Drzavno tekmovanje (national round) 2004 · Lemma

1Number theoryProofIs there a natural number \(n\) with the following property: if \(n\) is multiplied by the sum of its own digits, then the digits of the resulting product add up to \(3\)?2AlgebraFree answerA natural number is written on each face of a cube. At every vertex of the cube one writes the product of the numbers on the three faces that meet at that vertex. The eight numbers at the vertices add …3GeometryProofLet \(E\) and \(F\) be the midpoints of the sides \(AD\) and \(DC\) of a rectangle \(ABCD\), and let \(G\) be the point where the segments \(AF\) and \(EC\) cross. Prove that \[ \angle CGF = \angle FBE . \] …4Number theoryFree answerAna picked the eight digits \(1, 2, 3, 4, 5, 6, 7\) and \(9\). She then forms groups of four two-digit primes, each group using all of her chosen digits. What is the sum of the four primes in one such …

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