INMO 2001

Let $A B C$ be a triangle in which no angle is $90^{\circ}$. For any point $P$ in the plane of the triangle, let $A_{1}, B_{1}, C_{1}$ denote the reflections of $P$ in the sides $B C, C A, A B$ respectively. Prove the following statements:

(a) If $P$ is the incentre or an excentre of $A B C$, then $P$ is the circumcentre of $A_{1} B_{1} C_{1}$;

(b) If $P$ is the circumcentre of $A B C$, then $P$ is the orthocentre of $A_{1} B_{1} C_{1}$;

(c) If $P$ is the orthocentre of $A B C$, then $P$ is either the incentre or an excentre of $A_{1} B_{1} C_{1}$.

Show that the equation

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

has infinitely many solutions in integers $x, y, z$.

If $a, b, c$ are positive real numbers such that $a b c=1$, prove that

$$a^{b+c} b^{c+a} c^{a+b} \leq 1$$

Given any nine integers show that it is possible to choose, from among them, four integers $a, b, c, d$ such that $a+b-c-d$ is divisible by 20 . Further show that such a selection is not possible if we start with eight integers instead of nine.

Let $A B C$ be a triangle and $D$ be the mid-point of side $B C$. Suppose $\angle D A B=\angle B C A$ and $\angle D A C=15^{\circ}$. Show that $\angle A D C$ is obtuse. Further, if $O$ is the circumcentre of $A D C$, prove that triangle $A O D$ is equilateral.

Let $\mathbf{R}$ denote the set of all real numbers. Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$ satisfying the condition

$$f(x+y)=f(x) f(y) f(x y)$$

for all $x, y$ in $\mathbf{R}$.