INMO 1991

Find the number of positive integers $n$ for which

(i) $n \leq 1991$ ;

(ii) 6 is a factor of $(n^2 + 3n +2)$ .

Given an acute-angled triangle $ABC$ , let points $A' , B' , C'$ be located as follows: $A'$ is the point where altitude from $A$ on $BC$ meets the outwards-facing semicircle on $BC$ as diameter. Points $B', C'$ are located similarly. Prove that $A[BCA']^2 + A[CAB']^2 + A[ABC']^2 = A[ABC]^2$ where $A[ABC]$ is the area of triangle $ABC$ .

Given a triangle $ABC$ let \begin{eqnarray*} x &=& \tan\left(\dfrac{B-C}{2}\right) \tan \left(\dfrac{A}{2}\right) y &=& \tan\left(\dfrac{C-A}{2}\right) \tan \left(\dfrac{B}{2}\right) z &=& \tan\left(\dfrac{A-B}{2}\right) \tan \left(\dfrac{C}{2}\right). \end{eqnarray*} Prove that $x+ y + z + xyz = 0$ .

Let $a,b,c$ be real numbers with $0 < a< 1$ , $0 < b < 1$ , $0 < c < 1$ , and $a+b + c = 2$ . Prove that $\dfrac{a}{1-a} \cdot \dfrac{b}{1-b} \cdot \dfrac{c}{1-c} \geq 8$ .

Triangle $ABC$ has an incenter $I$ . Let points $X$ , $Y$ be located on the line segments $AB$ , $AC$ respectively, so that $BX \cdot AB = IB^2$ and $CY \cdot AC = IC^2$ . Given that the points $X, I, Y$ lie on a straight line, find the possible values of the measure of angle $A$ .

(i) Determine the set of all positive integers $n$ for which $3^{n+1}$ divides $2^{3^n} + 1$ ;

(ii) Prove that $3^{n+2}$ does not divide $2^{3^n} + 1$ for any positive integer $n$ .