OSN 2005

Let $n$ be a positive integer. Determine the number of triangles (non congruent) with integral side lengths and the longest side length is $n$ .

For an arbitrary positive integer $n$ , define $p(n)$ as the product of the digits of $n$ (in decimal). Find all positive integers $n$ such that $11p(n)=n^2-2005$ .

Let $k$ and $m$ be positive integers such that $\frac12\left(\sqrt{k+4\sqrt{m}}-\sqrt{k}\right)$ is an integer.

(a) Prove that $\sqrt{k}$ is rational.

(b) Prove that $\sqrt{k}$ is a positive integer.

Let $M$ be a point in triangle $ABC$ such that $\angle AMC=90^{\circ}$ , $\angle AMB=150^{\circ}$ , $\angle BMC=120^{\circ}$ . The centers of circumcircles of triangles $AMC,AMB,BMC$ are $P,Q,R$ , respectively. Prove that the area of $\triangle PQR$ is greater than the area of $\triangle ABC$ .

For an arbitrary real number $x$ , $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$ . Prove that there is exactly one integer $m$ which satisfy $m-\left\lfloor \frac{m}{2005}\right\rfloor=2005$ .

Find all triples $(x,y,z)$ of integers which satisfy

$x(y + z) = y^2 + z^2 - 2$

$y(z + x) = z^2 + x^2 - 2$

$z(x + y) = x^2 + y^2 - 2$ .

Let $ABCD$ be a convex quadrilateral. Square $AB_1A_2B$ is constructed such that the two vertices $A_2,B_1$ is located outside $ABCD$ . Similarly, we construct squares $BC_1B_2C$ , $CD_1C_2D$ , $DA_1D_2A$ . Let $K$ be the intersection of $AA_2$ and $BB_1$ , $L$ be the intersection of $BB_2$ and $CC_1$ , $M$ be the intersection of $CC_2$ and $DD_1$ , and $N$ be the intersection of $DD_2$ and $AA_1$ . Prove that $KM$ is perpendicular to $LN$ .

There are $90$ contestants in a mathematics competition. Each contestant gets acquainted with at least $60$ other contestants. One of the contestants, Amin, state that at least four contestants have the same number of new friends. Prove or disprove his statement.