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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
1 year, 6 months ago

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ooooh, I like question 3 :) I tend to have a soft spot for such functional equations. Staff · 1 year, 6 months ago

Sir try This and this $$\ddot\smile$$ · 1 year, 6 months ago