2012 SEEMOUS

Part of SEEMOUS

Subcontests

(4)

1

two integrals, limits of n to infinity

\lim_{n\to\infty}n\int^1_0\left(\frac{1-x}{1+x}\right)^ndx.

k\ge1

be an integer. Compute

\lim_{n\to\infty}n^{k+1}\int^1_0\left(\frac{1-x}{1+x}\right)^nx^kdx.

1

property of traces

a) Prove that if

k

is an even positive integer and

A

is a real symmetric

n\times n

matrix such that

\operatorname{tr}(A^k)^{k+1}=\operatorname{tr}(A^{k+1})^k

A^n=\operatorname{tr}(A)A^{n-1}.

b) Does the assertion from a) also hold for odd positive integers

k

1

convergent sum of angle measures

a_n>0

n\ge1

. Consider the right triangles

\triangle A_0A_1A_2

\triangle A_0A_2A_3,\ldots

\triangle A_0A_{n-1}A_n,\ldots,

as in the figure. (More precisely, for every

n\ge2

A_0A_n

\triangle A_0A_{n-1}A_n

\triangle A_0A_nA_{n+1}

with right angle

\angle A_0A_nA_{n+1}

, and the vertices

A_{n-1}

A_{n+1}

lie on the opposite sides of the straight line

A_0A_n

|A_{n-1}A_n|=a_n

n\ge1

.) https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi8yL2M1ZjAxM2I1ZWU0N2E4MzQyYWIzNmQ5OGM3NjJlZjljODdmMTliLnBuZw==&rn=U0VFTU9VUyAyMDEyLnBuZw== Is it possible for the set of points

\{A_n\mid n\ge0\}

to be unbounded but the series

\sum_{n=2}^\infty m\angle A_{n-1}A_0A_n

to be convergent? Note. A subset

B

of the plane is bounded if and only if there is a disk

D

B\subseteq D

1

A=a_ij=i^j+j^i mod 3, det A≠0

A=(a_{ij})

n\times n

a_{ij}

is the remainder of the division of

i^j+j^i

3

i,j=1,2,\ldots,n

. Find the greatest

n

\det A\ne0