1990 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

another nice 4-th problem proposed by Romania, not so hard

Find the least number of elements of a finite set

A

such that there exists a function

f : \left\{1,2,3,\ldots \right\}\rightarrow A

with the property: if

i

j

are positive integers and

i-j

is a prime number, then

f(i)

f(j)

are distinct elements of

A

1

euler lines coincide, an yugoslavian beauty

ABC

be an acute triangle and let

A_{1}, B_{1}, C_{1}

be the feet of its altitudes. The incircle of the triangle

A_{1}B_{1}C_{1}

touches its sides at the points

A_{2}, B_{2}, C_{2}

. Prove that the Euler lines of triangles

ABC

A_{2}B_{2}C_{2}

1

a polynomial equality

P(X)

P(X)=(X+2X^{2}+\ldots +nX^{n})^{2}=a_{0}+a_{1}X+\ldots +a_{2n}X^{2n}

a_{n+1}+a_{n+2}+\ldots +a_{2n}=\frac{n(n+1)(5n^{2}+5n+2)}{24}

1

classic and easy sequence

(a_{n})_{n\geq 1}

is defined by a_{1} \equal{} 1, a_{2} \equal{} 3, and a_{n \plus{} 2} \equal{} (n \plus{} 3)a_{n \plus{} 1} \minus{} (n \plus{} 2)a_{n}, \forall n \in \mathbb{N}. Find all values of

n

a_{n}

is divisible by

11