OBM 1991

At a party every woman dances with at least one man, and no man dances with every woman. Show that there are men M and M' and women W and W' such that M dances with W, M' dances with W', but M does not dance with W', and M' does not dance with W.

$P$ is a point inside the triangle $ABC$ . The line through $P$ parallel to $AB$ meets $AC$ $A_0$ and $BC$ at $B_0$ . Similarly, the line through $P$ parallel to $CA$ meets $AB$ at $A_1$ and $BC$ at $C_1$ , and the line through P parallel to BC meets $AB$ at $B_2$ and $AC$ at $C_2$ . Find the point $P$ such that $A_0B_0 = A_1B_1 = A_2C_2$ .

Given $k > 0$ , the sequence $a_n$ is defined by its first two members and $$a_{n+2} = a_{n+1} + \frac{k}{n}a_n$$ a)For which $k$ can we write $a_n$ as a polynomial in $n$ ?

b) For which $k$ can we write $\frac{a_{n+1}}{a_n} = \frac{p(n)}{q(n)}$ ? ( $p,q$ are polynomials in $\mathbb R[X]$ ).

Show that there exists $n>2$ such that $1991 | 1999 \ldots 91$ (with $n$ 9's).

$P_0 = (1,0), P_1 = (1,1), P_2 = (0,1), P_3 = (0,0)$ . $P_{n+4}$ is the midpoint of $P_nP_{n+1}$ . $Q_n$ is the quadrilateral $P_{n}P_{n+1}P_{n+2}P_{n+3}$ . $A_n$ is the interior of $Q_n$ .

Find $\cap_{n \geq 0}A_n$ .