International Mathematical Olympiad · 2008

Problems

Problem 1An acute-angled triangle $ABC$ has orthocentre $H$ . The circle passing through $H$ with centre the midpoint of $BC$ intersects the line $BC$ at $A_1$ and $A_2$ . Similarly, the circle passing through $H$ with centre the midpoint of $CA$ intersects the line $CA$ at $B_1$ and $B_2$ , and the circle passing through $H$ with centre the midpoint of $AB$ intersects the line $AB$ at $C_1$ and $C_2$ . Show that $A_1, A_2, B_1, B_2, C_1, C_2$ lie on a circle.Solutions: 1
Problem 2Let $x$ , $y$ , $z$ be real numbers, all different from $1$ , such that $xyz=1$ . Prove that $\frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1,$ and prove that equality holds for infinitely many triples of rational numbers $x$ , $y$ , $z$ .Solutions: 1
Problem 3Prove that there are infinitely many positive integers $n$ such that $n^2+1$ has a prime factor greater than $2n+\sqrt{2n}$ .Solutions: 1
Problem 4Find all functions $f: (0,\infty) \to (0,\infty)$ (so $f$ is a function from the positive real numbers) such that $\dfrac{(f(w))^2 + (f(x))^2}{f(y^2) + f(z^2)} = \dfrac{w^2 + x^2}{y^2 + z^2}$ for all positive real numbers $w, x, y, z$ , satisfying $wx = yz$ .Solutions: 1
Problem 5Let $n$ and $k$ be positive integers with $k \ge n$ and $k - n$ an even number. Let $2n$ lamps labelled $1, 2, \ldots, 2n$ be given, each of which can be either on or off. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on). Let $N$ be the number of such sequences consisting of $k$ steps and resulting in the state where lamps $1$ through $n$ are all on, and lamps $n+1$ through $2n$ are all off. Let $M$ be the number of such sequences consisting of $k$ steps, resulting in the state where lamps $1$ through $n$ are all on, and lamps $n+1$ through $2n$ are all off, but where none of the lamps $n+1$ through $2n$ is ever switched on. Determine $\dfrac{N}{M}$ .Solutions: 1
Problem 6Let $ABCD$ be a convex quadrilateral with $BA \ne BC$ . Denote the incircles of triangles $ABC$ and $ADC$ by $\omega_1$ and $\omega_2$ respectively. Suppose that there exists a circle $\omega$ tangent to ray $BA$ beyond $A$ and to the ray $BC$ beyond $C$ , which is also tangent to the lines $AD$ and $CD$ . Prove that the common external tangents to $\omega_1$ and $\omega_2$ intersect on $\omega$ .Solutions: 1