Compete · Serbia
Okruzno takmicenje (Regional round) 2001
1Find every triple \((x, y, z)\) of positive integers for which \[ xyz + xy + xz + yz + x + y + z = 2000 . \]2Determine every positive integer \(n\) for which \[ 5^n + 7^n + 11^n = 6^n + 8^n + 9^n . \]3Two equilateral triangles \(ABC\) and \(PQR\) lie in the plane so that \(R\) is an interior point of the segment \(AB\) and \(C\) is an interior point of the segment \(PQ\), the points \(A\) and \(P\) …4The points \(A, B, C, D, E\) lie on one circle in such a way that \(A\) and \(D\) are on opposite sides of the line \(BC\), and \(B\) and \(E\) are on opposite sides of the line \(CD\). Given that \[ \angle ABC = \angle BCD = \angle CDE = 45^\circ , \] …5Four vertices of a given regular octagon are to be coloured blue and the remaining four red. Two colourings are called equivalent if one of them is carried onto the other by a rotation of the octagon about …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.