Functional Equations #6 - The Symmetry Substitution

#IMO​ #FunctionalEquations​ #MathOlympiad​
In this video we learn a new important substitution, which is the symmetry substitution P(y,x).
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[adityagupta]

Do you mind telling us which year this problem is from? Great work!

[damiennortier]

Replace x with y - x, you got f(x^2 - 2x) = - x(f(y-x) + f(y))
Now, set x = - 1 :
F(3) = f(y+1) + f(y) and that it because f(3) is a constant then, you got f(y+1) = f(3) - f(y) and then, since f(x) = ax + b, just verify :
F(x^2 - y^2) = a(x^2 - y^2) + b
(x - y) (f(x) + f(y)) = (x - y)(a(x + y) + 2b) = ax^2 - ay^2 + 2bx + 2by
Now, we have ax^2 - ay^2 + b = ax^2 - ay^2 + 2bx + 2by.
Since we're solving for a and b, just set each a and b equal to each other (each term before x^2, each before y^2, ...), you got :
a = a
a = a
2b = 0
2b = 0
b = 0

So f(x) = ax

[uwomisakirins]

Haha, spider-man idea was really funny

[keyyyla]

Let x-y = z. Since x^2 - y^2 = (x-y)(x+y), we have x^2 - y^2 = z(x+y) so by assumption we have for all x and y: f(z(x+y)) = z(f(x) + f(y)). Since this holds for all x and y, we can choose z = 1. it follows that f(x+y) = f(x) + f(y), so f has to be a linear function, which means f(x) = mx. Done.

[ritsu]

Hello, i want to know if my solution is acceptable :

We want to find all function (whose domain and codomain are in R) such as f(x^2-y^2) = (x-y) (f(x)+f(y)).

If x = 0 : f(-y^2)= -y*f(y).
If y = 0 : f(x^2) = x*f(x).

Therefore we have : f(x^2) = -f(-x^2). (by multiplying by -1). Then : -f(-x^2)= f(x^2).
Now : If we replace x by sqrt(x) and by -sqrt(x) (afterwards - so that the domain of the function isn't changed by the fact that we have negative and positive real numbers.), we have :

/ -f(-(-sqrt(x))^2)) = f((-sqrt(x)^2)) <=> -f(-x) = f(x).

Consequently : f(x)+f(-x) = 0. This means that we want a function whose opposite values have opposite images. This is only possible if the function f is a line passing by the origin of the cartesian plan. Whence : f(x) = ax (with a, a real number).


Really helpful channel, thank you !!

[ddmm]

it is very difficult to read your scriptures because you use pale colors. please use only red and black.