1st OMCC 1999


A1. A, B, C, D, E each has a unique piece of news. They make a series of phone calls to each other. In each call, the caller tells the other party all the news he knows, but is not told anything by the other party. What is the minimum number of calls needed for all five people to know all five items of news? What is the minimum for n people?
A2. Find a positive integer n with 1000 digits, none 0, such that we can group the digits into 500 pairs so that the sum of the products of the numbers in each pair divides n.
A3. A and B play a game as follows. Starting with A, they alternately choose a number from 1 to 9. The first to take the total over 30 loses. After the first choice each choice must be one of the four numbers in the same row or column as the last number (but not equal to the last number): 7 8 9 4 5 6 1 2 3 Find a winning strategy for one of the players.
B1. ABCD is a trapezoid with AB parallel to CD. M is the midpoint of AD, ∠MCB = 150^o, BC = x and MC = y. Find area ABCD in terms of x and y.
B2. a > 17 is odd and 3a-2 is a square. Show that there are positive integers b ≠ c such that a+b, a+c, b+c and a+b+c are all squares.
B3. S ⊆ {1, 2, 3, ... , 1000} is such that if m and n are distinct elements of S, then m+n does not belong to S. What is the largest possible number of elements in S?

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.