MathDB

2

Part of 1989 Irish Math Olympiad

Problems(2)

3x3 Magic Square

Source: 1989 IrMO Paper 1 Problem 2

9/29/2017
A 3x3 magic square, with magic number mm, is a 3×33\times 3 matrix such that the entries on each row, each column and each diagonal sum to mm. Show that if the square has positive integer entries, then mm is divisible by 33, and each entry of the square is at most 2n−12n-1, where m=3nm=3n. An example of a magic square with m=6m=6 is
(213321132)\left( \begin{array}{ccccc} 2 & 1 & 3\\ 3 & 2 & 1\\ 1 & 3 & 2 \end{array} \right)
matrixcombinatoricsmagic square
Information Passing problem

Source: Irish MO

5/10/2012
2. Each of nn members of a club is given a different item of information. The members are allowed to share the information, but, for security reasons, only in the following way: A pair may communicate by telephone. During a telephone call only one member may speak. The member who speaks may tell the other member all the information (s)he knows. Determine the minimal number of phone calls that are required to convey all the information to each of the members.
Hi, from my sketches I'm thinking the answer is 2n−22n-2 but I dont know how to prove that this number of calls is the smallest. Can anyone enlighten me? Thanks
combinatorics unsolvedcombinatorics