MathDB

1

Inequality with powers

(k,n)

of non-negative integers such that the following inequality holds

\forall x,y>0

:

1+ \frac{y^n}{x^k} \geq \frac{(1+y)^n}{(1+x)^k}.

9

1

x^3+y^3+z^3=2

Consider the equation

x^3 + y^3 + z^3 = 2

. a) Prove that it has infinitely many integer solutions

x,y,z

. b) Determine all integer solutions

x, y, z

with

|x|, |y|, |z| \leq 28

.

8

1

System of linear equations in a square grid

Given the sets

R_{mn} = \{ (x,y) \mid x=0,1,\dots,m; y=0,1,\dots,n \}

, consider functions

f:R_{mn}\to \{-1,0,1\}

with the following property: for each quadruple of points

A_1,A_2,A_3,A_4\in R_{mn}

which form a square with side length

0<s<3

, we have

f(A_1)+f(A_2)+f(A_3)+f(A_4)=0.

For each pair

(m,n)

of positive integers, determine

F(m,n)

, the number of such functions

f

on

R_{mn}

.

7

1

System of equations, n variables, n^3-n^2

For each natural number

n\geq 2

, solve the following system of equations in the integers

x_1, x_2, ..., x_n

:

(n^2-n)x_i+\left(\prod_{j\neq i}x_j\right)S=n^3-n^2,\qquad \forall 1\le i\le n

where

S=x_1^2+x_2^2+\dots+x_n^2.

6

1

Cauchy with powers

Determine all monotone functions

f: \mathbb{Z} \rightarrow \mathbb{Z}

, so that for all

x, y \in \mathbb{Z}

f(x^{2005} + y^{2005}) = (f(x))^{2005} + (f(y))^{2005}

5

1

Three midpoints are collinear

Given is a convex quadrilateral

ABCD

with

AB=CD

. Draw the triangles

ABE

and

CDF

outside

ABCD

so that

\angle{ABE} = \angle{DCF}

and

\angle{BAE}=\angle{FDC}

. Prove that the midpoints of

\overline{AD}

,

\overline{BC}

and

\overline{EF}

are collinear.

4

1

FLT fraction is a square

Determine the smallest natural number

a\geq 2

for which there exists a prime number

p

and a natural number

b\geq 2

such that

\frac{a^p - a}{p}=b^2.

3

1

Inequality with a(j)-a(i)<=j-i

Let

a_0, a_1, a_2, ... , a_n

be real numbers, which fulfill the following two conditions: a)

0 = a_0 \leq a_1 \leq a_2 \leq ... \leq a_n

. b) For all

0 \leq i < j \leq n

holds:

a_j - a_i \leq j-i

.
Prove that

\left( \displaystyle \sum_{i=0}^n a_i \right)^2 \geq \sum_{i=0}^n a_i^3.

2

1

Iterated polynomial ratio is x^{28}

Determine all polynomials

P

with integer coefficients satisfying

P(P(P(P(P(x)))))=x^{28}\cdot P(P(x))\qquad \forall x\in\mathbb{R}

1

Number of diagonal-quadrangles

For a convex

n

-gon

P_n

, we say that a convex quadrangle

Q

is a diagonal-quadrangle of

P_n

, if its vertices are vertices of

P_n

and its sides are diagonals of

P_n

. Let

d_n

be the number of diagonal-quadrangles of a convex

n

-gon. Determine

d_n

for all

n\geq 8

.

2005 Austrian-Polish Competition

Subcontests

Inequality with powers

x^3+y^3+z^3=2

System of linear equations in a square grid

System of equations, n variables, n^3-n^2

Cauchy with powers

Three midpoints are collinear

FLT fraction is a square

Inequality with a(j)-a(i)<=j-i

Iterated polynomial ratio is x^{28}

Number of diagonal-quadrangles

2005 Austrian-Polish Competition

Subcontests

Inequality with powers

x^3+y^3+z^3=2

System of linear equations in a square grid

System of equations, n variables, n^3-n^2

Cauchy with powers

Three midpoints are collinear

FLT fraction is a square

Inequality with a(j)-a(i)&lt;=j-i

Iterated polynomial ratio is x^{28}

Number of diagonal-quadrangles

Inequality with a(j)-a(i)<=j-i