ZA-SAMO 2004

Let $a=1111\dots1111$ and $b=1111\dots1111$ where $a$ has forty ones and $b$ has twelve ones. Determine the greatest common divisor of $a$ and $b$ .

Fifty points are chosen inside a convex polygon having eighty sides such that no three of the fifty points lie on the same straight line. The polygon is cut into triangles such that the vertices of the triangles are just the fifty points and the eighty vertices of the polygon. How many triangles are there?

Find all real numbers $x$ such that $x\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor=88$ . The notation $\lfloor x\rfloor$ means the greatest integer less than or equal to $x$ .

Let $A_1$ and $B_1$ be two points on the base $AB$ of isosceles triangle $ABC$ (with $\widehat{C}>60^\circ$ ) such that $\widehat{A_1CB_1}=\widehat{BAC}$ . A circle externally tangent to the circumcircle of triangle $\triangle A_1B_1C$ is tangent also to rays $CA$ and $CB$ at points $A_2$ and $B_2$ respectively. Prove that $A_2B_2=2AB$ .

For $n\ge 2$ , find the number of integers $x$ with $0\le x<n$ , such that $x^2$ leaves a remainder of $1$ when divided by $n$ .

The numbers $a_1,a_2$ and $a_3$ are distinct positive integers, such that

(i) $a_1$ is a divisor of $a_2+a_3+a_2a_3$ ;

(ii) $a_2$ is a divisor of $a_3+a_1+a_3a_1$ ;

(iii) $a_3$ is a divisor of $a_1+a_2+a_1a_2$ .

Prove that $a_1,a_2$ and $a_3$ cannot all be prime.