CONO-SUR 2015

Show that, for any integer $n$ , the number $n^3 - 9n + 27$ is not divisible by $81$ .

$3n$ lines are drawn on the plane ( $n > 1$ ), such that no two of them are parallel and no three of them are concurrent. Prove that, if $2n$ of the lines are coloured red and the other $n$ lines blue, there are at least two regions of the plane such that all of their borders are red.

Note: for each region, all of its borders are contained in the original set of lines, and no line passes through the region.

Let $ABCD$ be a convex quadrilateral such that $\angle{BAD} = 90^{\circ}$ and its diagonals $AC$ and $BD$ are perpendicular. Let $M$ be the midpoint of side $CD$ , and $E$ be the intersection of $BM$ and $AC$ . Let $F$ be a point on side $AD$ such that $BM$ and $EF$ are perpendicular. If $CE = AF\sqrt{2}$ and $FD = CE\sqrt{2}$ , show that $ABCD$ is a square.

Determine if there exists an infinite sequence of not necessarily distinct positive integers $a_1, a_2, a_3,\ldots$ such that for any positive integers $m$ and $n$ where $1 \leq m < n$ , the number $a_{m+1} + a_{m+2} + \ldots + a_{n}$ is not divisible by $a_1 + a_2 + \ldots + a_m$ .

Let $S = \{1, 2, 3, \ldots , 2046, 2047, 2048\}$ . Two subsets $A$ and $B$ of $S$ are said to be friends if the following conditions are true:

• They do not share any elements.
• They both have the same number of elements.
• The product of all elements from $A$ equals the product of all elements from $B$ .

Prove that there are two subsets of $S$ that are friends such that each one of them contains at least $738$ elements.