MathDB

1

Part of 2014 European Mathematical Cup

Problems(2)

JEMC p1

Source: European Mathematical Cup 2014, Junior Division, Problem 1

12/22/2014
Which of the following claims are true, and which of them are false? If a fact is true you should prove it, if it isn't, find a counterexample. a) Let a,b,ca,b,c be real numbers such that a2013+b2013+c2013=0 a^{2013} + b^{2013} + c^{2013} = 0 . Then a2014+b2014+c2014=0 a^{2014} + b^{2014} + c^{2014} = 0 . b) Let a,b,ca,b,c be real numbers such that a2014+b2014+c2014=0 a^{2014} + b^{2014} + c^{2014} = 0 . Then a2015+b2015+c2015=0 a^{2015} + b^{2015} + c^{2015} = 0 . c) Let a,b,ca,b,c be real numbers such that a2013+b2013+c2013=0 a^{2013} + b^{2013} + c^{2013} = 0 and a2015+b2015+c2015=0 a^{2015} + b^{2015} + c^{2015} = 0 . Then a2014+b2014+c2014=0 a^{2014} + b^{2014} + c^{2014} = 0 .
Proposed by Matko Ljulj
algebra unsolvedalgebra
Number of positive divisors

Source: European Mathematical Cup 2014, Senior Division, P1

12/14/2014
Prove that there exist infinitely many positive integers which cannot be written in form ad(a)+bd(b)a^{d(a)}+b^{d(b)} for some positive integers aa and bb For positive integer d(a)d(a) denotes number of positive divisors of aa
Proposed by Borna Vukorepa
modular arithmeticnumber theoryprime factorizationnumber theory unsolved