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20 different companies, each hires 15 new employees, 300 applicants

Source: Mathematics Regional Olympiad of Mexico West 2019 P6

September 9, 2022
combinatorics

Problem Statement

In Occidentalia there are 2020 different companies, each looking to hire 1515 new employees. A group of 300300 applicants interview each of the companies. Each company qualifies each applicant as suitable or not suitable to work in it, in such a way that each of them finds exactly pp suitable applicants, with p>15p > 15. and each applicant is found suitable by at least one company. What is the smallest of pp f or which it is always possible to assign 1515 applicants to each company, given that each company is assigned only applicants that it considers appropriate, and that each of the 300300 applicants is assigned to a company?