TR-MO 2001

Let $ABCD$ be a convex quadrilateral. The perpendicular bisectors of the sides $[AD]$ and $[BC]$ intersect at a point $P$ inside the quadrilateral and the perpendicular bisectors of the sides $[AB]$ and $[CD]$ also intersect at a point $Q$ inside the quadrilateral. Show that, if $\angle APD = \angle BPC$ then $\angle AQB = \angle CQD$

$(x_{n})_{-\infty<n<\infty}$ is a sequence of real numbers which satisfies $x_{n+1}=\frac{x_{n}^2+10}{7}$ for every $n \in \mathbb{Z}$ . If there exist a real upperbound for this sequence, find all the values $x_{0}$ can take.

One wants to distribute $n$ same sized cakes between $k$ people equally by cutting every cake at most once. If the number of positive divisors of $n$ is denoted as $d(n)$ , show that the number of different values of $k$ which makes such distribution possible is $n+d(n)$

Find all ordered triples of positive integers $(x,y,z)$ such that $$3^{x}+11^{y}=z^{2}$$

Two nonperpendicular lines throught the point $A$ and a point $F$ on one of these lines different from $A$ are given. Let $P_{G}$ be the intersection point of tangent lines at $G$ and $F$ to the circle through the point $A$ , $F$ and $G$ where $G$ is a point on the given line different from the line $FA$ . What is the locus of $P_{G}$ as $G$ varies.

We wish to color the cells of a $n \times n$ chessboard with $k$ different colors such that for every $i\in \{1,2,...,n\}$ , the $2n-1$ cells on $i$ . row and $i$ . column have all different colors.

a) Prove that for $n=2001$ and $k=4001$ , such coloring is not possible.

b) Show that for $n=2^{m}-1$ and $k=2^{m+1}-1$ , such coloring is possible.