SMO 2008 Open Round 2

Find all pairs of positive integers $(n,k)$ such that $(n+1)^k-1=n!$.

In the acute triangle $ABC$, $M$ is a point in the interior of segment $AC$ and $N$ is a point on the extension of segment $AC$ such that $MN=AC$. Let $D,E$ be the feet of the perpendiculars from $M,N$ onto the lines $BC,AB$, respectively. Prove that the orthocentre of $\triangle ABC$ lies on the circumcircle of $\triangle BED$.

Editorial qualification: the source figure places $N$ beyond $C$. Even with that interpretation, the hypotheses can allow $E=B$, so the stated circumcircle is then undefined. The original statement is preserved; the proof below assumes $N$ beyond $C$ and $B,E,D$ distinct. This question requires review.

Let $n,m$ be positive integers with $m\gt n\ge5$, with $m$ depending on $n$. Consider the sequence $a_1,a_2,\ldots,a_m$, where
$$a_i=i\quad(1\le i\le n),$$
$$a_{n+j}=a_{3j}+a_{3j-1}+a_{3j-2}\quad(1\le j\le m-n),$$
with $m-3(m-n)=1$ or $2$, that is,
$$a_m=a_{m-k}+a_{m-k-1}+a_{m-k-2},\qquad k=1\text{ or }2.$$
Thus, if $n=5$, the sequence is $1,2,3,4,5,6,15$; if $n=8$, it is $1,2,3,4,5,6,7,8,6,15,21$. Find $S=a_1+a_2+\cdots+a_m$ if (i) $n=2007$ and (ii) $n=2008$.

Let $0\lt a,b\lt \pi/2$. Show that
$$\frac5{\cos^2a}+\frac5{\sin^2a\sin^2b\cos^2b}\ge27\cos a+36\sin a.$$

Consider a $2008\times2008$ chessboard. Let $M$ be the smallest number of rectangles that can be drawn on the chessboard so that the sides of every cell of the board are contained in the sides of one of the rectangles. Find $M$. For example, for a $2\times3$ chessboard, $M=3$.