Solving An Insanely Hard Problem For High School Students

Hey Presh Talwalkar, very nice solution! I find your video extremely clear and well presented, I should better learn some presentation skill from you.
BTW, thank you for mentioning my channel, I really appreciate it. Cheers!

I just solved for a=0 and got f(x)=2x then solved for b=0 and glt f(x)=2x and tried f(x)=2x for the entire thing and it worked.

Hey Presh, thanks for the video :) I have one question - How does one prove that those 2 solutions are the only ones? We did infer them from only 2 special cases for a = 0 and a = 1. How do we know there are no other solutions for f?

Since we tried out a bunch of special values of a and b, how can we guarantee that an AP is the only type of equations that satisfies the given equality?

I keep getting a third solution of f(x)=x-a-b. f(a+b)=0 so f(f(a+b))=-a-b and f(2a)=a-b with 2f(b)=-2a.

all of these are easy

This specific question would be considered pretty easy by IMO cotestants, and like others have mentioned this type of question appears a lot in olympiad settings and is called a functional equation. The vast majorty of these questions are pure algebra and test your algebraic ability

Multiplication by two is one answer. I did it in 2 mins. I believe besides f equal 0, f(x)=2x is unique solution. Indeed if f and g are two solutions, then f(f(2x)) = g(g(2x)) for all x, and if they weren’t equal that would be absurd so we’re done via proof by humiliation.

Me watching this video realizing I still can't think what 7*3 is on top of my head

Rocket science socks

I managed to guess the answers by assuming the function must be a linear equation but couldn’t show that those were the only ones.

A pretty easy problem tbh, gonna consider going to IMO for real

I managed to pull together that 2x was a solution fairly quick but proving anything else I was at a loss

I feel so proud of myself for having been able to Solve this, even if not this efficiently.

as someone who’s been doing a lot of linear algebra lately, i immediately picked up on the fact that this equation very closely resembled the definition of linearity, (i.e. f(a) + f(b) = f(a+b) and f(ca) = cf(a)). then assuming f was linear, it was very easy to solve from there. i was just going off my intuition though, so your solution was probably much more rigorous and would certainly stand up better under scrutiny!

I cant believe there was q point in my life i used to understand this sh*t. WHYYYYY WHAT WAS THE REASON.

What is horrible with maths is when you dont understand the question, someone explains you slowly...and you dont understant the explanation...

Ahh, my engineer mind finds only 2x in a very differenti way.

i can do it in 5min

As a JEE aspirant , I solved this problem in 20 minutes in 1st try

Honestly this is extremely easy for students who has experience in Olympiad math.

Thanks

The language of the universe

You know it's a challenging exam when the total mark is not a multiple of 10 or 5

Idk if it’s because I’m watching this at 1 am, but I found this was weirdly easy to understand. Usually I have trouble with math, since I never paid attention in school, or did any homework from grade 3 to grade 9. Well done on the video

My thinking ability stops when it gets way too complicated like in here... so my mind itself simply substituted f=2, a=3, b=4 and got 28=28... ha ha ha... 🤣🤣🤣🤣

I only plugged 0 my bad 💀

I tried to watch this video a couple of years ago, I didn't understand it at all. That goes for your other videos as well. But today I did get it, And I've been watching some others.

Oh joy of understanding, greater than that of imagining or feeling!

I don't know what to do anymore, I have problems way easier than this and I have no idea how to problem solve. I only ever understand a question when I see the solution and how they did it, but I can just never figure out a new question itself.

That's why i dont study math

If you look at the equation carefully then you would be able to get the solution (mx+c) without even doing a lot these in-between steps.

I agree that this problem is not that hard. But i did need knowledge i learned in University. Just using high high school maths it seems impossible to me.

Mathematicians on their way to solve the most useless problem on Earth:

Didn't take me 1.5 hours. But couldn't do it in 5 minutes either. Took like half an hour or longer maybe. I liked the problem overall.
I did- a= 0, b= x. Then b=0, a=x. Then combined these. Now do a=b=x. Now in this one, substitute value of 2 f(x) from previous equation. Finally, let f(x) be the variable y. You get the answer.

Im a little late to the party, but i wanted to share my way of solving this that was a bit more intuition than algorithm.

If you assume that f is both invertable and distributed, then the equation goes from f(2a) + 2f(b) = f(f(a+b)) to 2a + 2b = f(a+b). With some simple subututions from there you get f(x) = 2x. The only thing my first shot was missing is that when you are inverting in that first step, each functiob could have a term that cancells out.

The "therefore" is not obvious at all. f(x)=0 is rather obvious, but doesn't give much in terms of full answer. The "progression" part is something that fell off my memory since school.

Problems like this are great because they look impossible before you learn what's going on and trivial after you know the trick

"Students had 1.5 hours to solve this problem. Lets solve it in 5 minutes."

"Edison spent years to invent electrical bulb. I have spent just ₹ 100 to buy it."

I thought I had forgotten how to solve that kind of problem. Your explanation is very clear, now I remember how. Thanks a lot!😊

Actually... I would not consider this very hard, even for high school students. Currently, I'm prepping for JEE advanced as a 16 year old, and I believe I could've solved this had I tried, since I've worked on other problems equally hard if not harder on functional equations.

This is coming from someone who considers math his weak suite (physics being my strongest). I know at least 10-15 people in my grade could solve this first try under 20 or 30 minutes (some maybe under 5 minutes).

Not to discredit IMO or boast, I'm sure the other problems are much, much, much harder and way above the capacity of me and my peers.

Math problem soving is uses easy concepts everyone understands, and arranged into a hard logic brain-teaser, but in math.

every year I've qualified for the County Math Olympics and my classmates fail to underetand what I'm doing.
I once took a course for the IMO and understood absolutely nothing from it.

That just goes to show the magnitude of the IMO and this problem is easy comparing to others, even I could do it (took me a day but i still did it)

I've always had a knack for functional equations because they are more approachable and easy to manipulate

wow , that was cool

Wat

As far as olympiad problems go, this one is quite easy. Any contestant at the IMO could definitely solve this problem in less than 1.5 hours (as problems 2 and 3 are much harder).