8 (AIME 1996)
A bored student walks down a hall that contains a row of
closed lockers, numbered 1 to 1024. He opens the locker
numbered 1, and then alternates between skipping and
opening each closed locker thereafter. When he reaches the
end of the hall, the student turns around and starts back.
He opens the first closed locker he encounters, then
alternates between skipping and opening each closed
locker thereafter. The student continues wandering
back and forth in this manner until every locker is open.
What is the number of the last locker he opens?
9 (AIME 1994)
The function 18#18 has the property that, for each real number 13#13,
10 (AIME 1993)
Let
39#39 For integers 4#4,
define
40#40. What is the coefficient of 13#13
in 41#41?
11 (AIME 1992)
For any sequence of real numbers
42#42,
define 43#43 to be the sequence
44#44,
whose 21#21th term is 45#45. Suppose that all
of the terms of the sequence
46#46 are 1
and that
47#47. Find 48#48.
12 (British Math Olympiad, 1977, #1)
A non-negative integer 49#49 is assigned to each positive integer
21#21 in such a way that the following conditions are satisfied:
13 (Putnam, 1999, problem A-1)
Find polynomials 12#12, 54#54, and 55#55, if they exist,
such that, for all 13#13:
See Polya Contest 1995 Power Round on attached sheet.