# 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
6 years, 3 months ago

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ooooh, I like question 3 :) I tend to have a soft spot for such functional equations.

Staff - 6 years, 3 months ago

Sir try This and this $\ddot\smile$

- 6 years, 3 months ago