International Math Olympiad | 2024 Math Contest

You should learn this trick.

A great exponential equation! What do you think about this problem?
If you're reading this ❤️

Hello My Friend ! Have a Great Day:)
@higher_mathematics

#maths #algebra

[skaeggo]

If you assume your audience knows what Lamberts W function is without any further explanation, I would think you can safely assume you don't have to spell out the initial divisions by x and 2 in painstaking detail.

[JacquesLafont]

At the beginning is the equality ln(x^2)=2ln(x) which is correct only when x>0 and at the end this calculation gives x=--0, 559..That does’nt seem rigorous, even if the solution is right.

[DedenK]

WRONG!
You have to do 3 cases :
1) If x>0 : ln(x²) = 2*ln(x) and x = exp(ln(x)). Then, you have an equation without any solution since -ln(8)/2 < -1/e, so impossible to apply W on both sides.
2) If x<0 : ln(x²) = 2*ln(-x) and x = -exp(ln(-x)). Then, you have a unique correct solution : x = -exp(-W(ln(8)/2)) ≈ -0.559.
3) If x=0 : 0=0²≠8^0=1, so not a solution.

[MathematicFanatic]

Why is W more valid than J? J is a function I just made up which is the inverse of the function x^(-2)*8^x. So the answer is J(1). Easy

[vaibhawc]

There is a big mistake in very first step. it should 2ln(|x|) not 2ln(x). your negative fraction result won't hold otherwise.

[szjozsi]

W is an approximation so you could might as well just iterate the solution. it like solving x^x = 12

[runnow]

The lambert W function is so disappointing :/ I can't explain it but I always feel let down when it's that function

[nottraian]

Honestly I would have just solve it graphically from the beginning. Then I would have tried to insert some values in the beginning equation to see what’s the interval in which X lays. For example I would have tried values like 0, 1, -1, -2 and so on. I would have gotten a interval and not an exact value, of course. But at least it would have been much easier and faster

[tmjcbs]

I thought math olympiads only involved high school math and not university math. Here in the Netherlands, but I suspect elsewhere too, the Lambert function is definitely not a high school subject. So I wonder: was this really an Olympiad math question? The title suggests that it is, but I find that hard to believe.
If it is not a math olympiad question than you should clearly state this: a lot of viewers may waste their time trying to solve this algebraically while that is impossible without knowing the Lambert function.
And as somebody else already commented: for people who understand the Lambert function the rewriting in the first two minutes is trivial and could be done much faster.
In conclusion: a dubious problem: unsolvable if you don't know the Lambert function and quite straightforward if you do.

[adg]

Love the explanations thanks for your video ❤ just one feedback it’s not said “NAYtural” - Although the word Nature is distinctly so, the Na in ‘NAtural’ is pronounced same way as NASA, or Na X.. in russian 😇

[skateordiee]

How about…

1) When X represents a number being multiplied by an exponent, then X = 8 (so X^2 = 8^2 = 64)

2) When X represents a number’s exponent, then X = 2 ( so 8^X = 8^2 = 64)

Maybe I’m not answering the question, maybe I am… but prove me wrong if I’m not lol

[MichaelJamesActually]

I wonder if Lambert knew he'd become the most trendy math trick on YouTube.

[sidharthghoshal]

this is like a textbook trivial application of lambert-w. Why did you even make this video if THAT was the thing you wanted to show? Was the purpose of this to just raise awareness of the lambert-W function? What on earth does this have to do with the IMO?

[tasoskotaras]

Lamberts function is such jenious and all. I always wondered though, if there are real life problems that this function is supposed to solve (something more than Olympiad problems, I mean). Is there an engineering - per instance - problem that deals with such exponential functions?

[suntoli]

From graph plot x should be around -0.6. I have PhD in physics but never heard about Lambert function.

[jjs]

I think the creator of this video missed the point of this problem: you have to show that there is no closed-form solution for problems like this. Actually, you can only write the solution with Lambert's W function or in another way using functions that can't be expressed with a closed-form expression. This is the important thing you have to learn about problems, where a variable is in the basis and exponent at the same time.

[mrsoikawa]

Awesome. I'll show this to my Year 2 class tomorrow

[Governemntistheproblem]

Lambda w function? Never heard of this

[VanNguyen-kxgx]

It’s higher maths in university. Not for year 12.

[pinkusbotzo]

I have read that the Lambert function can be solved only if x is greater than -1/e, which is ≈ -0, 367879. But your solution x ≈ -0.559 is lower.