A couple of problems from old contests
I found some problems from old contests in my book and I need some help solving them since I could not find the solutions online.
Some USA contest problems:
1.(1977.)Let a and b be 2 solutions of .Prove that is the solution of
2.(1983.).Prove that the solutions of are real if
3.(1977.)How many pairs of numbers p,and q from which are smaller than 100 and for which has a rational solution exist?
4.(1951.) Dividing the polynomial with we get a quotient and remainder.What is the coefficient next to in the quotient?
5.(1955.)If is the root of the polynomial and p and q don't have common divisors.If f(x) has integer coefficients then prove that is a divisor of f(k) for every integer k.
6.(1973.) The polynomial P(x) with integer coefficients has the values 1,2,3 for some integer values of x.Prove that there is at most 1 integer x for which the polynomial has the value of 5.
Thank you very much!!!!