1989 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

a super nice and difficult 4-th problem proposed by Romania

The elements of the set

F

are some subsets of

\left\{1,2,\ldots ,n\right\}

and satisfy the conditions: i) if

A

F

A

has three elements; ii)if

A

B

are distinct elements of

F

A

B

have at most one common element. Let

f(n)

be the greatest possible number of elements of

F

\frac{n^{2}-4n}{6}\leq f(n) \leq \frac{n^{2}-n}{6}

1

geometrical inequality involving the centroid

G

be the centroid of a triangle

ABC

d

be a line that intersects

AB

AC

B_{1}

C_{1}

, respectively, such that the points

A

G

are not separated by

d

[BB_{1}GC_{1}]+[CC_{1}GB_{1}] \geq \frac{4}{9}[ABC]

1

polynomial

\overline{a_{n}a_{n-1}\ldots a_{1}a_{0}}

be the decimal representation of a prime positive integer such that

n>1

a_{n}>1

. Prove that the polynomial

P(x)=a_{n}x^{n}+\ldots +a_{1}x+a_{0}

cannot be written as a product of two non-constant integer polynomials.

1

n positive integer and d_{k}, k<=n its divisors

n

be a positive integer and let

d_{1},d_{2},,\ldots ,d_{k}

be its divisors, such that

1=d_{1}<d_{2}<\ldots <d_{k}=n

. Find all values of

n

k\geq 4

n=d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}