Easy illustrative examples

1 (AHSME 1999, #13) Define a sequence of real numbers 1#1 by 2#2 and 3#3 for all 4#4. Then 5#5 equals? The original problem was multiple choice.

2 (AHSME 1999, #20) The sequence 1#1 satisfies 6#6, 7#7, and for all 8#8, 9#9 is the arithmetic mean of the first 10#10 terms. Find 11#11. The original problem was multiple choice.

3 (AHSME 1998, #17) Let 12#12 be a function with the two properties:

(a) for any two real numbers 13#13 and 14#14, 15#15, and

(b) 16#16.

4 (AHSME 1997, #27) Consider those functions 18#18 that satisfy 19#19 for all real 13#13. Any such function is periodic and there is a least common positive period 20#20 for all of them. Find 20#20. The original problem was multiple choice.

5 (Common idea) The probability a team wins its next game is .75 if it won its last game and .25 if it lost its last game. What's the probability a team that wins game 1 will win game 10?

6 (Common) Into how many pieces can a pizza be divided by 21#21 straight vertical cuts? (Assume the pizza is essentially 2-dimensional - also convex. And no moving the pieces around between the cuts.)

7 (Variations of the pizza problem)

1. Into how many pieces can a cake be cut with 21#21 straight cuts (not necessarily vertical! The point is that the cake has thickness, so now the shape is 3-dimensional and the cuts are not lines, but planes!)
2. Go back to the essentially two-dimensional pizza - but now assume the cuts are not straight lines, but V-shaped (that is, a cut is made with a ``wedger'' - starting from a point, it generates two rays). How many
3. Go back to the two-dimensional pizza and 21#21 straight line cuts, but now count the maximum number of pieces that don't have any of the crust on the boundary.