A Functional Equation from 2019 IMO

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f(f(x+y))=f(2x)+2f(y)
#functions #algebra #trigonometry
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[SL-efjv]

It's amazing to me how you can take a problem from one of the hardest math competitions in the world and make the solution so easy to understand and also fun the watch. Love your content so much❤️

[tonyscott]

f(f(x+y)) on the LHS should be symmetric under exchange of 'x' and 'y'. If you do that exchange on the RHS, you can isolate U(x) = W(y) and for all 'x' and 'y' U(x)=W(y) = constant. You get to your linear result more rapidly.

[scottleung]

Nice! BTW u should make a lecture on functional equations.

[misterdubity]

The 2 solutions suggests f(0) can be both 0 and any b, ie any integer. So I think f(x) = 2x+b is for all x except when x=0. ... Otherwise ... If you plug x=y=0 gives f(f(0)) = 3f(0); if f(0)=k, then f(k) = 3k so either k=0 or 2k+b = 3k so b=k=f(0). Suggesting f(x) = 2x and b=0 ???

[angelishify]

Let's take y = −x and differentiate eq. (denote f'(x) = g(x)). Then we have

g(2x) = g(−x) and replacing x with −x: g(−2x) = g(x); and replacing x with x/2: g(x) = g(−x/2)

So we have g(x) = g(x/4) = g(x/16) = ... = g(x/4^n).

In the limit n → ∞ we get

g(x) = g(0) = a and f(x) = ax + b.

[icfj]

It can be shown that the function is linear also for real numbers, so the solution is also the same.

[rakenzarnsworld]

f(f(3))=f(2)+2f(2)=f(4)+2f(1)
f(f(5))=f(4)+2f(3)=f(6)+2f(2)

3f(2)=f(4)+2f(1)

[MrLidless]

ff(x + y) = f(2x) + 2f(y)
Let y = 0
ff(x) = f(2x) + 2f(0)
Let x = 0
ff(y) = f(0) + 2f(y), ie ff(x) = f(0) + 2f(x)
So f(2x) + f(0) = 2f(x)
Ie it’s linear and work from there.

[vaggelissmyrniotis]

I found f(x)=2x+c but i didnt even considered the obvious solution f(x)=0

[honestadministrator]

At x = 0. f ( f ( y)) = f (0) + 2 f (y)
f ( z ) = f(0) + 2 z

[Roq-stone]

So (ab) is 2b, or not 2b.
Sorry, I can't help myself.

[DergaZuul]

Now GPT AI can solve this (MathGPT), but it is wrong to call this a linear function it is not linear it is affine (form ax+b is affine not linear), pretty common mistake.