MathDB

1

\sum_{k=1}^{2n+1}(-1)^k [ k/2 ] P(x + k)=0 for infinitely many real x

P(x)

with real coefficients having the following property: There exists a positive integer n such that the equality

\sum_{k=1}^{2n+1}(-1)^k \left[\frac{k}{2}\right] P(x + k)=0

holds for infinitely many real numbers

x

.

10

1

plan of castle in Baranow Sandomierski as a graph with 16 vertices

The plan of the castle in Baranow Sandomierski can be presented as the graph with

16

vertices on the picture. A night guard plans a closed round along the edges of this graph. (a) How many rounds passing through each vertex exactly once are there? The directions are irrelevant. (b) How many non-selfintersecting rounds (taking directions into account) containing each edge of the graph exactly once are there? https://cdn.artofproblemsolving.com/attachments/1/f/27ca05fc689fd8d873130db9d8cc52acf49bb4.png

8

1

3 points forming a triangle with area not exceeding 1/48

In the plane are given

27

points, no three of which are collinear. Four of this points are vertices of a unit square, while the others lie inside the square. Prove that there are three points in this set forming a triangle with area not exceeding

1/48

.

6

1

filling space of 7 unit cubes by solids , no 2 of them have a common interior

Consider the solid

Q

obtained by attaching unit cubes

Q_1...Q_6

at the six faces of a unit cube

Q

. Prove or disprove that the space can be filled up with such solids so that no two of them have a common interior point.

5

1

color the vertices of a regular n-gon using at most 6 colors

For which integers

n \ge 5

is it possible to color the vertices of a regular

n

-gon using at most

6

colors in such a way that any

5

consecutive vertices have different colors?

3

1

x_{n-1}^3 = x_n+ x_1 + 1, nxn system

For each integer

n \ge 3

solve in real numbers the system of equations:

\begin{cases} x_1^3 = x_2 + x_3 + 1 \\...\\x_{n-1}^3 = x_n+ x_1 + 1\\x_{n}^3 = x_1+ x_2 + 1 \end{cases}

7

1

locus of circumcenters of similar triangles, Austrian-Polish 2000

Triangle

A_0B_0C_0

is given in the plane. Consider all triangles

ABC

such that: (i) The lines

AB,BC,CA

pass through

C_0,A_0,B_0

, respectvely, (ii) The triangles

ABC

and

A_0B_0C_0

ABC

.

2

1

minimum distance of circles in different planes of a cube

In a unit cube,

CG

is the edge perpendicular to the face

ABCD

. Let

O_1

be the incircle of square

ABCD

and

O_2

be the circumcircle of triangle

BDG

. Determine min

\{XY|X\in O_1,Y\in O_2\}

.

4

1

Austria-polish match(1)

Find all positive integers

N

having only prime divisors

2,5

such that

N+25

is a perfect square.

9

1

a+b+c=1

If three nonnegative reals

a

,

b

,

c

satisfy

a+b+c=1

, prove that

2 \leq \left(1-a^{2}\right)^{2}+\left(1-b^{2}\right)^{2}+\left(1-c^{2}\right)^{2}\leq \left(1+a\right)\left(1+b\right)\left(1+c\right)

.

2000 Austrian-Polish Competition

Subcontests

\sum_{k=1}^{2n+1}(-1)^k [ k/2 ] P(x + k)=0 for infinitely many real x

plan of castle in Baranow Sandomierski as a graph with 16 vertices

3 points forming a triangle with area not exceeding 1/48

filling space of 7 unit cubes by solids , no 2 of them have a common interior

color the vertices of a regular n-gon using at most 6 colors

x_{n-1}^3 = x_n+ x_1 + 1, nxn system

locus of circumcenters of similar triangles, Austrian-Polish 2000

minimum distance of circles in different planes of a cube

Austria-polish match(1)

a+b+c=1