International Math Olympiad | 2024 Math Contest
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- Опубликовано: 11 окт 2024
- 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
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.
Character development, we are growing with the video!😅
😂
Totally agree. There's this broad habit in lectures where the first half they treat you like an idiot and the second half like a genius.
Totally unbalanced
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.
You are absolutely right . Logarithms only result in real values for positive values . If you graph the two functions , you will see that there is only one solution and it is negative . I actually used Newton's method which is a numerical method to come up with a value to 12 decimal places ---> x = minus .559142092566 .
In the original formula, x^2 is positive for all x, so no limit on x. The fix would be to call it 2*ln (abs(x)) , x can still be negative.
I agree. Mathematica calculates a complex result, which is not a solution. Lambert W function yields complex results for arguments < 1/e. This result has W(-1.039), an argument less than -0.367. As others say, starting the solution by taking logarithms is to head down the road to complex numbers
What if x is 0
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
You are dead right. You clarified this and made it much more interesting. This is what I was unable to achieve. Thank you, I learned a lot
Good job clarifying this!
Excellent work in clarifying! A little side note If I understand the W function correctly: I think in case 1 you can get infinite many complex solutions, since the
What is W() 😅???
@@claudpiro6469W the function... WTF?!
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
I didn't know this function too. So after proving there is only one solution with the functions graphics I'd rather started approximating the answer with bisection and gave an answer ~-0.5 ))
There is a big mistake in very first step. it should 2ln(|x|) not 2ln(x). your negative fraction result won't hold otherwise.
Bingo
The lambert W function is so disappointing :/ I can't explain it but I always feel let down when it's that function
W is an approximation so you could might as well just iterate the solution. it like solving x^x = 12
I wonder if Lambert knew he'd become the most trendy math trick on RUclips.
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
You’re describing numerical methods, which are definitely the best way to solve this kind of thing. For example, bisection starts with an interval around the root and cuts the size of the interval in half at each iteration. I would recommend looking up bisection, Newton’s method, and fixed point iteration if you want to know more, and definitely consider taking numerical analysis in college if you get a chance.
@@louiscarl7629 nah thx I’d like to do chemistry in university. I know a bit of maths because my high school is based on science, and maths is always taught.
It’s higher maths in university. Not for year 12.
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.
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 😇
Please continue, how it sounds in Russian?😊
@@nickolson000 honestly, XZ how it sounds in Russian 😅🤷♂️
Lambda w function? Never heard of this
Lambert W function
What is lambda function? Pl define that clearly for me to understand. Thanks.
Lambert W function: "Then a miracle occurs."
- Pikachu!
- Lambert W function!
...
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.
Awesome. I'll show this to my Year 2 class tomorrow
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.
You aren't taking w(-.5) your taking w(-ln8/2), wait that's about -1 so not sure
From graph plot x should be around -0.6. I have PhD in physics but never heard about Lambert function.
Same PhD in theoretical physics. But could not solve it, without looking up solution.
Using Numerical Analysis
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?
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?
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
or you could use the constant of 1²= 9.87. so x = .5559. half of 1 if you include time
I don't think you've answered the question. If you can just denote any zero of the function ln(x)/x - a by Xa and then tell the solution as a function of Xa, then you have not really answered the problem !
You are absolutely right. Giving your unknown solution another name is not a solution.
by drawing the two original functions it was clear that x was smaller than zero. I would have started from there.
You can't use the rule ln(x^2)=2 ln(x) for x < 0.
The question is awesome.
Because (-ln8/2) is less than (-1/e), W(-ln8/2) is a complex number = -0.291 + 1.36 i. Can you show how you solved for x? e^(real number) cannot be negative.
En refaisant le calcul avec - x = 8^(x/2) on obtient x= - exp(- W(log(8)/2)) = -0.559 142 ...
Have a good day.
@@ferdinandrius6063 Thanks. I finally realized that I had to consider both x>0 and x
Or just use Newton's Method to approximate the root since you're gonna approximate it in the final answer anyway. Great video though.
The solution just before -0. 559 is not an approximation, but it is written in a well-defined function, i.e. the Lambert function.
Why do you say this is olympiad question??
Are you low-key bragging?
@@Lokie-cd2hwno it’s just not an Olympiad question nor is it hard enough to be one
Good point!
Thanks sir
lnx on a negative x?
x isnt negative. the power is negative
Ln = loge X , sendo Ln X = a ; E e^a= x
This is the pinnacle of a smart person thinking they are explaining something simply. Try again please, a little bit slower.
Using the Lamberts W is an abbreviation, but not a true solution...
You didn't discuss the conditions where x could be equal to or greater or lesser than zero.Sad.
La fonction W n'est pas sur les calculatrice !!! C'est comme si tu nous disais x=f(8,2)
Why is he making it so complex by assuming so many things without explaining why he's assuming them
I do not know what Lambert's function is I have a BA in English and very Little understanding of algebra why I left math at algebra to pursue English Up until algebra math seemed to solve real life problems algebra is just type of way it seems of solving a jigsaw puzzle and left the real world behind--What is this video about I suppose to understand a problem rewrite it so that you can eliminate one piece at a time until you have left something you can recognize. If the audience of the video is for a beginner I do not believe you have helped I tried but I am lost as to why you put in a lambert when and how and why
على ماذا تدل W وكيف علمت قيمتها
دي دالة خاصة اسمها lambert w تقدر تحسب قيمتها بآلة
ده قانونها:
W(🦔e^🦔)=🦔
الw بتاع أي حاجة مضروبة في e أس نفسها تساوي الحجة دي
ال🦔 هنا ممكن يبقى أي رمز
My biological science ass would just graph and find where they intersected. I like math and I took all the way to calculus 3 at the college level but I honestly never have to use it anything except algebra in my job.
To use the W function is not solving
I agree
The universe opened up when this guy took out his red pen
Why in the hell would anyone write an X like that? That is the most annoying thing I've ever seen.
So it is distinctive from a multiplication symbol. We were taught to write x that way.
Is W(z) transcendental?
The Lambert W function is a transcendental function if that’s what you’re asking
🙏🌹😍🙃😃 NEXT ?!
I'm not a fan of this method. Seems complicated/busy. But thank you for the lesson anyway.
One has infintely many complex solutions , using the W-function and its complex extensions .
x = 8 and 2 = x. ez
Who is Double You?
Bro’s l’s and e’s look exactly same lol (but fr it made it more confusing)
What if x is 0
❤ល្អណាស់🇰🇭🇰🇭🇰🇭🇰🇭🙏🙏🙏good
Those x’s really irk me
Nice, clever and all that... but how can anyone write x like that 😭
Это не красивое уравнение. И не красивый ответ
Просто посмотрев на него становится понятно, что х отрицательная величина. И она приблизительно -1/2 . Решил?
правда, он насрал на нас
That 'ln' kinda looks like 'e sub n' just saying.
At a glance it would appear confusing to the uninitiated Math student....but nice methodology all the same!
-0.567
Uhh I solved it in 1 min tho? Am i wrong? 8×8×8×8=64×64
Simple's the best- by a 11yo
dawg, what you wrote is 64^2=8^4 when the question is x^2=8^x. tragically it doesnt work because 64!=4. im p sure lambert W function is the only way to answer this.
x=0
Ne comprenant pas l'anglais, je vais devoir repasser cette vidéo en plusieurs fois... ça va être comme à la télé : des rediffusions.
0
I understood nothing.
meaningless...
ofcourse you are dumb if you think that you cannot raise a number to a power of -0.559.
Maths is beyond your imagination
Cosa sarebbe w ()???