Algebra (Thailand Math POSN 3rd round)

Write a full solution.

  1. Let P(x),Q(x)P(x),Q(x) be real polynomials (real coefficients) with leading coefficient 11 or 1-1 such that deg(P(x))>deg(Q(x))deg(P(x)) > deg(Q(x)), find the number of solutions (P(x),Q(x))(P(x),Q(x)) to P(x)2+Q(x)2=x8+1P(x)^{2}+Q(x)^{2} = x^{8}+1. If possible, find each forms of solution.

  2. Let f(x)=(x55+x42+x33130)x55+x42+x33130f(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)f(n) where nn is a positive integer. (Where x\lfloor x \rfloor is a floor function, and defined to be xx<x+1\lfloor x \rfloor \leq x < \lfloor x \rfloor +1)

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

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

Note by Samuraiwarm Tsunayoshi
4 years, 6 months ago

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

Calvin Lin Staff - 4 years, 6 months ago

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Sir try This and this ¨\ddot\smile

Parth Lohomi - 4 years, 6 months ago

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