1. Find all functions from the set of real numbers into which satisfy for all the identity

2. Consider the function , where is the set of all non-negative integers, defined by the following conditions

(i);

(ii) and

(iii) for all .

(a)Determine the three sets , , and .

(b)For each , find a formula for in terms of .

3. Let be a positive integer. Find the largest nonnegative real number (depending on ) with the following property: whenever are real numbers such that is an integer, there exists some such that .

4. Let denote the set of all real numbers. Find all functions from to satisfying

(i)There are only finitely many in such that ,

And

(ii) for all in .

5. Find all for which there exists a non-constant function such that

for all

6. Consider function which satisfies the conditions for any mutually distinct real numbers satisfying , are mutully different and

Prove that function is linear.

7. Find all complex polynomial such that for any three integers satisfying is an integer.

8. Find all functions such that

9. Let be given positive real number, find all the functions such that holds for any positive integers , satisfying .

10. Given non-zero reals , , find all functions , such that for every , , and

11. Prove that for all integers and there exists a function from the positive integers to the positive integers such that for all positive integer.

12. Find all functions such that for every reals .

13. Let be a real-valued function defined on the positive reals such that

(1)If , then ,

(2) for all .

Show that for some value of .

14. Define on the positive integers by , where for some nonnegative integers. Find the smallest $n$ such that

15. Find all functions satisfying, for all ,

16. a) Show that there are no functions such that and for all .

b)Let be the set of all real numbers greater than . Show that there are functions satsfying the condition above.

17. Let where is a rational constant. Show that exists only finitely many rationals such that is finite.

18. Find all real-valued functions on the positive integers such that for all , and for all .

19. Find at least one function such that and for any real .

20. Let , (with ). Suppose that , , and for some we have , show that (for any valid ).

21. Find all functions such that for all

22. Find all functions such that we have: If then

23. Let be a set of real numbers. We say that a pair of functions from into is a *Spanish Couple* on , if they satisfy the following conditions

(i) Both functions are strictly increasing, i.e. and for all , with ;

(ii) The inequality holds for all .

Decide whether there exists a Spanish Couple

a)On the set of positive integers;

b)On the set .

24. For every let denote the number of (positive) divisors of . Find all functions with the following properties:

a) for all , and

b) divides for all , .

25. Consider those functions which satisfy the condition for all Find all possible values of

26. Find all surjective functions such that for every and every prime the number is divisible by if and only if is divisible by

27. Find all real polynomials such that

28. Determine all functions with such that

29. Show that for positive integer , and for ,

30. Find all such that

for every pair of real numbers .