# Thread: A couple of problems from old contests

1. ## 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?

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!!!!

2. Originally Posted by myro111
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?

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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