2009 Polish MO Finals

Part of Polish MO Finals

Subcontests

(6)

1

Determine all possible values of c_2

n

be a natural number equal or greater than 3 . A sequence of non-negative numbers

(c_0,c_1,\ldots,c_n)

satisfies the condition: c_{p}c_{s}\plus{}c_{r}c_{t}\equal{} c_{p\plus{}r}c_{r\plus{}s} for all non-negative

p,q,r,s

such that p\plus{}q\plus{}r\plus{}s\equal{}n. Determine all possible values of

c_2

when c_1\equal{}1.

1

A sphere is inscribed in tetrahedron ABCD

A sphere is inscribed in tetrahedron

ABCD

and is tangent to faces

BCD,CAD,ABD,ABC

P,Q,R,S

respectively. Segment

PT

is the sphere's diameter, and lines

TA,TQ,TR,TS

BCD

A',Q',R',S'

. respectively. Show that

A

is the center of a circumcircle on the triangle

S'Q'R'

1

there exist numbers a_1,a_2,...,a_n

x_1,x_2,..,x_n

be non-negative numbers whose sum is

1

. Show that there exist numbers

a_1,a_2,\ldots ,a_n

chosen from amongst

0,1,2,3,4

a_1,a_2,\ldots ,a_n

are different from

2,2,\ldots ,2

and 2\leq a_1x_1\plus{}a_2x_2\plus{}\ldots\plus{}a_nx_n\leq 2\plus{}\frac{2}{3^n\minus{}1}.

1

polynomials P,Q,R are equal.

P,Q,R

be polynomials of degree at least

1

with integer coefficients such that for any real number

x

holds: P(Q(x))\equal{}Q(R(x))\equal{}R(P(x)). Show that the polynomials

P,Q,R

1

Find the smallest positive integer k

S

be a set of all points of a plane whose coordinates are integers. Find the smallest positive integer

k

for which there exists a 60-element subset of set

S

with the following condition satisfied for any two elements

A,B

of the subset there exists a point

C

S

such that the area of triangle

ABC

is equal to k .

1

hexagon (regular)

Each vertex of a convex hexagon is the center of a circle whose radius is equal to the shorter side of the hexagon that contains the vertex. Show that if the intersection of all six circles (including their boundaries) is not empty, then the hexagon is regular.