2002 Polish MO Finals

Part of Polish MO Finals

Subcontests

(3)

2

3 numbers written on blackboard

Three non-negative integers are written on a blackboard. A move is to replace two of the integers

k,m

k+m

|k-m|

. Determine whether we can always end with triplet which has at least two zeros

sequence and coprime elements

k

is a positive integer. The sequence

a_1, a_2, a_3, ...

a_1 = k+1

a_{n+1} = a_n ^2 - ka_n + k

a_m

a_n

are coprime (for

m \not = n

2

Collinear points

AC

BC

of acute-angled triangle

ABC

rectangles with equal areas

ACPQ

BKLC

were built exterior. Prove that midpoint of

PL

C

and center of circumcircle are collinear.

Triangle in a space

There is given a triangle

ABC

in a space. A sphere does not intersect the plane of

ABC

4

K, L, M, P

on the sphere such that

AK, BL, CM

are tangent to the sphere and

\frac{AK}{AP} = \frac{BL}{BP} = \frac{CM}{CP}

. Show that the sphere touches the circumsphere of

ABCP

1

a,b,c such that a^2+1 and b^2+1 are primes

Find all the natural numbers

a,b,c

a^2+1

b^2+1

(a^2+1)(b^2+1)=(c^2+1)