ققنوس

1	Let $H$ be the orthocenter of an acute-angled triangle $ABC$ . The circle $\Gamma_{A}$ centered at the midpoint of $BC$ and passing through $H$ intersects the sideline $BC$ at points $A_{1}$ and $A_{2}$ . Similarly, define the points $B_{1}$ , $B_{2}$ , $C_{1}$ and $C_{2}$ . Prove that six points $A_{1}$ , $A_{2}$ , $B_{1}$ , $B_{2}$ , $C_{1}$ and $C_{2}$ are concyclic.	S
2	(i) If $x$ , $y$ and $z$ are three real numbers, all different from $1$ , such that $xyz = 1$ , then prove that $\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1$ . (With the $\sum$ sign for cyclic summation, this inequality could be rewritten as $\sum \frac {x^{2}}{\left(x - 1\right)^{2}} \geq 1$ .) (ii) Prove that equality is achieved for infinitely many triples of rational numbers $x$ , $y$ and $z$ .	S
3	Prove that there are infinitely many positive integers $n$ such that $n^{2} + 1$ has a prime divisor greater than $2n + \sqrt {2n}$ .	S

Day 2 - 17 July 2008

1	Find all functions $f: (0, \infty) \mapsto (0, \infty)$ (so $f$ is a function from the positive real numbers) such that $\frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}$ for all positive real numbes $w,x,y,z,$ satisfying $wx = yz.$	S
2	Let $n$ and $k$ be positive integers with $k \geq n$ and $k - n$ an even number. Let $2n$ lamps labelled $1$ , $2$ , ..., $2n$ 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 sequeces 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 number of such sequences consisting of $k$ steps, resulting in the state where lamps 1 through $n$ are all on, and lamps $n + 1$ tgrough $2n$ are all off, but where none of the lamps $n + 1$ through $2n$ is ever switched on. Determine $\frac {N}{M}$	S
3	Let $ABCD$ be a convex quadrilateral with $BA$ different from $BC$ . Denote the incircles of triangles $ABC$ and $ADC$ by $k_{1}$ and $k_{2}$ respectively. Suppose that there exists a circle $k$ 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 $k_{1}$ and $k_{2}$ intersects on $k$ .

حرف های یه نوجوان