MathDB

1

integral inequality with integral condition, [0,1]->R

n

be a positive integer and

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

be a continuous function such that

\int^1_0x^kf(x)dx=1

for every

k\in\{0,1,\ldots,n-1\}

. Prove that

\int^1_0f(x)^2dx\ge n^2.

Problem 3

1

FE Mn(R)->[n], inequality f(XY)≤min{f(X),f(Y)}

Let

\mathcal M_n(\mathbb R)

denote the set of all real

n\times n

matrices. Find all surjective functions

f:\mathcal M_n(\mathbb R)\to\{0,1,\ldots,n\}

which satisfy

f(XY)\le\min\{f(X),f(Y)\}

for all

X,Y\in\mathcal M_n(\mathbb R)

.

Problem 2

1

sequences of polygons, vertices are midpoints of sides, convergence

Let

P_0,P_1,P_2,\ldots

be a sequence of convex polygons such that, for each

k\ge0

, the vertices of

P_{k+1}

are the midpoints of all sides of

P_k

. Prove that there exists a unique point lying inside all these polygons.

Problem 1

1

function analysis, f(x)=ax has &ge;1 solution for each a>0

Let

f:[1,\infty)\to(0,\infty)

be a continuous function. Assume that for every

a>0

, the equation

f(x)=ax

has at least one solution in the interval

[1,\infty)

. (a) Prove that for every

a>0

, the equation

f(x)=ax

has infinitely many solutions. (b) Give an example of a strictly increasing continuous function

f

with these properties.

2008 SEEMOUS

Subcontests

integral inequality with integral condition, [0,1]->R

FE Mn(R)->[n], inequality f(XY)≤min{f(X),f(Y)}

sequences of polygons, vertices are midpoints of sides, convergence

function analysis, f(x)=ax has &ge;1 solution for each a>0

2008 SEEMOUS

Subcontests

integral inequality with integral condition, [0,1]-&gt;R

FE Mn(R)-&gt;[n], inequality f(XY)&le;min{f(X),f(Y)}

sequences of polygons, vertices are midpoints of sides, convergence

function analysis, f(x)=ax has &amp;ge;1 solution for each a&gt;0

integral inequality with integral condition, [0,1]->R

FE Mn(R)->[n], inequality f(XY)≤min{f(X),f(Y)}

function analysis, f(x)=ax has ≥1 solution for each a>0