2011 SEEMOUS

Part of SEEMOUS

Subcontests

(4)

1

sequences of f:[0,1]->R, sum_k(f(k/n))

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

be a twice continuously differentiable increasing function. Define the sequences given by

L_n=\frac1n\sum_{k=0}^{n-1}f\left(\frac kn\right)

U_n=\frac1n\sum_{k=0}^nf\left(\frac kn\right)

n\ge1

. 1. The interval

[L_n,U_n]

is divided into three equal segments. Prove that, for large enough

n

I=\int^1_0f(x)\text dx

belongs to the middle one of these three segments.

1

vector inequality involving normed and inner products

\overline a,\overline b,\overline c\in\mathbb R^n

(\lVert\overline a\rVert\langle\overline b,\overline c\rangle)^2+(\lVert\overline b\rVert\langle\overline a,\overline c\rangle)^2\le\lVert\overline a\rVert\lVert\overline b\rVert(\lVert\overline a\rVert\lVert\overline b\rVert+|\langle\overline a,\overline b\rangle|)\lVert\overline c\rVert^2

\langle\overline x,\overline y\rangle

denotes the scalar (inner) product of the vectors

\overline x

\overline y

\lVert\overline x\rVert^2=\langle\overline x,\overline x\rangle

1

a_(ij)*a_(ji)&le;0, eigenvalues of A=(a_(ij))

A=(a_{ij})

n\times n

matrix such that

A^n\ne0

a_{ij}a_{ji}\le0

i,j

. Prove that there exist two nonreal numbers among eigenvalues of

A

1

Prove inequality

f:[0,1]\rightarrow R

be a continuous function and n be an integer number,n>0.Prove that

\int_0^1f(x)dx \le (n+1)*\int_0^1 x^n*f(x)dx