OIM 1989

Determine all triples of real numbers that satisfy the following system of equations: $$x+y-z=-1 x^2-y^2+z^2=1 -x^3+y^3+z^3=-1$$

Let $x,y,z$ be real numbers such that $0\le x,y,z\le\frac{\pi}{2}$ . Prove the inequality $$\frac{\pi}{2}+2\sin x\cos y+2\sin y\cos z\ge\sin 2x+\sin 2y+\sin 2z.$$

Let $a,b$ and $c$ be the side lengths of a triangle. Prove that: $$\frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a}<\frac{1}{16}$$

The incircle of the triangle $ABC$ is tangent to sides $AC$ and $BC$ at $M$ and $N$ , respectively. The bisectors of the angles at $A$ and $B$ intersect $MN$ at points $P$ and $Q$ , respectively. Let $O$ be the incentre of $\triangle ABC$ . Prove that $MP\cdot OA=BC\cdot OQ$ .

Let the function $f$ be defined on the set $\mathbb{N}$ such that $\text{(i)}\ \ f(1)=1$ $\text{(ii)}\ f(2n+1)=f(2n)+1$ $\text{(iii)} f(2n)=3f(n)$ Determine the set of values taken $f$ .

Show that the equation $2x^2-3x=3y^2$ has infinitely many solutions in positive integers.