Opštinsko takmičenje (Municipal round) 2015 · Lemma

1Number theoryProofProve that no integers \(m\) and \(n\) satisfy \[ (m + n + 2)^2 = 3(mn + 1). \]2Number theoryProofa) Suppose the ordered quadruple \((x, y, z, w)\) is a solution of the equation \[ x^2 + y^2 + z^2 + w^2 = xyzw . \] Prove that \((yzw - x,\, y,\, z,\, w)\) is a solution of the same equation. b) Prove …3CombinatoricsFree answerSeveral lines are drawn in the plane. Line \(a\) intersects exactly three of the other lines, and line \(b\) intersects exactly four of the other lines. Line \(c\) intersects exactly \(n\) of the other …4GeometryProofIn a convex hexagon \(ABCDEF\) the following lines are parallel: \[ AB \parallel FC \parallel DE, \qquad BC \parallel AD \parallel EF, \qquad CD \parallel BE. \] Prove that \(BE \parallel FA\).5Logic and setsFree answerAt a round table sit \(2014\) people. Each of them either always tells the truth or always lies. Every single person at the table said the following sentence: "Apart from me and my two immediate neighbours, …

Work through the paper in order - each problem opens in the workspace with this paper as its trail.