Compete · Serbia
Opštinsko takmičenje (Municipal round) 2001
1Real numbers \(a_1 < a_2 < \cdots < a_n\) are given. Find every real number \(x\) for which the expression \[ |x - a_1| + |x - a_2| + \cdots + |x - a_n| \] takes its smallest value.2Find all triples of pairwise distinct nonzero decimal digits \(a\), \(b\), \(c\) for which the fractions \[ \frac{\overline{ab}}{\overline{bc}} \qquad \text{and} \qquad \frac{a}{c} \] have the same value. …3In a trapezoid \(ABCD\) with \(AB \parallel CD\), the two angles at the base \(AB\) add up to \(90^\circ\). Prove that the segment joining the midpoints of the two bases has length equal to half the difference …4A class has \(30\) students, and every day exactly three of them are on duty in the school kitchen. Prove that the duty roster cannot be arranged so that every two students of the class are on duty together …5Words are built from the two letters \(A\) and \(B\) only. Is it possible to form a set of words containing \(3\) words of \(4\) letters, \(10\) words of \(5\) letters, \(30\) words of \(6\) letters and …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.