Compete · Serbia
Opštinsko takmičenje (Municipal round) 2000
1A triangle is cut into two triangles that are congruent to each other. Prove that the original triangle is isosceles.2The decimal representation of a positive integer \(n\) uses only the digits \(1\), \(3\), \(7\) and \(9\), and each of these four digits appears at least once. Prove that the digits of \(n\) can be rearranged …3In how many ways can \(1000\) numbers be chosen from the set \(\{1, 2, \dots, 1999\}\) so that no two of the chosen numbers have sum \(1999\) or sum \(2000\)?4Does there exist a triangle of area \(1\) whose sides \(b\) and \(c\) satisfy \(c \le b\) and \(b = 1.4\)?5Two operations \(F\) and \(G\) turn an ordered triple of real numbers into another triple by the following rules: \(F\) sends \((a, b, c)\) to \((a+1,\, b+c,\, c+1)\), and \(G\) sends \((a, b, c)\) to …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.