2011 El Salvador Correspondence / Qualifying NMO XI
Source:
October 16, 2021
algebrageometrynumber theorycombinatoricsel salvador NMO
Problem Statement
p1. In the month of January of a certain year, there were exactly four Mondays and four Fridays. What day of the week was February ?
p2. Determine all positive integers that have the following property: "Among the positive divisors of different than and , the largest is times the smallest."
p3. Vecindad Island has inhabitants, divided into three types: the innocent, the wicked and the capricious, the innocent always tell the truth, the wicked always lie, and the capricious alternately tell lies one day and truths the next. A reporter visits the island for two days.
On the first day, the reporter interviews all the inhabitants:
The first one says: "There is exactly one villain on the island."
The second says: "There are exactly two villains on the island."
And so on until he reaches inhabitant number , who says: "There are exactly villains on the island."
On the second day, the reporter interviews everyone again in the same order:
The first says: "There is exactly one innocent on the island."
The second says: "There are exactly two innocents on the island."
And so on until he reaches inhabitant number , who says: "There are exactly innocents on the island."
How many capricious are there on the island?
p4. Let be a regular pentagon such that the star has area . Let be the intersection point of and , let be the intersection point of and . Determine the area of quadrilateral APQD.
https://cdn.artofproblemsolving.com/attachments/0/3/dcb0609bc85699b600b61ed97f6345a6b2b832.png
p5. Determine all positive integers with , such that an integer.