2

Part of 2001 IMO Shortlist

Problems(2)

IMO ShortList 2001, number theory problem 2

Source: IMO ShortList 2001, number theory problem 2

Consider the system \begin{align*}x + y &= z + u,\\2xy & = zu.\end{align*} Find the greatest value of the real constant

m

m \leq x/y

for any positive integer solution

(x,y,z,u)

of the system, with

x \geq y

number theorycalculusIMO Shortlistalgebraquadratics

IMO ShortList 2001, algebra problem 2

Source: IMO ShortList 2001, algebra problem 2

a_0, a_1, a_2, \ldots

be an arbitrary infinite sequence of positive numbers. Show that the inequality

1 + a_n > a_{n-1} \sqrt[n]{2}

holds for infinitely many positive integers

n

inequalitiescalculusIMO Shortlist