# A couple of problems from old contests

• Jan 6th 2011, 12:20 PM
myro111
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?

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!!!!
• Jan 6th 2011, 12:34 PM
mr fantastic
Quote:

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 \$\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?

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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