OMM 2000

Circles $A,B,C,D$ are given on the plane such that circles $A$ and $B$ are externally tangent at $P, B$ and $C$ at $Q, C$ and $D$ at $R$ , and $D$ and $A$ at $S$ . Circles $A$ and $C$ do not meet, and so do not $B$ and $D$ . (a) Prove that the points $P,Q,R,S$ lie on a circle. (b) Suppose that $A$ and $C$ have radius $2, B$ and $D$ have radius $3$ , and the distance between the centers of $A$ and $C$ is $6$ . Compute the area of the quadrilateral $PQRS$ .

A triangle of numbers is constructed as follows. The first row consists of the numbers from $1$ to $2000$ in increasing order, and under any two consecutive numbers their sum is written. (See the example corresponding to $5$ instead of $2000$ below.) What is the number in the lowermost row?

1 2 3 4 5 3 5 7 9 8 12 16 20 28 4

Given a set $A$ of positive integers, the set $A'$ is composed from the elements of $A$ and all positive integers that can be obtained in the following way: Write down some elements of $A$ one after another without repeating, write a sign $+$ or $-$ before each of them, and evaluate the obtained expression. The result is included in $A'$ . For example, if $A = \{2,8,13,20\}$ , numbers $8$ and $14 = 20-2+8$ are elements of $A'$ . Set $A''$ is constructed from $A'$ in the same manner. Find the smallest possible number of elements of $A$ , if $A''$ contains all the integers from $1$ to $40$ .

Let $a$ and $b$ be positive integers not divisible by $5$ . A sequence of integers is constructed as follows: the first term is $5$ , and every consequent term is obtained by multiplying its precedent by $a$ and adding $b$ . (For example, if $a = 2$ and $b = 4$ , the first three terms are $5,14,32$ .) What is the maximum possible number of primes that can occur before encoutering the first composite term?

A board $n$ × $n$ is coloured black and white like a chessboard. The following steps are permitted: Choose a rectangle inside the board (consisting of entire cells)whose side lengths are both odd or both even, but not both equal to $1$ , and invert the colours of all cells inside the rectangle. Determine the values of $n$ for which it is possible to make all the cells have the same colour in a finite number of such steps.

Let $ABC$ be a triangle with $\angle B > 90^o$ such that there is a point $H$ on side $AC$ with $AH = BH$ and BH perpendicular to $BC$ . Let $D$ and $E$ be the midpoints of $AB$ and $BC$ respectively. A line through $H$ parallel to $AB$ cuts $DE$ at $F$ . Prove that $\angle BCF = \angle ACD$ .