MathDB

13

Part of 2015 Hanoi Open Mathematics Competitions

Problems(2)

(x^2 + y^2 - 2) (x + y)^2 +(xy + 1)^2 = 0, rational (HOMC 2015 J Q13)

Source:

8/6/2019
Give rational numbers x,yx, y such that (x2+y2−2)(x+y)2+(xy+1)2=0(x^2 + y^2 - 2) (x + y)^2 + (xy + 1)^2 = 0 Prove that 1+xy\sqrt{1 + xy} is a rational number.
algebrarationalequation
c=a^{2014}+b^{2014} ,d=a^{2015}+b^{2015}, rel. prime (HOMC 15 Q13)

Source:

9/7/2019
Let mm be given odd number, and let a,ba, b denote the roots of equation x2+mx−1=0x^2 + mx - 1 = 0 and c=a2014+b2014c = a^{2014} + b^{2014} , d=a2015+b2015d =a^{2015} + b^{2015} . Prove that cc and dd are relatively prime numbers.
number theoryrelatively primeexponential