Let $ABC$ be a triangle with orthocenter $H$. The external bisector of $\angle BHC$ meets sides $AB$ and $AC$ at $D$ and $E$, respectively. The internal bisector of $\angle BAC$ meets the circumcircle of triangle $ADE$ at $K$. Prove that line $HK$ passes through the midpoint of $BC$.
Find all pairs of natural numbers $(n;k)$, where $n$ is a nonnegative integer and $k$ is an integer greater than $1$, such that
$$A=17^{2006n}+4\cdot17^{2n}+7\cdot19^{5n}$$
can be expressed as a product of $k$ consecutive positive integers.
There are $2006$ points in space, no four of which are coplanar. Join every pair of points by a segment. A natural number $m$ is called good if each segment can be assigned a natural number not exceeding $m$ so that every triangle formed by any three of the points has two sides assigned equal numbers and its remaining side assigned a number greater than those two numbers. Find the smallest good number.
Prove that, for all real numbers $x,y,z\in[1,2]$,
$$(x+y+z)\left(\frac1x+\frac1y+\frac1z\right)\ge6\left(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}\right).$$
Determine exactly when equality holds.
Let $ABC$ be an acute, non-isosceles triangle inscribed in a circle with center $O$ and radius $R$. A variable line $d$ is always perpendicular to $OA$ and meets rays $AB$ and $AC$ at $M$ and $N$, respectively. Suppose that lines $BN$ and $CN$ intersect at $K$, and that line $AK$ meets line $BC$.
1. Let $P=AK\cap BC$. Prove that the circumcircle of triangle $MNP$ always passes through a fixed point as $d$ varies.
2. Let $H$ be the orthocenter of triangle $AMN$. Write $BC=a$, and let $l$ be the distance from $A$ to line $HK$. Prove that $HK$ always passes through the orthocenter of triangle $ABC$. Deduce that $l\le\sqrt{4R^2-a^2}$, and determine exactly when equality holds.
A real sequence $(a_n)$ is defined by
$$a_0=1,\qquad a_{n+1}=\frac12\left(a_n+\frac1{3a_n}\right)\quad\text{for every }n=1,2,3,\ldots.$$
Prove that, for every integer $n$, the number
$$A_n=\frac3{3a_n^2-1}$$
is a perfect square and has at least $n$ distinct prime divisors.