100 Functional Equations Problems.pdf

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Functional Equations Problems
AMIR HOSSEIN PARVARDI
JUNE 13, 2011
Dedicated to PCO .
email: ahpwsog@gmail.com , blog: http://math-olympiad.blogsky.com .
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DEFINITIONS
•N is the set of positive integers.
•N∪{ 0 } = N is the set of non-negative integers.
•Z is the set of integers.
•Q is the set of rational numbers.
•R +
is the set of positive real numbers.
•R is the set of real numbers.
• If a function f is defined on the set A to the set B , we write f : A→B
and read “ f is a function from the set A to the set B .”
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PROBLEMS
1.
Find all surjective functions f : N→N such that f (n)≥n + (−1) n ,∀n∈N.
2.
Find all functions g : R→R such that for any real numbers x and y
g(x + y) + g(x)g(y) = g(xy) + g(x) + g(y).
3.
Find all real valued functions defined on the reals such that for every real
x, y
f (x 2 −y 2 ) = xf (x)−yf (y).
4. Find all real valued functions defined on the reals such that for every real
x, y:
f (xf (x) + f (y)) = f (x) 2 + y.
Find all functions f : N→N such that f (f (n)) + (f (n)) 2
= n 2 + 3n + 3 for
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all positive integers n.
6. Let n be a positive integer. Find all strictly increasing functions f : N →N
such that the equation
f (x)
k n
= k−x
has an integral solution x for all k∈N.
Find all functions f : R + →R +
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such that
f ( x + y
2
2f (x)f (y)
f (x) + f (y) ∀x, y∈R + .
) =
8.
Find all functions f : R→R such that
f (1−x) = 1−f (f (x)) ∀x∈R.
Find all functions f : R + →R +
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such that
f (1 + xf (y)) = yf (x + y) ∀x, y∈R + .
10.
Find all functions f : R + →R +
such that
f (xf (y)) = f (x + y) ∀x, y∈R + .
11.
Find all functions f : R→R such that
f (f (x) + y) = f (x 2 −y) + 4yf (x) ∀x, y∈R.
12.
Find all functions f, g, h : R→R such that
f (x + y) + g(x−y) = 2h(x) + 2h(y) ∀x, y∈R.
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13.
Find all functions f : R→R such that
f ( x + y + z ) = f ( x ) f (1 −y ) + f ( y ) f (1 −z ) + f ( z ) f (1 −x ) ∀x, y, z∈R.
14. Find all functions f : R→R such that
f ( f ( x ) −f ( y )) = ( x−y ) 2 f ( x + y ) ∀x, y∈R.
15. Find all functions f, g : R→R such that
• If x < y , then f ( x ) < f ( y );
• for all x, y∈R , we have f ( xy ) = g ( y ) f ( x ) + f ( y ).
16. Find all functions f : R→R such that
f (( x + z )( y + z )) = ( f ( x ) + f ( z ))( f ( y ) + f ( z )) ∀x, y, z∈R.
17.
Find all functions f : R→R that satisfy
f ( x 3 + y 3 ) = x 2 f ( x ) + yf ( y 2 )
for all x, y∈R.
18.
Find all functions f : R→R that satisfy
f ( m + nf ( m )) = f ( m ) + mf ( n )
for all m and n .
19. Find all functions f : R→R such that f ( x ) f ( y ) = f ( x + y ) + xy for all
x, y∈R .
20. Find all functions f : N∪{ 0 }→N∪{ 0 } .Such that x 3 f (y) divides f ( x ) 3 y
for all x, yN∪{ 0 } .
21.
Find all continuous functions f : R→R such that
f ( x + y ) f ( x−y ) = ( f ( x ) f ( y )) 2 ∀x, y∈R.
22.
Find all functions f : R→R such that
( x + y )( f ( x ) −f ( y )) = ( x−y ) f ( x + y ) ∀x, y∈R.
23.
Find all functions f : R→R such that
f (( f ( x ) + y ) = f ( x 2 −y ) + 4 f ( x ) y ∀x, y∈R.
24.
Find all the functions f : Z→R such that
f ( m + n−mn ) = f ( m ) + f ( n ) −f ( mn ) ∀m, n∈Z
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Find all functions f : (0, 1)→(0, 1) such that f ( 2 ) = 2
25.
and
(f (ab)) 2 = (af (b) + f (a)) (bf (a) + f (b)) ∀a, b∈(0, 1).
26. Find all functions f : Q→Q such that
f (x + y + f (x)) = x + f (x) + f (y) ∀x, y∈Q.
27. Find all functions f : R→R such that
f (x 2 + f (y)) = (x−y) 2 f (x + y) ∀x, y∈R.
28. Find all functions f : R→R such that
•f (x + y) = f (x) + f (y) ∀x, y∈R,
•f (x) = x 2 f ( x ) ∀x∈R\{0}.
29. Let a >
3
4
be a real number. Find all functions f : R→R such that
f (f (x)) + a = x 2 ∀x∈R.
30.
Find all injective functions f : N→N which satisfy
f (f (n))≤ n + f (n)
2 ∀n∈N.
31.
Find all continuous functions f (x), g(x), q(x) : R→R such that
f (x 2 ) + f (y 2 ) = [q(x)−q(y)]g(x + y) ∀x, y∈R.
32.
Find all functions f : R→R so that
f (x + y) + f (x−y) = 2f (x) cos y ∀x, y∈R.
33.
Find all functions f : R→R such that
f (x−f (y)) = f (x) + xf (y) + f (f (y)) ∀x, y∈R.
Find all functions f : R + →R + such that
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f (f (x)) = 6x−f (x) ∀x∈R + .
35.
Find all functions f : R→R such that
f (x + y) + f (xy) + 1 = f (x) + f (y) + f (xy + 1) ∀x, y∈R.
36.
Find all functions f : R→R such that
f (x)f (yf (x)−1) = x 2 f (y)−f (x) ∀x, y∈R.
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