Subcontests
(5)A way to assign tasks
You are the boss. You have ten men and there are ten tasks. Your men have two parameters to each task, one is enthusiasm, the other is ability.Now you are to assign the tasks to your men one by one.When man A has more enthusiasm about task v than about task u, and man A has better ability in task v than man B does, though if you assign task u to man A and task v to man B, man A feel unsatisfied.Or, if there is a more efficient way than yours that you can assign each task to men with better ability, you will be brought to account by your employer.Prove that there is a way to assign tasks without any dissatisfaction or disadvantage. Number of lattice point
Let P(x,y),Q(x,y) be two-variable polynomials with the coefficients of integer.Supposed that when an,bn are determined for certain integers a0, b0 by an+1=P(an,bn), bn+1=Q(an,bn) (n=0,1,2,⋯) there existed positive integer k such that (a1,b1)=(a0,b0) and (ak,bk)=(a0,b0).Prove that the number of the lattice points on the segment with end points of (an,bn) and (an+1,bn+1) is indepedent of n.