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Part of 1995 Irish Math Olympiad
Problems(2)
students
Source: Ireland 1995
7/1/2009
There are students in a class. Each week all the students participate in a table quiz. Their teacher arranges them into teams of players each. For as many weeks as possible, this arrangement is done in such a way that any pair of students who were members of the same team one week are not in the same team in subsequent weeks. Prove that after at most n\plus{}2 weeks, it is necessary for some pair of students to have been members of the same team in at least two different weeks.
combinatorics proposedcombinatorics
inequality in integers
Source: Ireland 1995
7/1/2009
Prove that for every positive integer ,
n^n \le (n!)^2 \le \left( \frac{(n\plus{}1)(n\plus{}2)}{6} \right) ^n.
inequalitiesinductionlogarithmsinequalities proposed