2006 Austrian-Polish Competition

Part of Austrian-Polish

Subcontests

(10)

1

Pyramid

ABCDS

be a (not neccessarily straight) pyramid with a rectangular base

ABCD

and acute triangular faces

ABS,BCS,CDS,DAS

. We consider all cuboids which are inscribed inside the pyramid with its base being in the plane

ABCD

and its upper vertexes are in the triangular faces (one in each). Find the locus of the midpoints of these cuboids.

1

8x8 chessboard, 3x1 dominoes

We have an 8x8 chessboard with 64 squares. Then we have 3x1 dominoes which cover exactly 3 squares. Such dominoes can only be moved parallel to the borders of the chessboard and also only if the passing squares are free. If no dominoes can be moved, then the position is called stable. a. Find the smalles number of covered squares neccessary for a stable position. b. Prove: There exist a stable position with only one square uncovered. c. Find all Squares which are uncoverd in at least one position of b).

1

Set with infinitely many members

A\subset \{x|0\le x<1\}

with the following properties: 1.

A

has at least 4 members. 2. For all pairwise different

a,b,c,d\in A

ab+cd\in A

A

has infinetly many members.

1

sum, floors

Find all nonnegative integers

m,n

\sum_{k=1}^{2^{m}}\lfloor \frac{kn}{2^{m}}\rfloor\in \{28,29,30\}

1

Existence of triangle

D

be an interior point of the triangle

ABC

CD

AB

D_{c}

BD

AC

D_{b}

AD

BC

D_{a}

. Prove that there exists a triangle

KLM

with orthocenter

H

and the feet of altitudes

H_{k}\in LM, H_{l}\in KM, H_{m}\in KL

(AD_{c}D) = (KH_{m}H)

(BD_{c}D) = (LH_{m}H)

(BD_{a}D) = (LH_{k}H)

(CD_{a}D) = (MH_{k}H)

(CD_{b}D) = (MH_{l}H)

(AD_{b}D) = (KH_{l}H)

(PQR)

denotes the area of the triangle

PQR

1

Classical

Prove that for all positive integers

n

and all positive reals

a,b,c

the following inequality holds:

\frac{a^{n+1}}{a^{n}+a^{n-1}b+\ldots+b^{n}}+\frac{b^{n+1}}{b^{n}+b^{n-1}c+\ldots+c^{n}}+\frac{c^{n+1}}{c^{n}+c^{n-1}a+\ldots+a^{n}}\\ \ge \frac{a+b+c}{n+1}

1

"Nice" numbers

A positive integer

d

is called nice iff for all positive integers

x,y

d

(x+y)^{5}-x^{5}-y^{5}

d

(x+y)^{7}-x^{7}-y^{7}

. a) Is 29 nice? b) Is 2006 nice? c) Prove that infinitely many nice numbers exist.

1

Is the tetrahedron regular?

ABCD

is a tetrahedron. Let

K

be the center of the incircle of

CBD

M

be the center of the incircle of

ABD

L

be the gravycenter of

DAC

N

be the gravycenter of

BAC

AK

BL

CM

DN

have one common point. Is

ABCD

necessarily regular?

1

Dividing properties

M(n)=\{n,n+1,n+2,n+3,n+4,n+5\}

be a set of 6 consecutive integers. Let's take all values of the form

\frac{a}{b}+\frac{c}{d}+\frac{e}{f}

\{a,b,c,d,e,f\}=M(n)

\frac{x}{u}+\frac{y}{v}+\frac{z}{w}=\frac{xvw+yuw+zuv}{uvw}

be the greatest of all these values. a) show: for all odd

n

\gcd (xvw+yuw+zuv, uvw)=1

\gcd (x,u)=\gcd (y,v)=\gcd (z,w)=1

. b) for which positive integers

n

\gcd (xvw+yuw+zuv, uvw)=1

1

Find Polynomials

Find all polynomials

P(x)

with real coefficients satisfying the equation

(x+1)^{3}P(x-1)-(x-1)^{3}P(x+1)=4(x^{2}-1) P(x)

for all real numbers

x