OMCC 1999

Suppose that each of the 5 persons knows a piece of information, each piece is different, about a certain event. Each time person $A$ calls person $B$ , $A$ gives $B$ all the information that $A$ knows at that moment about the event, while $B$ does not say to $A$ anything that he knew.

(a) What is the minimum number of calls are necessary so that everyone knows about the event? (b) How many calls are necessary if there were $n$ persons?

Find a positive integer $n$ with 1000 digits, all distinct from zero, with the following property: it's possible to group the digits of $n$ into 500 pairs in such a way that if the two digits of each pair are multiplied and then add the 500 products, it results a number $m$ that is a divisor of $n$ .

The digits of a calculator (with the exception of 0) are shown in the form indicated by the figure below, where there is also a button +": ![Image](6965) Two players $A$ and $B$ play in the following manner: $A$ turns on the calculator and presses a digit, and then presses the button +". $A$ passes the calculator to $B$ , which presses a digit in the same row or column with the one pressed by $A$ that is not the same as the last one pressed by $A$ ; and then presses + and returns the calculator to $A$ , repeating the operation in this manner successively. The first player that reaches or exceeds the sum of 31 loses the game. Which of the two players have a winning strategy and what is it?

In the trapezoid $ABCD$ with bases $AB$ and $CD$ , let $M$ be the midpoint of side $DA$ . If $BC=a$ , $MC=b$ and $\angle MCB=150^\circ$ , what is the area of trapezoid $ABCD$ as a function of $a$ and $b$ ?

Let $a$ be an odd positive integer greater than 17 such that $3a-2$ is a perfect square. Show that there exist distinct positive integers $b$ and $c$ such that $a+b,a+c,b+c$ and $a+b+c$ are four perfect squares.

Denote $S$ as the subset of $\{1,2,3,\dots,1000\}$ with the property that none of the sums of two different elements in $S$ is in $S$ . Find the maximum number of elements in $S$ .