a functional equation

We look at a functional equation problem that was shortlisted for the 1995 International Mathematics Olympiad.

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[DavidSavinainen]

I love that it has the sum goes up to 19, the RHS has 95, and the sum is 1995. Brilliant!

[andreben]

That subtle use of the associativity of the addition on g is AWESOME!!! Such a nice trick!

[simoncopar]

It is obvious from 4:16 that g is an involution (exchanges n and m, so applying it twice gives back the same thing). This shortens the last part, as involution immediately leads to A=1.

[jamesfortune]

Michael should write a book with the subjects/problems from all his YouTube videos, perhaps categorized by college course and/or subject matter (e.g., Japanese temple problems). I'd buy it.

[jonathanbeeson]

At 13:45, didn't you forget to transcribe a "+95" in evaluating the left hand side ? I really enjoy your work Professor Penn, btw.

[dicksonchang]

13:38 you actually didn't write the +95 behind A(m+95)

[udic]

13:41 you forgot to add 95 to rhs.
But then you ignores the 95 on lhs.
Two wrongs make a right...

[synaestheziac]

“Peach parentheses” — love it. I wonder if that phrase has ever been uttered before

[rektator]

Small comment about induction:
Induction principle: Let S:N->N be the follower function of natural numbers. The induction axiom says that if A is a subset of N where
0 is in A and
the image of A under S is included in A,
then A = N.

In other words the induction hypothesis should be of the form that let any k be in A. The induction statement is then that S(k) is in A.

The point is that to assume that some k exists in A and S(k)\in A is not formally correct way to do induction. This property holds for the subset {0, 1} of natural numbers.

[dominiquelarchey-wendling]

The base case for the additive equation is not n = 1 but n = 0 instead. So proving g(0) = 0 which comes from g(0+0) = g(0)+g(0).

[Fun_maths]

Who knew that the solution of a problem would be the year of the problem

[shivansh]

Problem:

|2^n + 5^n - 65| is a perfect square.
Find all such positive Integers 'n' .

Here |x| is a absolute value of 'x'

Hey Michael, plz make a vedio on the solution of this problem 🙂

[debjitmullick]

Sir ... What are some good books for Functional Equations

[mcwulf]

A very cool answer. Who discovered that?

I wonder if there is a general proof that all f(X + f(Y)) type problems will be linear? I feel like jumping to f(X) = aX + b whenever I see one.

[envjvru]

Can anyone find a functional equation (the domain and codomain dont matter) that is satisfied by a linear function (ax+b) AND a nonlinear function?

Myself i haven’t seen any, so i think you could conjecture that if a linear function satisfies a functional equation, then only linear function(s) satisfy the functional equation. Then, memorize the proof so you can use it on olympiad problems and such whenever a linear function turns out to be a solution by inspection. That would help because on this problem it wasn’t hard to find that x+95 worked, but the proof that it was unique was tough.

[cHrtzbrg]

At 13:29 you forgot a +95 at the end (but in the next step you dropped it on the left hand side, so these errors cancelled out ...)

[luccavelier]

At 13:31 there is a mistake. On the right hand side, +95 is missing!!

[mathunt]

If you take his hint and look for a linear function which satisfies that functional equation you get the function very quickly.

[MrGyulaBacsi]

I'm not sure whether anybody else has mentioned this here before: I think Michael only proved that the linear function is a solution to that additive functional equation. He did not prove that this is the only possible solution. Or am I missing something?

[bigern]

Shouldn't there be a +95 at 13:44?