VN-TST 2002

Find all triangles $ABC$ for which $\angle ACB$ is acute and the interior angle bisector of $BC$ intersects the trisectors $(AX, (AY$ of the angle $\angle BAC$ in the points $N,P$ respectively, such that $AB=NP=2DM$ , where $D$ is the foot of the altitude from $A$ on $BC$ and $M$ is the midpoint of the side $BC$ .

Let $m$ be a given positive integer which has a prime divisor greater than $\sqrt {2m} +1$ . Find the minimal positive integer $n$ such that there exists a finite set $S$ of distinct positive integers satisfying the following two conditions:

I. $m\leq x\leq n$ for all $x\in S$ ;

II. the product of all elements in $S$ is the square of an integer.

Find all polynomials $P(x)$ with integer coefficients such that the polynomial $$Q(x)=(x^2+6x+10) \cdot P^2(x)-1$$ is the square of a polynomial with integer coefficients.

Prove that there exists an integer $n$, $n\geq 2002$, and $n$ distinct positive integers $a_1,a_2,\ldots,a_n$ such that the number $N= a_1^2a_2^2\cdots a_n^2 - 4(a_1^2+a_2^2+\cdots + a_n^2)$ is a perfect square.