2008 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

Sequece related to diophantine equation

c

be a positive integer. The sequence

a_1,a_2,\ldots

is defined as follows a_1\equal{}c, a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3 for all positive integers

n

c

so that there are integers

k\ge1

m\ge2

so that a_k^2\plus{}c^3 is the

m

th power of some integer.

1

number of 'lattice' lines is divisible by 4

n

be a positive integer. Consider a rectangle (90n\plus{}1)\times(90n\plus{}5) consisting of unit squares. Let

S

be the set of the vertices of these squares. Prove that the number of distinct lines passing through at least two points of

S

is divisible by

4

1

existence of a sequence satisfying two inequalities

Is there a sequence

a_1,a_2,\ldots

of positive reals satisfying simoultaneously the following inequalities for all positive integers

n

: a) a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le n^2 b) \frac1{a_1}\plus{}\frac1{a_2}\plus{}\ldots\plus{}\frac1{a_n}\le2008?

1

non-trivial first problem :) 4 concyclic points

Given a scalene acute triangle

ABC

AC>BC

F

be the foot of the altitude from

C

P

AB

, different from

A

so that AF\equal{}PF. Let

H,O,M

be the orthocenter, circumcenter and midpoint of

[AC]

X

be the intersection point of

BC

HP

Y

be the intersection point of

OM

FX

OF

AC

Z

F,M,Y,Z