MathDB

1

Part of 2012 Irish Math Olympiad

Problems(2)

Sets with Properties

Source: 2012 IrMO Paper 1 Problem 1

2/16/2018
Let C={1,22,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}C=\{1,22,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20\} and let S={4,5,9,14,23,37}S=\{4,5,9,14,23,37\} Find two sets AA and BB with the properties (a) A∩B=∅A\cap B=\emptyset. (b) A∪B=CA\cup B=C. (c) The sum of two distinct elements of AA is not in SS. (d) The sum of two distinct elements of BB is not in SS.
combinatorics
Sum of Digits

Source: 2012 Paper 2 Problem 1

2/16/2018
Let S(n)S(n) be the sum of the decimal digits of nn. For example. S(2012)=2+0+1+2=5S(2012)=2+0+1+2=5. Prove that there is no integer n>0n>0 for which n−S(n)=9990n-S(n)=9990.
number theory