A couple of not so simple problems from old contests

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I solved the rest myself.In summary, the conversation discusses various mathematical problems from different contests, including USA, German, and Moscow Olympiad. The problems involve finding solutions to polynomials and proving properties of them. The participants in the conversation share their thoughts and approaches to solving the problems, with one person eventually solving one of the problems using Vietes formulas.
  • #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!
 
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  • #2
What have you tried so far? Do you have ideas how to tackle them?
 
  • #3
1. Divide the second polynomial with x-ab and I get as a remainder [tex](ab)^6+(ab)^4+(ab)^3-(ab)^2-1[/tex] The remainder is the second polynomial but instead of x it is ab.I got this idea because if we have a polynomial P(x),a is the root only and only if P(x) is divisible by x-a,and that fact is well known.The other idea was Vietes formulas but couldn't get anything.

2.No idea.

3.No idea

4.No idea.

5.No idea.
 
  • #4
I'm kidding myself that I have anything close to the knowledge required to solve these, but...

Write x^6 + x^4 + x^3 - x^2 - 1 = 0 as x^4 + x^3 - 1 = x^2 (1 - x^4), substitute x = a, that should help you to get a range of values for ab.
 
  • #5
Dont worry I solved it another way.But thanks for the try.I used Vietes formulas
 

1. What are some examples of old contest problems that are not so simple?

Some examples of old contest problems that are not so simple could include the Traveling Salesman Problem, the Knapsack Problem, the Tower of Hanoi, the Monty Hall Problem, and the Prisoner's Dilemma.

2. How do these old contest problems differ from traditional scientific research problems?

Old contest problems often have very specific and well-defined parameters, making them easier to solve than open-ended scientific research problems. They also tend to have clear and objective solutions, whereas scientific research problems may have multiple interpretations and potential outcomes.

3. Why are old contest problems still relevant in the scientific community?

Old contest problems are still relevant because they often serve as benchmarks for testing new algorithms and problem-solving techniques. They also provide a way to compare and evaluate the effectiveness of different approaches to solving a problem.

4. What skills are necessary to successfully solve old contest problems?

Some key skills that are necessary to successfully solve old contest problems include critical thinking, problem-solving, mathematical and analytical abilities, and familiarity with various algorithms and techniques. Strong programming skills may also be beneficial, depending on the type of problem.

5. Are there any resources available for practicing and improving problem-solving skills for old contest problems?

Yes, there are many resources available for practicing and improving problem-solving skills for old contest problems. These include online forums and communities, books and articles on problem-solving strategies, and practice problems and competitions specifically designed for these types of problems.

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