MathDB

3

Part of 2010 Turkey Team Selection Test

Problems(2)

Turkey TST 2010 Q3

Source:

9/1/2010
A teacher wants to divide the 20102010 questions she asked in the exams during the school year into three folders of 670670 questions and give each folder to a student who solved all 670670 questions in that folder. Determine the minimum number of students in the class that makes this possible for all possible situations in which there are at most two students who did not solve any given question.
combinatorics proposedcombinatorics
Turkey TST 2010 Q6

Source:

9/1/2010
Let Λ\Lambda be the set of points in the plane whose coordinates are integers and let FF be the collection of all functions from Λ\Lambda to {1,−1}.\{1,-1\}. We call a function ff in FF perfect if every function gg in FF that differs from ff at finitely many points satisfies the condition ∑0<d(P,Q)<2010f(P)f(Q)−g(P)g(Q)d(P,Q)≥0 \sum_{0<d(P,Q)<2010} \frac{f(P)f(Q)-g(P)g(Q)}{d(P,Q)} \geq 0 where d(P,Q)d(P,Q) denotes the distance between PP and Q.Q. Show that there exist infinitely many perfect functions that are not translates of each other.
analytic geometryfunctioncombinatorics proposedcombinatorics