Gần 200 bài toán về hàm số ôn tập thi Olympic Toán Sinh Viên toàn quốc. Bài tập hàm số ôn thi Olympic Toán Sinh Viên với đầy đủ các dạng bài.. các đẳng thức, bất đẳng thức liên quan đến phương trình hàm
Trang 11 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(t2f(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(x2 + 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
Trang 216 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 < 2m is f(2m+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) =
yh f(x/2) + xk f(y/2) for all x, y
20 Let f(x) = (x2 + 1)/(2x) for x non-zero Define f0(x) = x and fn+1(x) = f( fn(x) ) Show that for x not
-1, 0 or 1 we have fn(x)/fn+1(x) = 1 + 1/f(y), where y = (x+1)N/(x-1)N and N = 2n
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 integervalued 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) = 4n+2f(n+1) - 16n+1f(n) + n 2sq(n), were sq(n) = n2 Show that f(1989), f(1990) and f(1991) are all divisible by 13
25 For a positive integer k, let f1(k) be the square of the sum of its digits Let fn+1(k) = f1( fn(k) ) Find the value of f1991(21990)
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 f1(n) = f(n), f2(n) = f( f1(n) ), f3(n) = f( f2(n) ) etc Find the smallest k such that fk(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)
Trang 329 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(x4 + y) = x3f(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 fi(x) on the reals such that for all x, y we have fk+1(x)
fk+1(y) = fk(xk) + fk(yk) for k = 1, 2 , n-1 and f1(x) + f1(y) = fn(xn) fn(yn)
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 = w1 + + wk when n is a product of the coprime prime powers w1, w2, , wk 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) = 3f(1), f(3) = 2f(2), f(4) = 3f(3), f(5) = 2f(4) and so on Similarly, define g(1) = 3, g(2) = 2g(1), g(3) = 3g(2), g(4) = 2g(3), g(5) = 3g(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 x0, x1, x2, satisfies xn = f(xn-1) for all positive n For some n > 0, xn - x0 = k, an integer Show that lim xn/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 x1 = 2, x2, , xn such that A < xn < B and xi = f(xi-1) or g(xi-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
Trang 444 Let n > 1 be an integer Let f(k) =1 + 3k/(3n - 1), g(k) = 1 - 3k/(3n - 1) Show that tan(f(1)/3) tan(f(2)/3) tan(f(n)/3) tan(g(1)/3) tan(g(2)/3) tan(g(n)/3) = 1
45 Let X be the closed interval [0, 1] Let f: X X be a function Define f1 = f, fn+1(x) = f( fn(x) ) For some n we have |fn(x) - fn(y)| < |x - y| for all distinct x, y Show that f has a unique fixed point
46 Let N0 = {0, 1, 2, 3, } and R be the reals Find all functions f: N0 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(x2 + y2) 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)) = x2f(xy)
56 Let f(x) = a1/(x + a1) + a2/(x + a2) + + an/(x + an), where ai 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(2n - 1) = 0 for n = 0, 1, 2, If m is not
of the form 2n - 1, then f(m) = f(m+1) + 1 Show that f(n) + n = 2k - 1 for some k, and find f(21990)
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
Trang 560 Find f(x) such that f(x)2f( (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 x2 for all x
64 Find all real valued functions f(x) on the reals such that f( (x - y)2) = x2 - 2y f(x) + f(y)2
65 Find all realvalued functions f defined on X, the set of all nonzero 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(x1993) + f(y1993) 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)/2 ) + f( sin( (x - y)/2 ) for all x, y Find f(1990 + 19901/2 + 19901/3)
72 Let N be the positive integers The function f : N N satisfies f(1) = 5, f( f(n) ) = 4n + 9 and f(2n)
= 2n+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) = k2 + k + 1, where 2k 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?
Trang 677 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(x2 + y2) 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) (2kx, 2k(kx + y) ), where k is real
81 f is a real-valued function on the reals It satisfies f(x3 + y3) = (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(xn)
<= f(x) f(x2)1/2 f(x3)1/3 f(xn)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) = xkf(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)
<= x2f(y/2) + y2f(x/2) for all x, y Show that f(x) <= x2
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(xayb) <= 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 f1 to be f and fk+1(n) = f(fk(n) ) Let g(n) be the smallest k such that fk(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(x2 - y2) = x f(x) - y f(y) for all x, y
Trang 793 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 f1(x) = (x2 + 48) and fn(x) = (x2 + 6fn-1(x) ) Find all real solutions to fn(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 - x2) f(2x/(1 +
x2) ) = (1 + x2)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 - x2) ) 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 : SS 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
Trang 8a 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
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
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,zR+
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 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 : An An and g: An An 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 : SS exits with the following property : f50 (x) = x for all xS
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 (no) =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)
Trang 9125 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 : R2 → R be a function such that for any regular n-gon
A1,A2, ,An : f (A1) + f (A2) + + f (An) = 0; Prove that f is the zero function
131 Given a R and f1 ,f2 ,fn : R R additive functions such that
f1(x)f2(x) .fn(x) = axn for all x R Prove that there exists b R and i {1; 2; : : : ; n} such that fi(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 s1,s2, ,sn be integers not less than k, and let pi be a prime divisor of f( 2si ) for i = 1; : : : ; k Prove that for t = 1, , k , ∑i=1,p pi | 2t iff k | 2t
139 Find all functions f :R R such that the equality f(f(x) + y) = f( x2- y) + 4f(x)y
Trang 10140 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 f1 ,f2 ,f3 :R R be functions such that a1f1 + a2f2 + a3f3 is monotonic for all
a1, a2,a3R Prove that there exist c1,c2, c3R, not all zero, such that c1f1(x)+
c2f2(x)+ c3f3(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
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 c1, c2, c3 in (a, b) such that the product f(c1) f(c2) f(c3) = 1