2017 Korea USCM

Part of South Korea USCM

Subcontests

(8)

1

damping oscillators

u(t)

is solution of the following initial value problem.

\begin{cases} u''(t) + u'(t) = \sin u(t) &\;\;(t>0),\\ u(0)=1,\;\; u'(0)=0 & \end{cases}

u(t)

u'(t)

t>0

\lim\limits_{t\to\infty} u(t)

1

Well-known trigonometric series bound

Prove the following inequality holds if

\{a_n\}

is a deceasing sequence of positive reals, and

0<\theta<\frac{\pi}{2}

\left|\sum_{n=1}^{2017} a_n \cos n\theta \right| \leq \frac{\pi a_1}{\theta}

1

matrix relation, every entry 1 matrix

Given a positive integer

n

and a real valued

n\times n

A

J

n\times n

matrix with every entry

1

A

satisfies the following relations. A+A^T = \frac{1}{n} J, AJ = \frac{1}{2} J Show that

A^m-I

is an invertible matrix for all positive odd integer

m

1

limit of integral

Evaluate the following limit.

\lim_{n\to\infty} \sqrt{n} \int_0^\pi \sin^n x dx

1

discriminant of cubic polynomial into matrix form

For a real coefficient cubic polynomial

f(x)=ax^3+bx^2+cx+d

, denote three roots of the equation

f(x)=0

\alpha,\beta,\gamma

. Prove that the three roots

\alpha,\beta,\gamma

are distinct real numbers iff the real symmetric matrix \begin{pmatrix} 3 & p_1 & p_2 \\ p_1 & p_2 & p_3 \\ p_2 & p_3 & p_4 \end{pmatrix}, p_i = \alpha^i + \beta^i + \gamma^i is positive definite.

1

Find limit, recurrence relation

\{a_n\}

defined by recurrence relation

a_{n+1} = 1+\frac{n^2}{a_n}

a_1>1

, find the value of

\lim\limits_{n\to\infty} \frac{a_n}{n}

1

A two var poly is linear comb of (x+ay)^k

Show that any real coefficient polynomial

f(x,y)

is a linear combination of polynomials of the form

(x+ay)^k

a

is a real number and

k

is a non-negative integer.)

1

sum of determinants

n(\geq 2)

is a given integer and

T

n\times n

matrices whose entries are elements of the set

S=\{1,\cdots,2017\}

. Evaluate the following value.

\sum_{A\in T} \text{det}(A)