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Thread: 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 $\displaystyle x^4+x^3-1=0$.Prove that $\displaystyle a*b$ is the solution of $\displaystyle x^6+x^4+x^3-x^2-1=0$

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

    German contest

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

    Moscow olympiad

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

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

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

    German contest

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

    Moscow olympiad

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

    5.(1955.)If $\displaystyle p/q$ is the root of the polynomial $\displaystyle f(x)=a[0]*x^n+a[1]*x^(n-1)+...+a[n]$and p and q don't have common divisors.If f(x) has integer coefficients then prove that $\displaystyle p-k*q$ 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.

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