MathDB

2

Part of 1992 Irish Math Olympiad

Problems(2)

Number of Solutions

Source: 1992 IrMO Paper 1 Problem 2

10/2/2017
How many ordered triples (x,y,z)(x,y,z) of real numbers satisfy the system of equations x2+y2+z2=9,x^2+y^2+z^2=9, x4+y4+z4=33,x^4+y^4+z^4=33, xyz=−4?xyz=-4?
algebra
Digital Root

Source: 1992 IrMO Paper 2 Problem 2

10/2/2017
If a1a_1 is a positive integer, form the sequence a1,a2,a3,…a_1,a_2,a_3,\dots by letting a2a_2 be the product of the digits of a1a_1, etc.. If aka_k consists of a single digit, for some k≥1k\ge 1, aka_k is called a digital root of a1a_1. It is easy to check that every positive integer has a unique root. ((For example, if a1=24378a_1=24378, then a2=1344a_2=1344, a3=48a_3=48, a4=32a_4=32, a5=6a_5=6, and thus 66 is the digital root of 24378.)24378.) Prove that the digital root of a positive integer nn equals 11 if, and only if, all the digits of nn equal 11.
number theory