14
Problems(2)
7 classmates are comparing their end-of-year grades in 12 subjects
Source: 2019 Dürer Math Competition Finals Day2 E14 https://artofproblemsolving.com/community/c1621835_2019_
1/5/2022
Seven classmates are comparing their end-of-year grades in subjects. They observe that for any two of them, there is some subject out of the where the two students got different grades. It is possible to choose n subjects out of the such that if the seven students only compare their grades in these subjects, it will still be true that for any two, there is some subject out of the n where they got different grades. What is the smallest value of for which such a selection is surely possible?Note: In Hungarian high schools, students receive an integer grade from to in each subject at the end of the year.
combinatorics
last four digits in base 2 are the same as its last four digits in base 5
Source: 2019 Dürer Math Competition Finals Day2 E+14 https://artofproblemsolving.com/community/c1621835_2019_
1/5/2022
Let be the set of all positive integers less than whose last four digits in base are the same as its last four digits in base . What remainder do we get if we divide the sum of all elements of by ?
number theorylast digitsDigits