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Num3ers Once A-Gain

(1)

Progressive Roots

If roots of equation x^2 - 2x + A = 0 and equation x^2 - 18x +B =0 are in geometric progression then find the values of A & B.

(2)

Rotating Boards

A chess board 8x8 is placed on top of an identical board and rotated through 45 degrees about its center. What is the area S which is black in both boards?

(3)

Boston Tea Party

Boxes of size 38 x 58 x 73 are thrown into the Boston Harbor. They have a density 97% that of water. We know that boxes have 3 axes of hydrostatic stability, but during the rolling around before settling, how high can any corner of a box reach above the water? Compute to 4 significant figures. Assume that the boxes are in hydrostatic balance but not necessarily stable.

(4)

Goal : Balls into Boxes

How many ways are there to put n balls into r boxes if: (*) (*) (*) (*)

Both the balls and the boxes are distinct The boxes are distinct but the balls are not distinct The balls are distinct but the boxes are not distinct Neither the balls nor the boxes are distinct

You can put more than one ball into a box. You can put zero balls into box(es). Note that each (*) is separate question from other cases. I mean we have 4 problems in 1 question.

(5)

Going to the Market (Carrying a Basket)

As we all know, because of market psychology when stocks increase in price on one day, they tend to decrease in price the next day. A payoff matrix shows the pertinent percents. Now here's the payoff matrix. If today the stock price increases, then the likelihood it will increase tomorrow is 0.2 and the likelihood it will decrease tomorrow is 0.8.

If today the stock price decreases, then the likelihood it will increase tomorrow is 0.6 and the likelihood it will decrease tomorrow is 0.4. So the question is, in the long run, what fraction of the time will the stock increase in price?