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Source: Azerbaijan IMO TST 2016,D1 P2

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IstekOlympiadTeam

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IstekOlympiadTeam

#1 h May 26, 2018, 10:23 AM • 3 Y

Y by Smita, Adventure10, Mango247

Find all polynomials $P(x)$ with real coefficents, such that for all $x,y,z$ satisfying $x+y+z=0$ , the equation below is true: $P(x+y)^3+P(y+z)^3+P(z+x)^3=3P((x+y)(y+z)(z+x))$

This post has been edited 2 times. Last edited by IstekOlympiadTeam, May 26, 2018, 11:03 AM

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Smita

#2 h May 26, 2018, 11:02 AM • 2 Y

Y by Adventure10, Mango247

Does p^3(x) mean it is multiplied 3 times or it is a composite function

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#3 h May 28, 2018, 3:13 PM • 3 Y

Y by Pluto1708, Adventure10, Mango247

It is same as

Quote:

Find all polynomials $P(x)$ with real coefficents, such that for all $x,y,z$ satisfying $x+y+z=0$ , the equation below is true: $P(x)^3+P(y)^3+P(z)^3=3P(xyz)$

$z=0,y=-x: P(x)^3+P(-x)^3=3P(0)-P(0)^3=c \to P(-x)^3=c-P(x)^3$
$y=1-x,z=-1:P(x)^3+P(1-x)^3+P(-1)^3=3P(x(x-1))=P(x)^3+c-P(x-1)^3+P(-1)^3$
Let $deg(P(x))=n>0$ then $deg(P(x)^3-P(x-1)^3)= 3n-1$ and $deg(P(x(x-1))=2n \to n=1$
For $n=0:P=0,-1,1$
For $n=1:P=ax+b$ and so $(ax+b)^3+(ay+b)^3+(-a(x+y)+b)^3=3(-axy(x+y)+b)$
$x=y=0 \to 3b^3=3b \to b=0,-1,1$
Case 1: $b=0: a^3x^3+a^3y^3-a^3(x+y)^3=-3axy(x+y)$
$3a^3(xy(x+y))=3axy(x+y) \to a=\pm 1$

Case 2: $b= \pm 1: (ax \pm 1)^3+(ay \pm 1)^3+(-a(x+y) \pm 1)^3=3(-axy(x+y)\pm 1)$
$y=0,x=1: (a \pm1)^3 \pm1+(-a\pm1)^3=\pm 3 \to 6a^2 =0$ - contradiction
So answer: $P(x)=0, \pm 1, \pm x$

This post has been edited 1 time. Last edited by RagvaloD, May 28, 2018, 3:17 PM

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#4 h Aug 15, 2019, 11:57 AM • 1 Y

Y by Adventure10

$x, y, z \to 0$ : $P(0)=0, \pm 1$ .
$x \to (y+z)$ : $P(-y)^3 + P(-z)^3 + P(y+z)^3 = 3P(yz(y+z)$ for all $y,z$ .
$y \to -z$ : $(P(z)+P(-z))(P(z)^2 - P(z)P(-z)+P(-z)^2)=2P(0)$ .
Note that the degree of $P(z)^2 - P(z)P(-z)+P(-z)^2$ is equal to twice of $P(z)$ .
Therefore, $P(z)+P(-z)=0$ and $P(0)=0$ .

Let $P(x)=a_n x^{2n+1} +... + a_0 x$ .
Then, $(a_n)^3 ((y+z)^{6n+3} - y^{6n+3} - z^{6n+3})$ are the greatest degree terms on the left side
and $(a_n)(yz(y+z))^{2n+1}$ are the greatest degree terms on the right side.
If $n>0$ , the term $y^{6n+2} z$ survives on the left but not on the right.
Therefore, $n=0$ and ${a_n}^3 =a_n$ . Thus, we get $P(x)=\pm x , 0$ .

This post has been edited 4 times. Last edited by bumjoooon, Aug 15, 2019, 12:00 PM

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