Bài tập hàm số ôn thi Olympic Toán Sinh Viên

Function

1. Determine all functions f: R -> R such that f(x - f(y) ) = f( f(y) ) + x f(y) + f(x) - 1 for all x, y in R.
[R is the reals.]
2. Consider all functions f from the set of all positive integers into itself satisfying f(t
2
f(s)) = s f(t)
2
for all s and t. Determine the least possible value of f(1998).
3. Let be the set of non-negative integers. Find all functions f from S to itself such that f(m + f(n)) =
f(f(m)) + f(n) for all m, n.
4. Let S be the set of all real numbers greater than -1. Find all functions f from S into S such that f(x
+ f(y) + xf(y)) = y + f(x) + yf(x) for all x and y, and f(x)/x is strictly increasing on each of the
intervals -1 < x < 0 and 0 < x.
5. Does there exist a function f from the positive integers to the positive integers such that f(1) = 2,
f(f(n)) = f(n) + n for all n, and f(n) < f(n+1) for all n?
6. Find all functions f defined on the set of all real numbers with real values, such that f(x
2
+ f(y)) = y
+ f(x)
2
for all x, y.
7. Construct a function from the set of positive rational numbers into itself such that f(x f(y)) = f(x)/y
for all x, y.
8. A function f is defined on the positive integers by: f(1) = 1; f(3) = 3; f(2n) = f(n), f(4n + 1) = 2f(2n
+ 1) - f(n), and f(4n + 3) = 3f(2n + 1) - 2f(n) for all positive integers n. Determine the number of
positive integers n less than or equal to 1988 for which f(n) = n.
9. Prove that there is no function f from the set of non-negative integers into itself such that f(f(n)) =
n + 1987 for all n.
10. The function f(x,y) satisfies: f(0,y) = y + 1, f(x+1,0) = f(x,1), f(x+1,y+1) = f(x,f(x+1,y)) for all
non-negative integers x, y. Find f(4, 1981).

11. The set of all positive integers is the union of two disjoint subsets {f(1), f(2), f(3), }, {g(1), g(2),
g(3), }, where f(1) < f(2) < f(3) < , and g(1) < g(2) < g(3) < , and g(n) = f(f(n)) + 1 for n =
1, 2, 3, . Determine f(240
12. The function f is defined on the set of positive integers and its values are positive integers. Given
that f(n+1) > f(f(n)) for all n, prove that f(n) = n for all n.
13. f and g are real-valued functions defined on the real line. For all x and y, f(x + y) + f(x - y) =
2f(x)g(y). f is not identically zero and |f(x)| <= 1 for all x. Prove that |g(x)| <= 1 for all x.
14. Find all real-valued functions f(x) on the reals such that f(f(x) + y) = 2x + f(f(y) - x) for all x, y.
15. Find all pairs of real-valued functions f, g on the reals such that f(x + g(y) ) = x f(y) - y f(x) + g(x)
for all real x, y.
16. The function f on the non-negative integers takes non-negative integer values and satisfies f(4n) =
f(2n) + f(n), f(4n+2) = f(4n) + 1, f(2n+1) = f(2n) + 1 for all n. Show that the number of non-
negative integers n such that f(4n) = f(3n) and n < 2
m
is f(2
m+1
).
17. f is a real-valued function on the reals such that | f(x) | <= 1 and f(x + 13/42) + f(x) = f(x + 1/6) +
f(x + 1/7) for all x. Show that there is a real number c > 0 such that f(x + c) = f(x) for all x.
18. X is a finite set and f, g are bijections on X such that for any point x in X either f( f(x) ) = g( g(x) )
or f( g(x) ) = g( f(x) ) or both. Show that for any x, f( f( f(x) ) ) = g( g( f(x) ) ) iff f( f( g(x) ) ) =
g( g( g(x) ) ).
19. h and k are reals. Find all real-valued functions f defined on the positive reals such that f(x) f(y) =
y
h
f(x/2) + x
k
f(y/2) for all x, y.
20. Let f(x) = (x
2

+ 1)/(2x) for x non-zero. Define f
0
(x) = x and f
n+1
(x) = f( f
n
(x) ). Show that for x not
-1, 0 or 1 we have f
n
(x)/f
n+1
(x) = 1 + 1/f(y), where y = (x+1)
N
/(x-1)
N
and N = 2
n
.
21. a and b are positive reals. Show that there is a unique real-valued function f defined on the positive
reals such that f( f(x) ) = b(a + b) f(x) - a f(x) for all x.
22. f(n) is an integer-valued function defined on the integers which satisfies f(m + f( f(n) ) ) = -
f( f(m+1)) - n for all m, n. The polynomial g(n) has integer coefficients and g(n) = g( f(n) ) for all
n. Find f(1991) and the most general form for g.
23. Find all pairs of real-valued functions f, g on the reals such that f(x + g(y) ) = x f(y) - y f(x) + g(x)
for all real x, y.
24. Define f(0) = 0, f(1) = 0, and f(n+2) = 4
n+2
f(n+1) - 16
n+1
f(n) + n 2

sq(n)
, were sq(n) = n
2
. Show that
f(1989), f(1990) and f(1991) are all divisible by 13.
25. For a positive integer k, let f
1
(k) be the square of the sum of its digits. Let f
n+1
(k) = f
1
( f
n
(k) ). Find
the value of f
1991
(2
1990
).
26. a <= b are positive integers, m = (a + b)/2. Define the function f on the integers by f(n) = n + a if n
< m, n - b if n >= m. Let f
1
(n) = f(n), f
2
(n) = f( f
1
(n) ), f
3
(n) = f( f
2

(n) ) etc. Find the smallest k such
that f
k
(0) = 0.
27. f has positive integer values and is defined on the positive integers. It satisfies f( f(m) + f(n) ) = m
+ n for all m, n. Find all possible values for f(1988).
28. f is a real-valued function on the reals such that:
(1) if x >= y and f(y) - y >= v >= f(x) - x, then f(z) = v + z for some z between x and y;
(2) for some k, f(k) = 0 and if f(h) = 0, then h <= k;
(3) f(0) = 1;
(4) f(1987) <= 1988;
(5) f(x) f(y) = f(x f(y) + y f(x) - xy) for all x, y.
Find f(1987).
29. c is a positive real constant and b = (1 + c)/(2 + c). f is a real-valued function defined on the
interval [0, 1] such that f(2x) = b f(x) for 0 <= x <= 1/2 and f(x) = b - (1 - b) f(2x - 1) for 1/2 <= x
<= 1. Show that 0 < f(x) - x < c for all 0 < x < 1.
30. Find all real-valued functions f on the reals which have at most finitely many zeros and satisfy f(x
4
+ y) = x
3
f(x) + f(f(y)) for all x, y.
31. Find all real-valued functions f on the reals such that (1) f(1) = 1, (2) f(-1) = -1, (3) f(x) <= f(0) for
0 < x < 1, (4) f(x + y) >= f(x) + f(y) for all x, y, (5) f(x + y) <= f(x) + f(y) + 1 for all x, y.
32. f is a strictly increasing real-valued function on the reals. It has inverse f
-1
. Find all possible f such
that f(x) + f
-1
(x) = 2x for all x.
33. Given n > 1, find all real-valued functions f

i
(x) on the reals such that for all x, y we have f
k+1
(x)
f
k+1
(y) = f
k
(x
k
) + f
k
(y
k
) for k = 1, 2 , n-1 and f
1
(x) + f
1
(y) = f
n
(x
n
) f
n
(y
n
).
34. f is a real-valued function on the reals such that f(x + 19) <= f(x) + 19 and f(x + 94) >= f(x) + 94
for all x. Show that f(x + 1) = f(x) + 1 for all x.
35. Let N be the set of positive integers. Define f: N → N by f(n) = n+1 if n is a prime power and = w

1
+ + w
k
when n is a product of the coprime prime powers w
1
, w
2
, , w
k
. For example f(12) = 7.
Find the smallest term of the infinite sequence m, f(m), f(f(m)), f(f(f(m))), .
36. Does there exist a function f: Z → Z (where Z is the set of integers) such that: (1) f(Z) includes the
values 1, 2, 4, 23, 92; (2) f(92 + n) = f(92 - n) for all n; (3) f(1748 + n) = f(1748 - n) for all n; (4)
f(1992 + n) = f(1992 - n) for all n?
37. Let X be the set {1, 2, 3, , 2n}. g is a function X → X such that g(k) ≠ k and g(g(k)) = k for all
k. How many functions f: X → X are there such that f(k) ≠ k and f(f(f(k))) = g(k) for all k?
38. Define f(1) = 2, f(2) = 3
f(1)
, f(3) = 2
f(2)
, f(4) = 3
f(3)
, f(5) = 2
f(4)
and so on. Similarly, define g(1) = 3,
g(2) = 2
g(1)
, g(3) = 3
g(2)
, g(4) = 2

g(3)
, g(5) = 3
g(4)
and so on. Which is larger, f(10) or g(10)?
39. Find all strictly increasing real-valued functions on the reals such that f( f(x) + y) = f(x + y) + f(0)
for all x, y.
40. f is a real-valued function on the reals such that f(x+1) = f(x) + 1. The sequence x
0
, x
1
, x
2
,
satisfies x
n
= f(x
n-1
) for all positive n. For some n > 0, x
n
- x
0
= k, an integer. Show that lim x
n
/n
exists and find it.
41. Find all continuous real-valued functions f on the reals such that (1) f(1) = 1, (2) f(f(x)) = f(x)
2
for
all x, (3) either f(x) >= f(y) for all x >= y, or f(x) <= f(y) for all x <= y.
42. Let X be the set of real numbers > 1. Define f: X → X and g: X → X by f(x) = 2x and g(x) = x/(x-

1). Show that given any real numbers 1 < A < B we can find a finite sequence x
1
= 2, x
2
, , x
n
such that A < x
n
< B and x
i
= f(x
i-1
) or g(x
i-1
).
43. Find all functions f which are defined on the rationals, take real values and satisfy f(x + y) = f(x)
f(y) - f(xy) + 1 for all x, y.
44. Let n > 1 be an integer. Let f(k) =1 + 3
k
/(3
n
- 1), g(k) = 1 - 3
k
/(3
n
- 1). Show that tan(f(1)π/3)
tan(f(2)π/3) tan(f(n)π/3) tan(g(1)π/3) tan(g(2)π/3) tan(g(n)π/3) = 1.
45. Let X be the closed interval [0, 1]. Let f: X → X be a function. Define f
1
= f, f

n+1
(x) = f( f
n
(x) ). For
some n we have |f
n
(x) - f
n
(y)| < |x - y| for all distinct x, y. Show that f has a unique fixed point.
46. Let N
0
= {0, 1, 2, 3, } and R be the reals. Find all functions f: N
0
→ R such that f(m + n) + f(m -
n) = f(3m) for all m, n.
47. Find all real-valued functions f on the positive reals which satisfy f(x + y) = f(x
2
+ y
2
) for all x, y.
48. Find all real-valued functions f(x) on the rationals such that:
(1) f(x + y) - y f(x) - x f(y) = f(x) f(y) - x - y + xy, for all x, y
(2) f(x) = 2 f(x+1) + 2 + x, for all x and
(3) f(1) + 1 > 0.
49. N is the set of positive integers. Find all functions f: N → N such that f( f(n) ) + f(n) = 2n + 2001
or 2n + 2002.
50. Find all real-valued functions on the reals which satisfy f( xf(x) + f(y) ) = f(x)
2
+ y for all x, y.
51. A is the set of positive integers and B is A ∪ {0}. Prove that no bijection f: A → B can satisfy

f(mn) = f(m) + f(n) + 3 f(m) f(n) for all m, n.
52. The function f is defined on the positive integers and f(m)≠ f(n) if m - n is prime. What is the
smallest possible size of the image of f.
53. f is a real valued function on the reals satisfying (1) f(0) = 1/2, (2) for some real a we have f(x+y)
= f(x) f(a-y) + f(y) f(a-x) for all x, y. Prove that f is constant.
54. f is a function defined on the positive integers with positive integer values. Use f
m
(n) to mean
f(f( f(n) )) = n where f is taken m times, so that f
2
(n) = f(f(n)), for example. Find the largest
possible 0 < k < 1 such that for some function f, we have f
m
(n) ≠ n for m = 1, 2, , [kn], but f
m
(n) = n for some m (which may depend on n).
55. Find all functions f on the positive integers with positive integer values such that (1) if x < y, then
f(x) < f(y), and (2) f(y f(x)) = x
2
f(xy).
56. Let f(x) = a
1
/(x + a
1
) + a
2
/(x + a
2
) + + a
n

/(x + a
n
), where a
i
are unequal positive reals. Find the
sum of the lengths of the intervals in which f(x) >= 1.
57. f is a function defined on all reals in the interval [0, 1] and satisfies f(0) = 0, f(x/3) = f(x)/2, f(1 - x)
= 1 - f(x). Find f(18/1991).
58. The function f is defined on the non-negative integers. f(2
n
- 1) = 0 for n = 0, 1, 2, . If m is not
of the form 2
n
- 1, then f(m) = f(m+1) + 1. Show that f(n) + n = 2
k
- 1 for some k, and find f(2
1990
).
59. The function f on the positive integers satisfies f(1) = 1, f(2n + 1) = f(2n) + 1 and f(2n) = 3 f(n).
Find the set of all m such that m = f(n) for some n.
60. Find f(x) such that f(x)
2
f( (1-x)/(1+x) ) = 64x for x not 0, ±1.
61. The function f(n) is defined on the positive integers and takes non-negative integer values. It
satisfies (1) f(mn) = f(m) + f(n), (2) f(n) = 0 if the last digit of n is 3, (3) f(10) = 0. Find f(1985).
62. Let R be the real numbers and S the set of real numbers excluding 0 and 1. Find all functions f : S
→ R such that f(x) + 1/(2x) f( 1/(1-x) ) = 1 for all x.
63. Find all real-valued functions f(x) on the reals such that f(2002x - f(0) ) = 2002 x
2
for all x.

64. Find all real valued functions f(x) on the reals such that f( (x - y)
2
) = x
2
- 2y f(x) + f(y)
2
.
65. Find all real-valued functions f defined on X, the set of all non-zero reals, such that (1) f(-x) =
-f(x), (2) f(1/(x+y)) = f(1/x) + f(1/y) + 2(xy-1000) (or all x, y in X such that x + y is in X.
66. Find all real-valued functions f on the positive reals such that f(1) = 1/2 and f(xy) = f(x) f(3/y) +
f(y) f(3/x).
67. Q is the rationals and R is the reals. Find all functions f : Q → R such that f(x + y) = f(x) + f(y) +
2xy for all x, y.
68. The real-valued function f is defined on the reals and satisfies f(xy) = x f(y) + y f(x) and f(x + y) =
f(x
1993
) + f(y
1993
) for all x, y. Find f(√5753).
69. R is the real numbers and S is R excluding the point 2/3. Find all functions f : S → R such that 2
f(x) + f(2x/(3x - 2) ) = 996x for all x.
70. R is the reals. S is R excluding 0. Show that there is just one function f : S → R such that f(x) = x
f(1/x) and f(x + y) = f(x) + f(y) - 1 for all x, y (with x + y non-zero).
71. A real-valued function f is defined on the reals and satisfies f(xy) = x f(y) + y f(x) and f(2x) =
f( sin( (x + y)π/2 ) + f( sin( (x - y)π/2 ) for all x, y. Find f(1990 + 1990
1/2
+ 1990
1/3
).
72. Let N be the positive integers. The function f : N → N satisfies f(1) = 5, f( f(n) ) = 4n + 9 and f(2

n
)
= 2
n+1
+ 3 for all n. Find f(1789).
73. Q is the rationals and R the reals. The function f : Q → R satisfies f(x + y) = f(x) f(y) - f(xy) + 1
for all x, y and f(1988) ≠ f(1987). Show that f(-1987/1988) = 1/1988.
74. N is the set of positive integers. M is the set of non-negative integers. f: N → M is a function such
that f(10) = 0, f(n) = 0 if the last digit of n is 3, f(mn) = f(m) + f(n). Find f(1984) and f(1985).
75. Define f on the positive integers by f(n) = k
2
+ k + 1, where 2
k
is the highest power of 2 dividing n.
Find the smallest n such that f(1) + f(2) + + f(n) >= 123456.
76. Let X be the set of non-negative integers and f : X → X a map such that ( f(2n+1) )
2
- ( f(2n) )
2
= 6
f(n) + 1 and f(2n) >= f(n) for all n in X. How many numbers in f(X) are less than 2003?
77. Let X be the set of non-negative integers. Find all functions f: X → X such that x f(y) + y f(x) = (x
+ y) f(x
2
+ y
2
) for all x, y.
78. 0 < k < 1 is a real number. Define f: [0, 1] → [0, 1] by f(x) = 0 for x <= k, 1 - (√(kx) + √( (1-k)(1-
x) ) )
2

for x > k. Show that the sequence 1, f(1), f( f(1) ), f( f( f(1) ) ), eventually becomes zero.
79. N is the positive integers, R is the reals. The function f : N → R satisfies f(1) = 1, f(2) = 2 and
f(n+2) = f(n+2 - f(n+1) ) + f(n+1 - f(n) ). Show that 0 <= f(n+1) - f(n) <= 1. Find all n for which
f(n) = 1025
80. Find all real-valued functions f on the reals whose graphs remain unchanged under all
transformations (x, y) → (2
k
x, 2
k
(kx + y) ), where k is real.
81. f is a real-valued function on the reals. It satisfies f(x
3
+ y
3
) = (x + y)(f(x)
2
- f(x) f(y) + f(y)
2
) for all
x, y. Prove that f(1996x) = 1996 f(x) for all x.
82. N
+
is the set of positive integers. f: N
+
→ N
+
satisfies f(1) = 1, f(2n) < 6 f(n), and 3 f(n) f(2n+1) =
f(2n) + 3 f(2n) f(n) for all n. Find all m, n such that f(m) + f(n) = 293.
83. X is the interval [1, ∞). Find all functions f: X → X which satisfy f(x) <= 2x + 2 and x f(x + 1) =
f(x)

2
- 1 for all x.
84. R
+
is the positive reals. f: R
+
→ R
+
satisfies f(xy) <= f(x) f(y) for all x, y. Prove that for any n: f(x
n
)
<= f(x) f(x
2
)
1/2
f(x
3
)
1/3
f(x
n
)
1/n
.
85. k is a positive real. X is the closed interval [0, 1]. Find all functions f: X x X → X such that f(x, 1)
= f(1, x) = x for all x, f(xy, xz) = x
k
f(y, z) for all x, y, z, and f( f(x, y), z) = f(x, f(y, z) ) for all x, y,
z.
86. Let X be the non-negative reals. f: X → X is bounded on the interval [0, 1] and satisfies f(x) f(y)

<= x
2
f(y/2) + y
2
f(x/2) for all x, y. Show that f(x) <= x
2
.
87. Let X be the set of real numbers greater than 1. Find all functions f on X with values in X such
that f(x
a
y
b
) <= f(x)
1/(4a)
f(y)
1/(4b)
for all x, y and all positive real a, b.
88. Define f on the positive integers as follows: f(1) = f(2) = f(3) = 2. For n > 3, f(n) is the smallest
positive integer which does not divide n. Define f
1
to be f and f
k+1
(n) = f(f
k
(n) ). Let g(n) be the
smallest k such that f
k
(n) = 2. Determine g(n) as explicitly as possible.
89. Let R be the reals and R
+

the positive reals. Show that there is no function f : R
+
→ R such that
f(y) > (y - x) f(x)
2
for all x, y such that y > x.
90. Let Q be the rationals. Find all functions f : Q → Q such that f(x + f(y) ) = f(x) + y for all x, y.
91. The function f assigns an integer to each rational. Show that there are two distinct rationals r and s,
such that f(r) + f(s) <= 2 f(r/2 + s/2).
92. Find all real-valued functions f on the reals such that f(x
2
- y
2
) = x f(x) - y f(y) for all x, y.
93. Show that there is no real-valued function f on the reals such that ( f(x) + f(y) )/2 >= f( (x+y)/2 ) +
|x - y| for all x, y.
94. Let S be the set of functions f defined on reals in the closed interval [0, 1] with non-negative real
values such that f(1) = 1 and f(x) + f(y) <= f(x + y) for all x, y such that x + y <= 1. What is the
smallest k such that f(x) <= kx for all f in S and all x?
95. Define f
1
(x) = √(x
2
+ 48) and f
n
(x) = √(x
2
+ 6f
n-1
(x) ). Find all real solutions to f

n
(x) = 2x.
96. Let R
+
be the set of positive reals and let F be the set of all functions f : R
+
→ R
+
such that f(3x)
>= f( f(2x) ) + x for all x. Find the largest A such that f(x) >= A x for all f in F and all x in R
+
.
97. Find all real-valued continuous functions defined on the interval (-1, 1) such that (1 - x
2
) f(2x/(1 +
x
2
) ) = (1 + x
2
)
2
f(x) for all x.
98. Find all functions f(n) defined on the non-negative integers with values in the set {0, 1, 2, ,
2000} such that: (1) f(n) = n for 0 <= n <= 2000; and (2) f( f(m) + f(n) ) = f(m + n) for all m, n.
99. How many functions f(n) defined on the positive integers with positive integer values satisfy f(1)
= 1 and f(n) f(n+2) = f(n+1)
2
+ 1997 for all n?
100. Find all functions f(n) on the positive integers with positive integer values, such that f(n) +
f(n+1) = f(n+2) f(n+3) - 1996 for all n

101. Find all functions f(n) on the positive integers with positive integer values, such that f(n) +
f(n+1) = f(n+2) f(n+3) - 1996 for all n.
102. f : [√1995, ∞) → R is defined by f(x) = x(1993 + √(1995 - x
2
) ). Find its maximum and
minimum values.
103. For any positive integer n, let f(n) be the number of positive divisors of n which equal ±1
mod 10, and let g(n) be the number of positive divisors of n which equal ±3 mod 10. Show that
f(n) >= g(n).
104. Find all real-valued functions f(x) on the reals such that f(xy)/2 + f(xz)/2 - f(x) f(yz) >=
1/4 for all x, y, z.
105. The function f(x) is defined and differentiable on the non-negative reals. It satisfies
| f(x) | <= 5, f(x) f '(x) >= sin x for all x. Show that it tends to a limit as x tends to infinity.
106. Find all real-valued functions f(n) on the integers such that f(1) = 5/2, f(0) is not 0, and
f(m) f(n) = f(m+n) + f(m-n) for all m, n.
107. Let S be the set of all positive real number. Prove that there is no function f : SS such
that (f(x))
2
>=f(x+y) (f(x)+y) for arbitrary positive real numbers x and y
108. Let A={1,2,3, ,m+n} , where m and n are positive integers and let the function f : A A
be defined by the equations: f( i )= i+1 for i = 1,2, ,m-1,m+1, ,m+n-1 and
f(m)=1,f(m+n)=m+1
a. Prove that if m and n are odd then there exist a function g: A A such that g(g(a))=f(a)
for all a ∈ Α
b. Prove that if m is even then m=n if there exist a function g: A A such that g(g(a))=f(a)
for all a ∈ Α
109. Let R
+
be the set of all positive real number. Find all functions f : R
+

→ R
+
that satisfy
the following conditions
a. f(xyz) + f(x) + f(y) + f(z)=f(√xy) f(√yz) f(√xz) for all x,y,z ∈ R
+

b. f(x) < f(y) for all 1 ≤ x <y
110. Find all nondecreasing functions f : R → R such that
a. f(0)=0, f(1)=1
b. f(a) + f(b) = f(a)f(b)+ f(a+b-ab) for all real numbers a,b such that a <1<b
111. f is a function defined on all reals in the interval [0, 1] and satisfies f(0) = 0, f(x/3) =
f(x)/2, f(1 - x) = 1 - f(x). Find f(18/1991).
112. Determine all functions f : [1;∞) → [1;∞) satisfying the following tow conditions:
a. f(x+1) =( ( f(x) )
2
-1)/x for x ≥ 1
b. the function( g(x)/x ) is bounded
113. Find all functions f defined on the set of positive reals which take positive real values and
satisfy: f(x f(y))=yf(x) for all x,y ,and f(x) tends to 0 ax x tends to infinity
114. The function f(n) is defined on the positive integers and takes non-negative integer values.
f(2) = 0, f(3) > 0, f(9999) = 3333 and for all m, n: f(m+n)-f(m)-f(n)=0 or 1. Determine f(1982)
115. Given positive integer m, n. Set A={1,2, ,n}. Determine the number of functions f: A →
A attaining exactly m values and satisfying the condition if k , l ∈Α , k ≤ l then f(f(k))=f(k) ≤ f(l)
116. Let = 1,2, ,n. Prove or disprove the following statement : for all integer n ≥ 2 there
exits function f : A
n
→

A

n
and g: A
n
→

A
n
which satisfy
a. f(f(k)) = g(g(k))=k for k=1,2, ,n.
b. g(f(k))=k+1 for k=1,2, ,n-1
117. Let f : (0,1)→ R be a function, such that f(1/n) = (-1)
2
for n = 1,2, Prove that there do
not exist increasing function g : (0,1) → R , h : (0,1) → R, such that f = g-h
118. Let S = {1,2,3,4,5}.Find out how many functions f : S → S exits with the following
property : f
50
(x) = x for all x ∈ S
119. Let N denote the set of all positive integers .Prove or disprove that : there exits a function
f : N→ N such that the equality f(f(n)) = 2n holds for all n ∈ N
120. Prove that all functions f : R → R satisfying :∀ x ∈ R f (x) = f (2x) = f (1-x) are
periodic.
121. Determine all the possible integer k such that there is a function f : N→ Z such that
a. f (1997) = 1998,
b. f (ab) = f (a) + f (b) + k f (d (a, b)),∀ a, b ∈ N, where d(a,b) denotes the greatest common
divisor of a and b
122. Let a be rational number , b,c,d be real , and the function f : R → [-1,1] satisfy : f ( x + a
+ b)-f ( x + b ) = c.[ x + 2a + [x] - 2[ x + a] - [b]] + d for each x ∈ R. Show that f is a periodic
function
123. Let f : N→ N be a function satisfying

a. For every n N, f ( n + f (n)) = f (n);
b. f (n
o
) =1 for some positive integer, where N denote is the set of all nature numbers .Show
that f(n) ≡ n
124. Find all pairs of functions f; g : R → R such that
a. if x < y, then f (x) < f (y);
b. for all x; y ∈ R, f (xy) = g(y) f(x) + f(y)
125. Show that there is no function f : R → R such that f ( x+y ) > f (x) ( 1+y f(x)) for all
positive real x,y
126. Find all subjective f : N→ N satisfying the condition m | n iff f (m) | f (n) for all m ,n in N
127. Let f : R
+
→ R
+
be un increasing function. for each u ∈ ,denote the greatest lower
bound of the set { f(t) +(u/t) : t > 0} by g(u) .Show that
a. If x ≤ g (xy) , then x ≤ 2 f(2y)
b. If x ≤ f(y) ,then x ≤ g (xy)
128. Find all functions f : Q
+
→ Q
+
such that for all x ∈ Q
+

a. f (x + 1) = f (x) + 1
b. f (x
2
) = f (x)

2

129. Determine the number of functions f : {1; 2; : : : ; n} →{1995; 1996}which satisfy the
condition that f(1) + f(2) + + f(1996) is odd.
130. Let n > 2 be an integer and f : R
2
→ R be a function such that for any regular n-gon
A
1,
A
2,
,A
n
: f (A
1
) + f (A
2
) + + f (A
n
) = 0; Prove that f is the zero function.
131. Given a ∈ R and f
1 ,
f
2
,f
n
: R → R additive functions such that f
1
(x)f
2

(x) f
n
(x) =
ax
n
for all x ∈ R. Prove that there exists b ∈ R and i ∈ {1; 2; : : : ; n} such that f
i
(x) = bx for all x
∈ R.
132. Let n be a positive integer and D a set of n concentric circles in the plane. Prove that if the
function f : D →D satisfies d(f(A); f(B)) ≥ d(A;B) for all A;B ∈ D, then d(f(A); f(B)) = d(A;B)
for every A;B ∈ D.
133. Determine whether there exists a function f : Z → Z such that for each k = 0; 1; : : : ; 1996
and for each m ∈ Z the equation f(x)+bx =m has at least one solution x ∈ Z.
134. Find all continuous functions f : R→ R such that for all

x ∈ R : f (x) = f( x
2
+ 1/4)
135. Let f : (0;1) → R be a function such that
a. f is strictly increasing
b. f(x) > -1/x for all x > 0
c. f(x)f(f(x) + 1=x) = 1 for all x > 0. Find f(1).
136. Suppose f : R
+
→ R
+
is a decreasing continuous function such that for all x; y ∈ R
+
, f(x

+ y) + f(f(x) + f(y)) = f(f(x + f(y))) + f(y + f(x)). Prove that f(f(x)) = x.
137. For which does there exist a nonconstant function f : R → R such that f(α(x + y)) = f(x) +
f(y).
138. For k ∈ N, let s
1
,s
2
, ,s
n
be integers not less than k, and let p
i
be a prime divisor of f( 2
si
) for i = 1; : : : ; k. Prove that for t = 1, , k , ∑
i=1,p
p
i
| 2
t
iff k | 2
t

139. Find all functions f :R →R such that the equality f(f(x) + y) = f( x
2
- y) + 4f(x)y
140. Let f be a function defined on {0; 1; 2; } such that f(2x) = 2f(x); f(4x + 1) = 4f(x) + 3;
f(4x -1) =2f(2x-1)-1 . Prove that f is injective (if f(x) = f(y), then x = y).
141. Let f
1
,

f
2
,f
3
:R → R be functions such that a
1
f
1
+ a
2
f
2
+ a
3
f
3
is monotonic for all a
1
,

a
2
,
a
3
∈ R . Prove that there exist c
1
,

c
2
,

c
3
∈ R , not all zero, such that c
1
f
1
(x)

+ c
2
f
2
(x)

+ c
3
f
3
(x) = 0 for
all

x ∈ R

142. Find all functions u : R → R for which there exists a strictly monotonic function f : R →
R such that f(x + y) = f(x)u(y) + f(y) ∀ x; y ∈ R

143. Let A = {1; 2; 3; 4; 5}. Find the number of functions f from the set of nonempty subsets of
A to A for which f(B) ∈ B for any B A and f(B  C) ∈ {f(B); f(C)} for any B; C A.
144. A real function f defined on all pairs of nonnegative integers is given. This function
satisfies the following conditions :
a. f(0; 0) = 0,
b. f(2x; 2y) = f(2x + 1; 2y + 1) = f(x; y),
c. f(2x + 1; 2y) = f(2x; 2y + 1) = f(x; y) + 1for all nonnegative integers x; y.
145. Let n be a nonnegative integer and a; b be nonnegative integers such that f(a; b) = n. Find
out how many nonnegative integers x satisfy the equation f(a; x) + f(b; x) = n
146. Show that there is no function f : R → R such that f(0) > 0 ; f( x + y ) ≥ f( x ) + y f(f( x ))
147. Prove that if 0< a ≤ 1 then there is no function f : R
+
→ R
+
satisfy f ( f(x) + 1/f(x) ) = x
+a ∀ x ∈ (0,+
148. Find all continuous functions f : R → R such that f(x + 2002) ( f(x) + √2003) = -2004 for
all x.
149. Find all continuous functions f : [0, 1] → R which are differentiable on the open interval
(0, 1) and satisfy f(0) = f(1) = 1, and 2003 f ' (x) + 2004 f(x) >= 2004 for all x in (0, 1).
150. Given a < b, we are given any continuous functions f, g : [a, b] → [a, b] such that f( g(x) )
= g( f(x) ) for all x, and f is monotonic. Show that f(z) = g(z) = z for some z in [a, b].
151. Given a < b, and a differentiable function f : [a, b] → R such that f(a) = - (b - a)/2, f(b) =
(b - a)/2, f( (a+b)/2 ) ≠ 0, prove that there are three distinct numbers c
1
, c
2
, c
3
in (a, b) such that the

product f(c
1
) f(c
2
) f(c
3
) = 1.
152. Let X be the set of rationals excluding 0, ±1. Let f: X → X be defined as f(x) = x - 1/x. Let
f
1
(x) = f(x), f
2
(x) = f(f(x)), f
3
(x) = f(f
2
(x)) etc. Does there exist a value x in X such that for any
positive integer n, we can find y in X with f
n
(y) = x
153. f is a continuous real-valued function on the reals such that for some 0 < a, b < 1/2, we
have f(f(x)) = a f(x) + b x for all x. Show that for some constant k, f(x) = k x for all x
154. f(x) is a continuous real function satisfying f(2x
2
- 1) = 2 x f(x). Show that f(x) is zero on
the interval [-1, 1].
155. Let P be the set of all subsets of {1, 2, , n}. Show that there are 1
n
+ 2
n

+ + m
n
functions f : P → {1, 2, , m} such that f(A ∩ B) = min( f(A), f(B) ) for all A, B.
156. Let Z be the integers. Prove that if f : Z → Z satisifies f( f(n) ) = f( f(n+2) + 2 ) = n for all
n, and f(0) = 1, then f(n) = 1 - n.
157. Let N be the positive integers. Define f : N → {0, 1} by f(n) = 1 if the number of 1s in the
binary representation of n is odd and 0 otherwise. Show that there do not exist positive integers k
and m such that f(k + j) = f(k + m + j) = f(k + 2m + j) for 0 <= j < m.
158. R is the real line. f, g: R -> R are non-constant, differentiable functions satisfying: (1) f(x
+ y) = f(x)f(y) - g(x)g(y) for all x, y; (2) g(x + y) = f(x)g(y) + g(x)f(y) for all x, y; and (3) f '(0) =
0. Prove that f(x)
2
+ g(x)
2
= 1 for all x.
159. R
+
denotes the positive reals. Prove that there is a unique function f : R
+
→ R
+
satisfying f(
f(x) ) = 6x - f(x) for all x.
160. Let A be a finite set with at least two elements and f : A → A such that for all sets B ∈ A
with at least two elements we have f(B) ≠ B. Prove that there is one and only one a ∈ A such that
f(a) = a.
161. Find all functions f : N → N the have the property that: f(1) + f(2) + . . . + f(n) is a perfect
cube less or equal with n
3
for all positive integers n

162. Find all the functions f : N →M with the property: 1 + f(n)f(n + 1) = 2n
2
(f(n + 1) - f(n)), ∀
n ∈ N in each of the following cases: a) M = N ;b) M = Q.
163. Consider f a subjective real function such that for any sequence of real numbers (xn)n > 0
such that ( f(xn) )n > 0 converges, ( xn )n > 0 itself should also converge. Prove that f is
continuous.
164. Consider f a real continuous function such that it exists M > 0 such that for any real x and
y : | f ( x + y ) - f ( x ) - f( y ) | < M.
a. Prove that for any reals x the limit lim n -> infinity ( f ( nx ) / n ) exists and it is finite.
b. Call this limit g( x ). Prove that g is continuous in 0
c. Prove that limx - > infinity ( f(x) / x ) exists and it is finite.
165. Prove or disprove that there exists a function from the positive integers to the positive
integers such that for any positive integer n we have
a. f(n) < n
8

b. for any a
1
,a
2
, ,a
k
distinct positive integers, f( n ) is not equal to f(a
1
) + f(a
2
) + . . . +f (a
k
).

166. Let f be a monotone function f: [a, b]->R such that for any x
1
and x
2
in [a ,b] there exists a
real number c such that the integral of f taken between x
1
and x
2
is f(c)(x
1
- x
2
).
a. Prove that f is continuous.
b. Is f necessarily continuous if we do not assume that f is monotone?
167. Consider f: [0, 1) a monotone function. Prove that the limits below exist and are equal:
limx -> 1 of the integral of f taken from 0 to x and lim n -> infinity (1 / n) [ f( 0 ) +f ( 1/n ) + f ( 2/n )
+ . . . +f ( (n-1) / n ) ].
168. Let p and n be positive integers, p > 2n, p prime and M be a set of n points in the plane
such that any three points of M are not collinear. The function f : M -> { 0, 1, . . . p - 1 } has the
following properties:
a. f - 1( 0 ) has exactly one element
b. for any circle C passing through at least three points of M, the sum of all f( P ) when P is a
point of both M and C, is divisible by p.
169. Find f : (0;1)→ (0;1) so that for all positive real x; y we have f(xf(y)) = yf(x) and when x
tends to 1 then f(x) tends to 0.
170. Let f be a polynomial with real coefficients such that for each positive integer n the
equation f(x) = n has at least one rational solution. Find f.
171. a. Let f and g be integer valued functions defined in the set of all integers. Prove that the

function h=fg is not surjective.
172. b. Let f be a surjective, integer valued function defined on the integers. Prove that there
exist two surjective, integer valued functions g, h, defined on the integers so that f=gh.
173. Find the number of functions f :{1,2, ,n}→{1,2,3,4,5} so that for any k=1,2, ,n-1 we
have f(k+1)- f(k) ≥ 3.
174. If f: [-1 , 1] -> R is a continuous real function and a, b, c are the Riemann integrals of the
functions f(x)
2
, f(x) and xf(x) then 2a ≥

b
2
+ 3c
2
. When does the equality occur?
©Bùi Hoàng Giang
30 Aug 2003
Last updated/corrected 4 Sep 2003