- #1
myro111
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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 [tex] x^4+x^3-1=0 [/tex] .Prove that [tex] a*b [/tex] is the solution of [tex] x^6+x^4+x^3-x^2-1=0 [/tex]
2.(1983.).Prove that all the solutions of [tex] x^5+ax^4+bx^3+cx^2+dx+e=0 [/tex] are real if [tex] 2a^2<5b [/tex]
German contest
3.(1977.)How many pairs of numbers p,and q from [tex] N [/tex] which are smaller than 100 and for which [tex] x^5+px+q=0 [/tex] has a rational solution exist ?
Moscow olympiad
4.(1951.) Dividing the polynomial [tex] x^1^9^5^1-1 [/tex] with [tex] P(x)=x^4+x^3+2*x^2+x+1 [/tex] we get a quotient and remainder.What is the coefficient next to [tex] x^1^4 [/tex] in the quotient?
5.(1955.)If [tex] p/q [/tex] is the root of the polynomial [tex] f(x)=a[0]*x^n+a[1]*x^n^-^1+...+a[n] [/tex] and p and q don't have common divisors.If [tex] f(x) [/tex] has integer coefficients then prove that [tex] p-k*q [/tex] is a divisor of [tex] f(k) [/tex] for every integer k.
Thank you very much!
Some USA contest problems:
1.(1977.)Let a and b be 2 solutions of [tex] x^4+x^3-1=0 [/tex] .Prove that [tex] a*b [/tex] is the solution of [tex] x^6+x^4+x^3-x^2-1=0 [/tex]
2.(1983.).Prove that all the solutions of [tex] x^5+ax^4+bx^3+cx^2+dx+e=0 [/tex] are real if [tex] 2a^2<5b [/tex]
German contest
3.(1977.)How many pairs of numbers p,and q from [tex] N [/tex] which are smaller than 100 and for which [tex] x^5+px+q=0 [/tex] has a rational solution exist ?
Moscow olympiad
4.(1951.) Dividing the polynomial [tex] x^1^9^5^1-1 [/tex] with [tex] P(x)=x^4+x^3+2*x^2+x+1 [/tex] we get a quotient and remainder.What is the coefficient next to [tex] x^1^4 [/tex] in the quotient?
5.(1955.)If [tex] p/q [/tex] is the root of the polynomial [tex] f(x)=a[0]*x^n+a[1]*x^n^-^1+...+a[n] [/tex] and p and q don't have common divisors.If [tex] f(x) [/tex] has integer coefficients then prove that [tex] p-k*q [/tex] is a divisor of [tex] f(k) [/tex] for every integer k.
Thank you very much!