jibz

Two rival elementary school dodgeball teams just finished their thrilling season-capping bout.

With the game over, each player shakes hands with each of the other team's N players, and, as is customary with sports teams, they do this in an orderly manner, by having each team form a line, and then having the two lines walk past each other, resulting in N^2 handshakes.

On the bus ride home word gets around among the visiting team that the home team had the palm cooties, and that each handshake caused the palm cooties to be equally redistributed between the pair of palms involved, i.e. if one of them had an X amount and the other a Y amount of cooties, then the handshake changed both of those amounts to (X+Y)/2. Assuming that prior to shaking hands the visitors had no palm cooties, while the home team the cooties evenly distributed amongst its players, how much of the cooties gets transferred to the visiting team? What can you say about this amount as N → ∞?

The 5-gon doesn't have to be regular or convex. It just can't self-intersect.What's the infimum of the possible areas and can it be attained in a non-degenerate way?

You are given two containers A and B with equal amounts of liquid in each. The liquid in A and B is identical in every way except that the one in B is at a higher temperature than the one in A. Your task is to raise the temperature of the liquid in A as much as possible --- at the expense of the temperature of the liquid in B --- by applying the following operations any number of times:

>> You are allowed to partition any parcel of A or B into parcels of any number and size.

>> You are also allowed to put any two parcels into contact with each other thereby bringing their contents into thermal equilibrium --- without mixing their contents.

>> You may combine (mix) the contents of any set parcels of A or any set of parcels of B into one parcel --- but you are never to combine a parcel of A with a parcel of B. i.e. you must keep liquids A and B from mixing!

At the end you will recombine all parcels of A and all parcels of B. Obviously the final temperature of A depends on what you did.

Q: What is the supremum of the set of possible final temperatures of A and why?

_________

Assume ideal conditions: the combined thermal energy of the two liquids stays constant; thermal equilibrium is attained instantaneously; specific heat, density, etc. stay fixed, so when parcels x and y are brought into thermal equilibrium, their final temperature will be

T = (T_x⋅V_x + T_y⋅V_y) / (V_x + V_y),

where T_i and V_i is the temperature and volume of parcel i, respectively.