CONO-SUR 2003

Define the sequence $\{a_n\}$ in the following manner: $a_1=1$ $a_2=3$ $a_{n+2}=2a_{n+1}a_{n}+1$ ; for all $n\geq1$ Prove that the largest power of $2$ that divides $a_{4006}-a_{4005}$ is $2^{2003}.$

Let $ABC$ be an acute triangle such that $\angle{B}=60$ . The circle with diameter $AC$ intersects the internal angle bisectors of $A$ and $C$ at the points $M$ and $N$ , respectively $(M\neq{A},$ $N\neq{C})$ . The internal bisector of $\angle{B}$ intersects $MN$ and $AC$ at the points $R$ and $S$ , respectively. Prove that $BR\leq{RS}$ .

In an acute triangle $ABC$ , the points $H$ , $G$ , and $M$ are located on $BC$ in such a way that $AH$ , $AG$ , and $AM$ are the height, angle bisector, and median of the triangle, respectively. It is known that $HG=GM$ , $AB=10$ , and $AC=14$ . Find the area of triangle $ABC$ .

Let $n=3k+1$ , where $k$ is a positive integer. A triangular arrangement of side $n$ is formed using circles with the same radius, as is shown in the figure for $n=7$ . Determine, for each $k$ , the largest number of circles that can be colored red in such a way that there are no two mutually tangent circles that are both colored red.

Attachments:

Show that there exists a sequence of positive integers $x_1, x_2,…x_n,…$ that satisfies the following two conditions: (i) Every positive integer appears exactly once, (ii) For every $n=1,2,…$ the partial sum $x_1+x_2+…+x_n$ is divisible by $n^n$ .