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2019 Chile Classification / Qualifying NMO Juniors XXXI

Source:

October 11, 2021
algebrageometrynumber theorycombinatoricschilean NMO

Problem Statement

p1. Consider the sequence of positive integers 2,3,5,6,7,8,10,11...2, 3, 5, 6, 7, 8, 10, 11 .... which are not perfect squares. Calculate the 20192019-th term of the sequence.
p2. In a triangle ABCABC, let DD be the midpoint of side BCBC and EE be the midpoint of segment ADAD. Lines ACAC and BEBE intersect at FF. Show that 3AF=AC3AF = AC.
p3. Find all positive integers nn such that n!+2019n! + 2019 is a square perfect.
p4. In a party, there is a certain group of people, none of whom has more than 33 friends in this. However, if two people are not friends at least they have a friend in this party. What is the largest possible number of people in the party?
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