Compete · Slovenia
Izbirno tekmovanje (selection round) 2001
1Let \(d\) denote the greatest common divisor and \(v\) the least common multiple of the natural numbers \(m\) and \(n\). Prove that if \[ 3m + n = 3v + d , \] then \(n\) divides \(m\).2Find all pairs of integers \(x\) and \(y\) that satisfy the equation \[ x^2 + xy + y^2 = 1 . \]3A circle of radius \(\sqrt{2}\) is given. A second circle, of radius \(2\), has its centre on the first circle. Find the area of the shaded region, that is, of the part of the smaller disc that lies outside …4Jure wanted to prepare presents for nine friends, giving each friend two chocolate bars. In the shop he saw that a hazelnut chocolate bar costs \(6\) tolars more than a milk one, and that each of them …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.