MathDB

(1,1),(2,3),(4,5) and (999,111) exists point on y=2x

Source: Saint Petersburg MO 2020 Grade 9 Problem 6

May 7, 2020

algebracombinatoricsinvariantanalytic geometrynumber theory

Problem Statement

The points

(1,1),(2,3),(4,5)

and

(999,111)

are marked in the coordinate system. We continue to mark points in the following way :
[*]If points

(a,b)

are marked then

(b,a)

and

(a-b,a+b)

can be marked [*]If points

(a,b)

and

(c,d)

are marked then so can be

(ad+bc, 4ac-4bd)

.
Can we, after some finite number of these steps, mark a point belonging to the line

y=2x