INMO 2000

The in-circle of triangle $A B C$ touches the sides $B C, C A$ and $A B$ in $K, L$ and $M$ respectively. The line through $A$ and parallel to $L K$ meets $M K$ in $P$ and the line through $A$ and parallel to $M K$ meets $L K$ in $Q$. Show that the line $P Q$ bisects the sides $A B$ and $A C$ of triangle $A B C$.

Solve for integers $x, y, z$ :

$$x+y=1-z, x^{3}+y^{3}=1-z^{2}$$

If $a, b, c, x$ are real numbers such that $a b c \neq 0$ and

$$\frac{x b+(1-x) c}{a}=\frac{x c+(1-x) a}{b}=\frac{x a+(1-x) b}{c}$$

then prove that either $a+b+c=0$ or $a=b=c$.

In a convex quadrilateral $P Q R S, P Q=R S,(\sqrt{3}+1) Q R=S P$ and $\angle R S P-\angle S P Q=$ $30^{\circ}$. Prove that

$$\angle P Q R-\angle Q R S=90^{\circ}$$

Let $a, b, c$ be three real numbers such that $1 \geq a \geq b \geq c \geq 0$. Prove that if $\lambda$ is a root of the cubic equation $x^{3}+a x^{2}+b x+c=0$ (real or complex), then $|\lambda| \leq 1$.

For any natural number $n,(n \geq 3)$, let $f(n)$ denote the number of non-congruent integer-sided triangles with perimeter $n$ (e.g., $f(3)=1, f(4)=0, f(7)=2$ ). Show that

(a) $f(1999)>f(1996)$

(b) $f(2000)=f(1997)$.