2014 SEEMOUS

Part of SEEMOUS

Subcontests

(4)

1

limit integral

\lim_{n\to\infty}n\int^n_0\frac{\operatorname{arctan}\frac xn}{x(x^2+1)}dx=\frac\pi2

. b) Find the limit

\lim_{n\to\infty}n\left(m\int^n_0\frac{\operatorname{arctan}\frac xn}{x(x^2+1)}dx-\frac\pi2\right)

1

convergence of recurrence

Consider the sequence

(x_n)

x_1=2,\enspace x_{n+1}=\frac{x_n+1+\sqrt{x_n^2+2x_n+5}}2,\enspace n\ge2.

Prove that the sequence

y_n=\sum_{k=1}^n\frac1{x_k^2-1},\enspace n\ge1

is convergent and find its limit.

1

determinant of matrix, values of function

n

be a nonzero natural number and

f:\mathbb R\to\mathbb R\setminus\{0\}

be a function such that

f(2014)=1-f(2013)

x_1,x_2,x_3,\ldots,x_n

be real numbers not equal to each other. If

\begin{vmatrix}1+f(x_1)&f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&1+f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&f(x_2)&1+f(x_3)&\cdots&f(x_n)\\\vdots&\vdots&\vdots&\ddots&\vdots\\f(x_1)&f(x_2)&f(x_3)&\cdots&1+f(x_n)\end{vmatrix}=0,

f

is not continuous.

1

SEEMOUS 2014(P3)

A\in M_n(\mathbb{C})

a\in \mathbb{C}

A-A^*=2aI_n

A^*=(\overline{A})^T

I_n

is identity matrix. (i) Show that

|\det A|\ge |a|^n

. (ii) Show that if

|\det A|=|a|^n

A=aI_n