Please answer all the parts or skip the question

One hundred people line up to board an airplane. Each has a boarding pass with

assigned seat. However, the first person to board has lost his boarding pass and takes a random

seat. After that, each person takes the assigned seat if it is unoccupied, and one of unoccupied

seats at random otherwise. What is the probability that the last person to board gets to sit in

his assigned seat?

Let k be a non-zero integer. There are only a finite

number of solutions in integers p, q, x, y, each greater

than 1, of the equation x

p - y

q = k.

(On Catalan's conjecture) [For k = 1 this was

conjectured by Cassels (1953) and proved by Tijdeman

(1976).]

(Gamma 2/1986)

The function f satisfies 0 1, then n(n - t) 0 nel-n.A single stream of cars, each of width a and exactly in line, is passing along a straight road of breadth b with speed V. The distance between successive cars (i.e. the distance between back of one car and the front of the following car) is c. A chicken crosses the road in safety at a constant speed u in a straight line making an angle 0 with the direction of traffic. Show that Va (# ) csind + a cos 0 Show also that if the chicken chooses 0 and u so that she crosses the road at the least possible (constant) speed, she crosses in timeHank's Gold Mine has a very long vertical shaft of height . A light chain of length / passes over a small smooth light fixed pulley at the top of the shaft. To one end of the chain is attached a bucket A of negligible mass and to the other a bucket B of mass m. The system is used to raise ore from the mine as follows. When bucket A is at the top it is filled with mass 2m of water and bucket B is filled with mass Am of ore, where 0