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Math Help - A couple of problems from old contests

  1. #1
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    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 x^4+x^3-1=0.Prove that a*b is the solution of x^6+x^4+x^3-x^2-1=0

    2.(1983.).Prove that the solutions of x^5+a*x^4+b*x^3+c*x^2+d*x+e=0 are real if 2*a^2<5*b

    German contest

    3.(1977.)How many pairs of numbers p,and q from N which are smaller than 100 and for which x^5+p*x+q=0 has a rational solution exist?

    Moscow olympiad

    4.(1951.) Dividing the polynomial x^1951-1 with P(x)=x^4+x^3+2*x^2+x+1 we get a quotient and remainder.What is the coefficient next to x^14 in the quotient?

    5.(1955.)If p/q is the root of the polynomial f(x)=a[0]*x^n+a[1]*x^(n-1)+...+a[n]and p and q don't have common divisors.If f(x) has integer coefficients then prove that p-k*q 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!!!!
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  2. #2
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    Quote Originally Posted by myro111 View Post
    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 x^4+x^3-1=0.Prove that a*b is the solution of x^6+x^4+x^3-x^2-1=0

    2.(1983.).Prove that the solutions of x^5+a*x^4+b*x^3+c*x^2+d*x+e=0 are real if 2*a^2<5*b

    German contest

    3.(1977.)How many pairs of numbers p,and q from N which are smaller than 100 and for which x^5+p*x+q=0 has a rational solution exist?

    Moscow olympiad

    4.(1951.) Dividing the polynomial x^1951-1 with P(x)=x^4+x^3+2*x^2+x+1 we get a quotient and remainder.What is the coefficient next to x^14 in the quotient?

    5.(1955.)If p/q is the root of the polynomial f(x)=a[0]*x^n+a[1]*x^(n-1)+...+a[n]and p and q don't have common divisors.If f(x) has integer coefficients then prove that p-k*q 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!!!!
    Please don't post more than two questions in a thread. Otherwise the thread can get convoluted and difficult to follow. See rule #8: http://www.mathhelpforum.com/math-he...ng-151418.html.

    Also, these questions are too advanced for the PRE-algebra and Algebra subforum. Try posting in the Other Topics subforum. Thankyou.

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