Compete · Serbia
Opštinsko takmičenje (Municipal round) 1998
1How many three-digit numbers written using the digits \(0, 1, 2, 3, 4, 5\) are divisible by \(15\), if (a) all digits must be distinct? (b) digits may repeat?2Write each of the numbers \(1, 2, 3, \dots, 9\) into exactly one of the nine shapes in the figure - odd numbers into the triangles, even numbers into the squares - so that all \(12\) of the inequality …3Find a five-digit natural number whose half is the square of a natural number and whose third is the cube of a natural number.4Several numbers, all different from \(0\), are written on a board. Each of them is equal to half the sum of the remaining ones. How many numbers are written on the board?5The numbers \(1, 2, 3, 4, 5\) are divided into two groups so that each group contains at least one of them. Prove that one of the groups contains two numbers whose difference also belongs to that same …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.