Write a full solution.

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.

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 \(\lfloor x \rfloor \leq x < \lfloor x \rfloor +1\))

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

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

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

Log in to reply

Sir try This and this \(\ddot\smile\)

Log in to reply