India National Olympiad 2000 Problems
Problem 1
The incircle of $ABC$ touches $BC$, $CA$, $AB$ at $K$, $L$, $M$ respectively. The line through $A$ parallel to $LK$ meets $MK$ at $P$, and the line through $A$ parallel to $MK$ meets $LK$ at $Q$. Show that the line $PQ$ bisects $AB$ and bisects $AC$.
Problem 2
Solve for integers $x,y,z$:\[ \{ \begin{array}{ccc} x + y &=& 1 – z \\ x^3 + y^3 &=& 1 – z^2 . \end{array} \]
Problem 3
If $a,b,c,x$ are real numbers such that $abc \not= 0$ and\[ \frac{xb + (1-x)c}{a} = \frac{xc + (1-x)a}{b} = \frac{xa + (1-x) b }{c}, \]then prove that $a = b = c$.
Problem 4
In a convex quadrilateral $PQRS$, $PQ =RS$, $(\sqrt{3} +1 )QR = SP$ and $\angle RSP – \angle SQP = 30^{\circ}$. Prove that $\angle PQR – \angle QRS = 90^{\circ}.$
Problem 5
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 $x^3 + ax^2 + bx + c = 0$ (real or complex), then $| \lambda | \leq 1.$
Problem 6
For any natural numbers $n$, ( $n \geq 3$), let $f(n)$ denote the number of congruent integer-sided triangles with perimeter $n$. Show that
(i) $f(1999) > f (1996)$;
(ii) $f(2000) = f(1997)$.
