Bài tập hàm số ôn thi Olympic Toán Sinh Viên

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

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 [...]... 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 . 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. 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. 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