FKMO 2007

Let $O$ be the circumcenter of an acute triangle $ABC$ and let $k$ be the circle with center $P$ that is tangent to $O$ at $A$ and tangent to side $BC$ at $D$ . Circle $k$ meets $AB$ and $AC$ again at $E$ and $F$ respectively. The lines $OP$ and $EP$ meet $k$ again at $I$ and $G$ . Lines $BO$ and $IG$ intersect at $H$ . Prove that $\frac{{DF}^2}{AF}=GH$ .

Given a $4\times 4$ squares table. How many ways that we can fill the table with $\{0,1\}$ such that two neighbor squares (have one common side) have product which is equal to $0$ ?

Find all triples of $(x, y, z)$ of positive intergers satisfying $1+{4}^{x}+{4}^{y}=z^2$ .

Find all pairs $(p, q)$ of primes such that ${p}^{p}+{q}^{q}+1$ is divisible by $pq$ .

For the vertex $A$ of a triangle $ABC$ , let $l_a$ be the distance between the projections on $AB$ and $AC$ of the intersection of the angle bisector of ∠ $A$ with side $BC$ . Define $l_b$ and $l_c$ analogously. If $l$ is the perimeter of triangle $ABC$ , prove that $\frac{l_a l_b l_c}{l^3}\le\frac{1}{64}$ .

Let f:N→N be a function satisfying $kf(n)\le f(kn)\le kf(n) + k - 1$ for all $k, n\in N$ .

(a)Prove that $f(a) + f(b)\le f(a + b)\le f(a) + f(b) + 1$ for all $a, b\in N$ .

(b)If $f$ satisfies $f(2007n)\le 2007f(n) + 200$ for every $n\in N$ , show that there exists $c\in N$ such that $f(2007c) = 2007f(c)$ .