Functional equation, dysfunctional mind?

by wu2481632, Jan 30, 2017, 3:59 AM

Darn I actually can't think straight today. Wasted way too much time on AoPS and spent too little time doing HW. To make up for that, here is a functional equation:

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x^3+y^3)=xf(x^2)+yf(y^2)$$for all $x,y\in\mathbb{R}.$ (Romania 2009)

Solution

Small Piece of Tasty Food

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9 Comments

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Oops I had the same problem today. I should probably do homework first and then do math.

by blue8931, Jan 30, 2017, 5:03 AM

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Isn't
Quote:
It follows that if $f$ is a solution of the functional equation, then $cf$ for some constant $c$ must be a solution of the functional equation. Therefore there exists some function satisfying the given conditions such that $f(1) = 1$, unless $f(1) = 0$.
only true over rationals?

by FlyingCucumber, Jan 30, 2017, 4:35 PM

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Nvm, I didn't read entire thing.

by FlyingCucumber, Jan 30, 2017, 4:37 PM

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No @FC. Suppose that $f$ is a solution of the functional equation. Then $cf$ gives us $cf(x^3+y^3) = cxf(x^2) + cyf(y^2) \implies f(x^3+y^3)=xf(x^2) + yf(y^2)$ and so if $f$ is a solution then $cf$ is a solution.

by wu2481632, Jan 30, 2017, 4:39 PM

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Wait I thought you assumed it was linear for some reason ^^

by FlyingCucumber, Jan 30, 2017, 4:49 PM

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Everyone is solving this and I feel bad for being stuck on it for so long... is homogenizing a common technique or do I just lack ingenuity?

by aftermaths, Jan 30, 2017, 10:16 PM

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WOW your sol was...complicated

Establish additivity and then
\[f((x+1)^3)=(x+1)f((x+1)^2)=(x+1)^2f(x+1) \Rightarrow f((x+1)^2)=(x+1)f(x+1) \Rightarrow f(x^2)=xf(x)\]so by 2002 USAMO P4 we're done.

by shiningsunnyday, Jan 31, 2017, 4:52 AM

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wait ssd how do you get from
$$(x+1)f((x+1)^2)$$to
$$(x+1)^2f(x+1)$$?

NVM that doesn't work oops
This post has been edited 1 time. Last edited by shiningsunnyday, Feb 1, 2017, 4:50 AM

by wu2481632, Jan 31, 2017, 1:42 PM

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Hey, that's pretty good!

less complicatedness

:10: :winner_first: :weightlift: :nhl:
This post has been edited 1 time. Last edited by shiningsunnyday, Feb 2, 2017, 3:05 AM

by zephyrcrush78, Feb 1, 2017, 3:10 AM

To share with readers my favorite problem I came across today :) (Shout for contrib.)

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