OMO 2012Number Theory Level 4
The numbers \(1,2,3,4,\ldots,2012\) are written on a blackboard. Each minute, a student goes to the blackboard,chooses two numbers \(x\) and \(y\), erases them and writes a new number \(2x+2y\) on the blackboard. This process continues till only one number \(N\) remains on the blackboard. Find the remainder when the maximum possible value of \(N\) is divided by \(1000\).