MathDB

4

Part of 2002 May Olympiad

Problems(2)

coloring vertices of regular 2002-gon

Source: VIII May Olympiad (Olimpiada de Mayo) 2002 L2 P4

9/19/2022
The vertices of a regular 20022002-sided polygon are numbered 11 through 20022002, clockwise. Given an integer n n, 1≤n≤20021 \le n \le 2002, color vertex nn blue, then, going clockwise, countn n vertices starting at the next of nn, and color nn 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 nn is colored blue. When the vertex to be colored is already blue, the process stops. We denote P(n)P(n) to the set of blue vertices obtained with this procedure when starting with vertex nn. For example, P(364)P(364) is made up of vertices 364364, 728728, 10921092, 14561456, 18201820, 182182, 546546, 910910, 12741274, 16381638, and 20022002. Determine all integers nn, 1≤n≤20021 \le n \le 2002, such that P(n)P(n) has exactly 1414 vertices,
combinatoricscombinatorial geometry
5 digit combination in bank safe

Source: VIII May Olympiad (Olimpiada de Mayo) 2002 L1 P4

9/22/2022
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: 0734407344, 1409814098, 2735627356, 3642936429, 4537445374, 5220752207, 6382263822, 7055870558, 8523785237, 9766597665. What is the combination to the safe?
number theoryDigits