MathDB

3

Part of 1991 Cono Sur Olympiad

Problems(2)

Prove solutions of system

Source: Cono Sur 1991-problem 3

5/29/2006
It is known that the number of real solutions of the following system if finite. Prove that this system has an even number of solutions: (y2+6)(x−1)=y(x2+1)(y^2+6)(x-1)=y(x^2+1) (x2+6)(y−1)=x(y2+1)(x^2+6)(y-1)=x(y^2+1)
algebra unsolvedalgebra
F(n)

Source: Cono Sur 1991-problem 6

5/29/2006
Given a positive integrer number nn (n≠0n\ne 0), let f(n)f(n) be the average of all the positive divisors of nn. For example, f(3)=1+32=2f(3)=\frac{1+3}{2}=2, and f(12)=1+2+3+4+6+126=143f(12)=\frac{1+2+3+4+6+12}{6}=\frac{14}{3}. a Prove that n+12≥f(n)≥n\frac{n+1}{2} \ge f(n)\ge \sqrt{n}. b Find all nn such that f(n)=919f(n)=\frac{91}{9}.
inequalitiesnumber theoryprime numbersalgebra unsolvedalgebra