Subcontests
(5)coloring vertices of regular 2002-gon
The vertices of a regular 2002-sided polygon are numbered 1 through 2002, clockwise. Given an integer n, 1≤n≤2002, color vertex n blue, then, going clockwise, countn vertices starting at the next of n, and color n blue. And so on, starting from the vertex that follows the last vertex that was colored, n vertices are counted, colored or uncolored, and the number n is colored blue. When the vertex to be colored is already blue, the process stops. We denote P(n) to the set of blue vertices obtained with this procedure when starting with vertex n. For example, P(364) is made up of vertices 364, 728, 1092, 1456, 1820, 182, 546, 910, 1274, 1638, and 2002.
Determine all integers n, 1≤n≤2002, such that P(n) has exactly 14 vertices, 5 digit combination in bank safe
In a bank, only the manager knows the safe's combination, which is a five-digit number. To support this combination, each of the bank's ten employees is given a five-digit number. Each of these backup numbers has in one of the five positions the same digit as the combination and in the other four positions a different digit than the one in that position in the combination. Backup numbers are: 07344, 14098, 27356, 36429, 45374, 52207, 63822, 70558, 85237, 97665. What is the combination to the safe? list of ten numbers game
Let k be a fixed positive integer, k≤10. Given a list of ten numbers, the allowed operation is: choose k numbers from the list, and add 1 to each of them. Thus, a new list of ten numbers is obtained. If you initially have the list 1,2,3,4,5,6,7,8,9,10, determine the values of k for which it is possible, through a sequence of allowed operations, to obtain a list that has the ten equal numbers. In each case indicate the sequence.