is a function defined on collection of bounded measurable subsets of

\mathbb{R}

defined as

\Phi(S) = \iint_S (1-5x^2 + 4xy-5y^2 ) dx dy

Find the maximum value of

\Phi

.

2

1

rank 1 projection

Suppose a

n\times n

real matrix

A

satisfies

\text{tr}(A)=2018

,

\text{rank}(A)=1

. Prove that

A^2=2018 A

.

1

Sequence of vector defined by exterior product

Given vector

\mathbf{u}=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right)\in\mathbb{R}^3

and recursively defined sequence of vectors

\{\mathbf{v}_n\}_{n\geq 0}

\mathbf{v}_0 = (1,2,3), \mathbf{v}_n = \mathbf{u}\times\mathbf{v}_{n-1} Evaluate the value of infinite series

\sum_{n=1}^\infty (3,2,1)\cdot \mathbf{v}_{2n}

.

2018 Korea USCM

Subcontests

Sequence with strange inequality relation

Non-symmetric but positive definite

Easy gronwall type inequality

Range of spectral radius

Find determinant of matrix obtained by two permuations

Maximum value of Integral on subset of plane

rank 1 projection

Sequence of vector defined by exterior product