Regijsko tekmovanje (regional round) 2014 · Lemma

1Number theoryMultiple choiceOne digit of the seven-digit number \(2345678\) is to be deleted, so that the six-digit number left behind is divisible by \(9\). Which digit must it be? A \(8\) B \(7\) C \(6\) D \(5\) E \(4\)2Number theoryMultiple choiceThere are \(17\) girls and \(12\) boys on a playground. At least how many more children must arrive so that everybody present can then be divided into two groups of the same size, in such a way that each …3GeometryMultiple choiceJanez drew a pattern on a sheet of paper, made up of congruent squares and congruent hexagons. On top of the pattern he then drew two dashed lines perpendicular to each other, as in the figure. What is …4Number theoryFree answerFind every natural number \(n\) with the following property: there is a rectangle whose side lengths are natural numbers, whose perimeter equals \(n\), and whose area is numerically equal to \(n\) as well. …5GeometryFree answerLet \(ABC\) be a right triangle with its right angle at \(C\), and write \(|BC| = a\), \(|AC| = b\). Let \(D\) be a point on the opposite side of the line \(AC\) from \(B\) for which the triangle \(ACD\) …6AlgebraFree answerAna and Beno cycle clockwise along the rectangular track shown in the figure, on which the length of every section is marked. Ana rides the shorter of the two loops, Beno the longer one, and their speeds …

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