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Algebra (Thailand Math POSN 3rd round)

Write a full solution.

  1. Let \(P(x),Q(x)\) be real polynomials (real coefficients) with leading coefficient \(1\) or \(-1\) such that \(deg(P(x)) > deg(Q(x))\), find the number of solutions \((P(x),Q(x))\) to \(P(x)^{2}+Q(x)^{2} = x^{8}+1\). If possible, find each forms of solution.

  2. Let \(f(x) = \displaystyle \left(\frac{x^{5}}{5} + \frac{x^{4}}{2} + \frac{x^{3}}{3} - \frac{1}{30}\right) - \left\lfloor \frac{x^{5}}{5} + \frac{x^{4}}{2} + \frac{x^{3}}{3} - \frac{1}{30}\right\rfloor \). Find all possible values of \(f(n)\) where \(n\) is a positive integer. (Where \(\lfloor x \rfloor \) is a floor function, and defined to be \(\lfloor x \rfloor \leq x < \lfloor x \rfloor +1\))

  3. Find all real polynomials \(P(x)\) that satisfy \(P(a-b)+P(b-c)+P(c-a) = 2P(a+b+c)\) for all reals \(a,b,c\) that satisfy \(ab+bc+ca = 0\).

This note is part of Thailand Math POSN 3rd round 2015

Note by Samuraiwarm Tsunayoshi
2 years, 1 month ago

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ooooh, I like question 3 :) I tend to have a soft spot for such functional equations. Calvin Lin Staff · 2 years, 1 month ago

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@Calvin Lin Sir try This and this \(\ddot\smile\) Parth Lohomi · 2 years, 1 month ago

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