If s² > 4p, then there are reals a,b with a b = p and a + b = s (namely, the roots of x² - s x + p). Let p= -1 and s arbitrary irrational. Then a and b must both be irrational, hence the functional equation gives us f(s)=f(-1). Now let p = -sqrt(2) and s any rational. Again, a and v must both be irrational, and the functional equation gives us f(s) = f(-sqrt(2)), and by the previous, we know already that f(-sqrt(2)) = f(-1).
Would it not be enough to say that f(a+b+0)=f(a+b)=f((a+b)0)=f(0), and that a+b can add to make any real number, therefore f(X)=k for all real numbers???
Isnt it possible to first ignore the irrationality condition, then find all functions for all reals such that f(ab) = f(a+b). The only such function satisfying this is f(x) = c, a constant. So it also satisfies for any irrational a and b since the irrationals are a subset of the reals. ?
I did something weird but I think it is correct, so: Sub a = x/k, b = yk where x/k & yk is irational => f(xy) = f(x/k + yk) sub k = sqrt(m+1) - sqrt(m) which should satisfy the above condition => f(xy) = f(x(sqrt(m+1) + sqrt(m)) y(sqrt(m+1) - sqrt(m))), put x=y => f(x^2) = f(x(2sqrt(m+1))) , sub any constant fo x => f(c^2) = f(c(2sqrt(m+1))) , and note that c(2sqrt(m+1)) can be any real number including -ve as c can be -ve and (-c)^2 = c^2 => thus f(x) = k tho still incredible solution
Great video! From you accent I guess you are from Hong Kong or similar places. Did I guess it right? Cuz I’m from Hong Kong and I absolutely love the channel
{rational} + {irrational} = {irrational}. Can we not let a = 1 and b be any irrational. Then: f(b) = f(b+1), both sides mapping irrational to irrational and true for all b. Does this prove that f maps to a constant?
The statement only says that f(a+b) = f(ab) when a and b are irrational, so you can't choose a=1. On top of that, even if you could, f(b) = f(b+1) is not a sufficient condition to conclude that f is constant, f could be any function with period 1. For example, cos(2*pi*x) or frac(x) would be non-constant functions that satisfy your condition.
There was nothing in the problem that said a and b could ONLY be irrational. If you showed that all reals mapped to f(0) then it follows that all irrational s do.
There could be a function that works only for irrational numbers but not for rationale, for example the piecewise f(x) = a solution proposed here if x irrational and 1 otherwise
Assuming you mean e^a e^b = e^(a+b) , this is an example of f(a)f(b) = f(a+b), but not f(ab) = f(a+b) as asked. Moreover, it wouldn't show these were *all* such functions.
Even though the question is so trivial, the proof is so beautiful. Its questions like these that gives you the 'aha' moments, and make you love maths.
I think the condition for irrational numbers has the porpoise of avoiding nice numbers like 0 or 1.
If s² > 4p, then there are reals a,b with a b = p and a + b = s (namely, the roots of x² - s x + p).
Let p= -1 and s arbitrary irrational. Then a and b must both be irrational, hence the functional equation gives us f(s)=f(-1).
Now let p = -sqrt(2) and s any rational. Again, a and v must both be irrational, and the functional equation gives us f(s) = f(-sqrt(2)), and by the previous, we know already that f(-sqrt(2)) = f(-1).
Yes!!
Hi, could you please walk me through how did you conclude that f(x) is constant using this argument?
maaaan, i cant belive i was able to do it, thank you!
Thanks for the solution, I'm Brazilian and I'm studying for this contest.
My pleasure!
Orgem de Progresso
You can try this too if you are preparing for Olympiads
ruclips.net/video/igdy05LZj90/видео.html
@མགུལ་སྙིང Thank you!
Boa sorte
thanks for solving theses beautiful problems.
You’re welcome :)
How and with which support do you think I should train to solve this kind of problems ?
Love your channel btw, wondeful work !
Would it not be enough to say that f(a+b+0)=f(a+b)=f((a+b)0)=f(0), and that a+b can add to make any real number, therefore f(X)=k for all real numbers???
since 0 is not irrational, then we can't do f((a+b)+0) = f((a+b)0), since f(a + b) = f(ab) holds for all a and b that are irrationals.
@@giovanicampos4120 yes you are correct. Thanks a lot
Isnt it possible to first ignore the irrationality condition, then find all functions for all reals such that f(ab) = f(a+b). The only such function satisfying this is f(x) = c, a constant. So it also satisfies for any irrational a and b since the irrationals are a subset of the reals. ?
I did something weird but I think it is correct, so:
Sub a = x/k, b = yk where x/k & yk is irational
=> f(xy) = f(x/k + yk) sub k = sqrt(m+1) - sqrt(m) which should satisfy the above condition
=> f(xy) = f(x(sqrt(m+1) + sqrt(m)) y(sqrt(m+1) - sqrt(m))), put x=y
=> f(x^2) = f(x(2sqrt(m+1))) , sub any constant fo x
=> f(c^2) = f(c(2sqrt(m+1))) , and note that c(2sqrt(m+1)) can be any real number including -ve as c can be -ve and (-c)^2 = c^2
=> thus f(x) = k
tho still incredible solution
Beautiful work
I don't get it. Why are you assuming all those functions are equal to f(0)?
Great video!
From you accent I guess you are from Hong Kong or similar places. Did I guess it right? Cuz I’m from Hong Kong and I absolutely love the channel
{rational} + {irrational} = {irrational}.
Can we not let a = 1 and b be any irrational. Then:
f(b) = f(b+1), both sides mapping irrational to irrational and true for all b. Does this prove that f maps to a constant?
The statement only says that f(a+b) = f(ab) when a and b are irrational, so you can't choose a=1.
On top of that, even if you could, f(b) = f(b+1) is not a sufficient condition to conclude that f is constant, f could be any function with period 1. For example, cos(2*pi*x) or frac(x) would be non-constant functions that satisfy your condition.
There was nothing in the problem
that said a and b could ONLY be
irrational. If you showed that all
reals mapped to f(0) then it
follows that all irrational s do.
nice video I'm from brazil
irrational :無理数
Milk pudding
thank you sir ~
So awesome
Cool problem
香港口音好親切
But if it holds for all real numbers, then it holds for all irrational numbers. So this condition is useless.
But you don't know before you solve the problem.
It doesn't say it holds for all real numbers.
There could be a function that works only for irrational numbers but not for rationale, for example the piecewise f(x) = a solution proposed here if x irrational and 1 otherwise
You stupid?
A familiar accent🤔
Chinese + British maybe
@@taopaille-paille4992 To me it's classical Hong Konger speaking English
@@OOO-ef1ps Nope that's not hk accent
He has some British accent that for me means he has spent multiple years in UK
e^a e^b=e^a+b
Assuming you mean e^a e^b = e^(a+b) , this is an example of f(a)f(b) = f(a+b), but not f(ab) = f(a+b) as asked. Moreover, it wouldn't show these were *all* such functions.
What a terrible calligraphy
Cool problem