Some of these are quite difficult! I can't guarantee that they are in order of difficulty; in fact, I'm rather sure they aren't. Try a few; next week, I'll go over some of them, have hints for the others, and cover some more advanced ideas.
14 (Manhattan (Kansas) Math Olympiad 1999)
In the sequence
77#77
every term is equal to the product of the previous two terms plus 1.
Prove that there are no terms in the sequence which are divisible by 4.
15 (Leningrad Math Olympiad 1988 Grade 10 Main Round)
The functions 12#12 and 54#54 are defined on the real
axis so that they satisfy the following condition:
for any real numbers 13#13 and 14#14,
78#78. Find an
explicit expression for the function 79#79.
16 (Leningrad Math Olympiad 1987 Grade 10 Elimination Round)
The continuous functions
80#80
satisfy the following condition
81#81 for every
82#82. It is known that 18#18 is an increasing function.
Prove that there exists an 83#83 such that
84#84.
17 (Leningrad Math Olympiad 1987 Grade 9 Elimination Round)
Let 85#85 be a sequence of natural numbers such that
86#86 and
87#87 for any natural number
88#88. Prove that if 89#89 and 90#90 are divisible by
1999, then 21#21 is odd.
18 (Leningrad Math Olympiad 1988 Grade 10 Elimination Round)
The function
91#91
is continuous and
92#92 for all real 13#13. It
is known that 93#93. Find 94#94.
19 (Leningrad Math Olympiad 1990 Grade 11 Elimination Round)
A continuous function
95#95
satisifes equality
96#96 for all real 13#13. Prove that
18#18 is constant.
20 (Leningrad Math Olympiad 1991 Grades 9-10 Elimination Round)
Does there exist a function
97#97
such that for any natural number 13#13,
21 (Leningrad Math Olympiad 1989 Grade 9 Elimination Round)
A sequence of real numbers
1#1 has the property that
101#101 for any natural number 102#102. Prove
that this sequence contains infinitely many postitive terms and
infinitely many negative terms.
22 (Leningrad Math Olympiad 1989 Grade 10 Elimination Round)
A sequence of real numbers
1#1 has the property that
103#103 for all 51#51 and 21#21. Prove
that this sequence is an arithmetic progression.
23 (Leningrad Math Olympiad 1991 Grade 11 Elimination Round)
The finite sequence
104#104 is called 20#20-balanced
if any sum of the form
105#105
is the same for any
106#106. Prove that if a sequence with
50 members is 107#107 for
108#108, then all its
members are equal to 0.
24 (Int. Math Olympiad 1977) Let 49#49 be a function defined
on the set of all positive integers and having all its values in the
same set. Prove that if
109#109 for each positive
integer 21#21, then 110#110 for each 21#21.
25 (Int. Math Olympiad 1976). A sequence 111#111 is defined
by 112#112, 113#113,
114#114
for 115#115. Prove that for positive integers 21#21,
26 (Bratislava Correspondence Seminar, Fall 1999 3rd series):
Find all functions
95#95
that satisfy:
118#118 for all real 13#13.
27 (Bratislava Correspondence Seminar, Fall 1999 3rd series):
Let
119#119 be the elements of the Fibonacci sequence
(that is, 120#120 and
121#121 for
all positive integers 21#21). Prove that if 122#122 is a a polynomial
of degree 998 for which 123#123 for
124#124,
then
125#125.
28 (Bratislava Correspondence Seminar, Fall 1998 3rd series):
For a function
126#126,
the following statement is true:
29 (Bratislava Correspondence Seminar, Fall 1998 3rd series --
but I'm sure this problem is not original):
95#95 is continuous and
130#130 for all real 13#13. Prove that 131#131 for all real 13#13.
30 (British Math Olympiad 1999).
Any positive integer 51#51 can be written uniquely in base 3 as
a string of 0s, 1s, and 2s (not beginning with a zero). For example:
31 (British Math Olympiad 1999)
Consider all functions 18#18 from the positive integers to the
positive integers such that:
32 (Putnam, 1999, problem A-6)
The sequence
145#145 is defined by 2#2,
146#146, 147#147, and for 148#148,
33 (Putnam 1990) Let 150#150, 151#151, 152#152 and
for 8#8,
34 (Putnam 1980) For which real numbers 158#158 does the
sequence defined by the initial condition 159#159 and
the recursion
160#160 have 161#161 for all
162#162?
35 (USAMO 1993) Consider functions
163#163 which satisfy:
36 (USAMO 1993) Let 158#158, 171#171 be odd positive integers. Define
the sequence 172#172 by putting 173#173, 174#174, and by letting
172#172 for 8#8 be the greatest odd divisor of
175#175.
Show that 172#172 is constant for 21#21 sufficiently large and determine the
eventual value as a function of 158#158 and 171#171.
37 (India, 1998)
Let 176#176 be a positive integer such that 177#177 is prime. Choose 178#178
from 179#179 for
180#180. Suppose that the 178#178 are not all
equal, and let 12#12 be a polynomial such that 181#181 for
180#180. Prove that
the degree of 12#12 is at least 176#176.