MathDB

4

Part of 2009 Irish Math Olympiad

Problems(2)

Operations on n-tuples

Source: 2009 IrMO Paper 1 Problem 4

1/30/2018
Given an nn-tuple of numbers (x1,x2,…,xn)(x_1,x_2,\dots ,x_n) where each xi=+1x_i=+1 or −1-1, form a new nn-tuple (x1x2,x2x3,x3x4,…,xnx1),(x_1x_2,x_2x_3,x_3x_4,\dots ,x_nx_1), and continue to repeat this operation. Show that if n=2kn=2^k for some integer k≥1k\ge 1, then after a certain number of repetitions of the operation, we obtain the nn-tuple (1,1,1,…,1).(1,1,1,\dots ,1).
combinatorics
Combinatorics problem

Source: TST

3/3/2017
At a strange party, each person knew exactly 2222 others. For any pair of people XX and YY who knew each other, there was no other person at the party that they both knew. For any pair of people XX and YY who did not know one another, there were exactly 66 other people that they both knew. How many people were at the party?
combinatoricscombinatorics unsolved