Math Olympiad Exam | Lambert W Function 

We can just notice that 3power(3)+3=30. Since the function x------3power(x)+x is increasing. Then 3 is the unique solution.

I hope my car navigation NEVER gives me the directions like this

Only mathematicians appreciate this elegant solution! Nice introduction to Lambert function.

This is my approach: Plot a graph using y=3^x and y=30-x, the solution can be found where the curves meet. Simple and easy.

Everyone is acting like what he did in this video uses unnecessary steps, but the goal of this video is not just to teach you how to solve 3^x+x=30, it is to teach you how to use the Lambert function to solve for x in similar situations.

First glance you all can notice X= 3, and its done. Fact 😂

X=3 Bc what's the number that power the number 3 and the addition number It's coming by 3³ and this equal to 27 but 27+3 = 30 So 3x +x =30 is 3³ + 3 =30 So x=3

Moving x to the right side, we get 3^x = -x+30. The exponential function is increasing while the linear function is decreasing and there is only one intersection. It's obvious that x=3 is the solution.

Am i the only one who can calculate the answer 3 without even thinking?

Well...As a high school student in Asia,I know that x=3 immediately😂😂

In Russia I haven't even been told about the Lambert W Function. Instead I've been taught multiple times about Gauss functions, and many other fundamentals of mathematical framework in many aspects (calculus, vector calculus, complex analysis, tensor analysis)... but W-function is what I've never heard of before...

No need completely to introduce Lambert's function. If (30-x) · 3^(30-x) = 3^30 from a simple algebraical transformation, as shown on the screen, so (30-x) · 3^(30-x) = 3^3 · 3^27, so (30-x) · 3^(30-x) = 27 · 3^27, so (30-x) = 27, so x = 3, end. The function is very interesting and useful (see even Wikipedia for examples), but not here.

а че типа сразу не очевиден результат был? Тут надо лишь доказывать, что других корней нет.

3 to the power of 3 = 27 + 3. Pretty easy X = 3. Dont need a big ass page explaining it all re writing it multiple times. Just work through to the powers 3x0, 3x3, 3x3x3 = Remainder left 3.

Or you just eyeball it and instantly see that the (an?) answer is three. The only tricky part is testing whether there are additional solutions.

I went to the rest room to pee, before I could pee, I cleaned up all the house. accidently peed in my pants.

More unnecessary steps.

Someone please explain what the "w" is all about. What is "w" and why did he multiple both sides by it?

3^x + x = 3^3 + 3 x=3 30 const ; 3^x + x is increasing func -> hence 1 intersection and 1 solution selection method 3 root.

3が解になって左辺は明らかに単調増加関数だから3しか解になり得ないっていうのが一番楽

3 to the power x +3=30 3 to the power x +3=27+3 3 to the power x= 27+3-3 3 to the power x = 27 3 to the power x = 3 to the power 3 Then X= 3......

3^x+x=3^3+3 so, x=3😑🙁 Is this right?

I just saw the answer as soon as I saw the problem and I don't understand the need for all the faff.

Very easy 3³+3=30, some problems don't need to work

Is x= 1? No Is x = 2? No Is x=3? YES. Solved in 15 seconds. It took me longer to write this then solve the equation. Then I sat through 10 minutes of this plus ads

Pl explain the letter w You wrote. I did not understand.

Зачем вообще все это? Ответ сразу виден. 3

7:00 this step, knowing to break 3^30 into 3^27 * 3^3 is equivalent to solving the equation in the first place 😂 to solve equation using this method you need to know the solution first 😂

3^3+3=3^×+× X=3. Simple

Me solving it just by looking at it 🤣🤣🤣

This is a really interesting way to solve that :) thank you for the video

Because of the sum (30), x should be either 2 or 3. As 3^2+2=11 and 3^3+3=30, then by inspection x = 3.. It won't take more than 1' to get the solution. You can get there easily just by trial..

what it just 3*3*3 + 3 = 30???????!!! why bother so many steps

3^×+×=30 x

sir we an get quick answer by putting value of x from 1,2,3 ...

و فى الآخر تروح تلاقى واحد بتكون تعرفه و تلاقيه مستنيك هناك و تكتشف إنك افترضت افتراضات قد علمها لك هذا الشخص و أنكما قد قمتما بحساب المقادير بناءا على قواعد منطقية مغلوطة و أنه قد غاب عنكما الكثير من هذا العلم المنطقى مع احتمال أن يكون هذا العلم و هو علم الحساب علم موضوع لكما مسبقا قد تخيلتما أنه علم صحيح و لكن صحته تستنبط من منطق لا تعتمد صحته إلا على قواعد هذا العالم المكون من إبداع الخالق عز و جل سبحانه و هو بديع السماوات و الأرض سبحانه ... قال تعالى " بديع السماوات و الأرض " ... و قال أيضا " و لتعلموا عدد السنين و الحساب " . و لم أجد خلقا أكثر خشية لله و تمجيدا له - و بالحق - من الملائكة . قال الله عز و جل سبحانه على لسان الملائكة " قالوا سبحانك لا علم لنا إلا ما علمتنا إنك أنت العليم الحكيم " . فتبا للإنس و الجن ما أكفرهم و ما أجهلهم إلا ما رحم ربى و هم قليل . اللهم اجعلنا لك ذكارين لك شكارين لك رهابين إليك منيبين لك خاشعين بك موحدين . و أشهد أن لا إله إلا الله وحده لا شريك له و أشهد أن محمدا عبد الله و رسوله . و أنه قد بلغ الرسالة و أدى الأمانة و نصح الأمة و كشف الله به الغمة . اللهم صل على محمد و على آل محمد كما صليت على إبراهيم و على آل إبراهيم و بارك على محمد و على آل محمد كما باركت على إبراهيم و على آل إبراهيم إنك حميد مجيد . اللهم أحينا مسلمين و توفنا مسلمين و ألحقنا بالصالحين . اللهم آمين . و الحمد لله رب العالمين .

By inspection x=3 is one solution and it is the only one because 3^x in increasing and 30-x is decreasing. No need to over elaborate!!!

What everyone is failing to understand is the video is not supposed to be about solvimg the equation easily, but rather be able to solve an exponential-linear function with the help of W Lambert Function. The function is useful for a lot of other problems. Can you not understand that he tried to start off with an easier problem to explain the practical use of the function.

이걸푸는데 10분이다 걸린다는것에 참 대단하다고 생각한다 ... 그냥 보면 답나오는 문제아니냐;;

After all that, you shortcut the ending lol... too funny. 30-x=27 -x= 27-30 -x= -3 -x/-1= -3/-1 X=3

Giải phương trình :3^x+x=30 +) Tập xác định:D=R +) Phương trình trên tương đương với phương trình sau: 3^x+x-30=0 Xét hàm số f(x) =3^x+x-30, x∈R =>f'(x) =(3^x)ln3+1>0∀x =>f(x) đồng biến trên R =>phương trình f(x) =0 có tối đa 1 nghiệm trên tập số thực +) Mà f(3) =0 =>X=3 là 1 nghiệm duy nhất của phương trình f(x) =0 Kết luận: tập nghiệm của phương trình là S={3}

Easier to do it with mental math, like I did very quickly, but I suppose that wouldn’t work in school lol.

That's a long ass solution, though my dumbass would just write 3^X+X=30 3^X+X=27+3 3^X+X=3^3+3 Therefore X=3

only reason it looks hard is because “Show your work.” exists

I solved it in 5 sec

Your solution is unfortunately circular. When you split 30 into 27 and 3 in the last steps, to obtain 30 = 27 + 3, you use the solution 3^3 + 3 = 30 in its derivation. You can see that by doing all the derivations on the general equation a^x + x = b. The general solution to equation a^x + x = b is x = b - W(a^b * ln(a))/ln(a) [1]. If c is s.t. a^c + c = b (i.e. c is a solution of the equation), then a^c = b-c [2] Looking at the factor a^b of the argument of W(), we see that a^b = a^c*a^(b-c) -- This is what you do when you split 3^30 into 3^3*3^27 i.e., using [2], a^b = (b-c)*a^(b-c) which in turn equals (b-c)*e^(ln(a)*(b-c)) -- In your particular case, 27*e^(ln(3)*27) Plugging inside W(), we get W(ln(a)*(b-c)*e^(ln(a)*(b-c))) = ln(a)*(b-c) With the latter result [1] becomes x = b - ln(a)*(b-c)/ln(a) = b - b + c = c. The solution we find is the one we have used in the calculation: circular derivation!

3 değeri verince eşit oluyor. 3 saniyede çözdüm.

Wow, this video on Math Olympiad Exam | Lambert W Function is so informative! I appreciate the detailed breakdown and the insights you've shared here.

Btw x is congruent to zero mod three, so you can rewrite it as 27^x+3x=30, and immediately we see using the canonical homomorphism between Z[3] and Z/3Z, that the new x must be one, and the original must have been 3.

Guess x=3 as solution. f(x)=3^x+x is strictly monoton increasing, so x=3 is the only real solution. That saves you about 9 1/2 minutes.

Very interesting way to solve the problem. Thanks for sharing. I just tried some numbers and got the answer very quickly.

You can graph the function f(x)= exp(x ln3) and g(x) =x-30 and see where f(x)=g(x)

I’m sure the video offers an interesting solution, but it took me 10-15 seconds to think of the answer

Wow, what is W function? First time seeing one. (It's been 30 years since highschool graduate, and never heard of it even in college freshman math class. Is W a basic term these days?)

Its like proving 1+1

The Lambert W function.

I have not found the part 3^30 ln(3)= 3^3 ln(3) 3^27, you had to think about it, well done

I gave up on contest math years ago being not nearly brilliant enough for it, just got recommended this and it absolutely blew my mind❤

The answer is three. These small examples are great for learning lambert W, but it’s an unnecessary bit of math baggage when the trial substitution is the best approach.

Que gran tutoriual tan excelenye de examen de olimpiadas

I realized 3 worked as a solution but Idk the math involved in solving the problem. 😢

Sudah jelas jawabnnya 3 ...mengapa harus di perpanjang..harusnya cari contoh lain yang lebih rumit

To guess is not a mathematik solution. If you use a instead 30 you will not find an answer. Also the trick in the video will not work.

Your solution make we lost the way. Cannot navigate wit you confusing methods.

There is a simple solution that giving x an integer from 1 to 3. You can find the answer in three seconds. But if you must prove it, this equation may be solved by your way.

W what is that?

My logic: X has to be a small number so whats yhe biggest x can be before 3 to the power of x is greater than 30. 3 ok lets try three. First try

3^x+x=30 30-x=3^x (30-x)/3^x=1 (30-x)3^(30-x)=3^30 (30-x)ln(3)3^(30-x)=3^(30)ln(3)=3^3×3^(27)×ln(3)=27×3^(27)×ln(3)=3^(27)ln(3^(27)) (30-x)ln(3)=27ln(3) 30-x=27 x=3

Why do you write the X, that X seems bad x, or fi, or z....it is very annoying... :(

please, show me more long-winded ways to solve that

yo viendo la miniatura dandome cuenta que la respuesta era 3 con el ojimetro (ni pienso ver el video asi que si hay mas de una respuesta o en numeros complejos (cosa que dudo pq no es con incognica con exponente) perdon)

解题步骤如下：因为3power(x)+x=30，所以，3power(x)

Or maybe just guess 3 since it is obvious. And no other solutions can exist because 3^x + x is monotonous

3^x = 30 - x Since 30 - x must be > 0 -> x < 30 x = 3 verifies the equation. 3^x increases while 30 - x decreases, so 3 is unique solution.

The answer is 3. I looked at the problem immediately and found the answer with no math at all.

Found the solution literally in 2 seconds, but can't explain properly

x = 3. 3^3+3 = 27+3 =30

Holy hell, never thought Lambert W function would be THAT useful...

Смешно....методом подбора за 15 секунд и так ясно что 3...😂

It is simpler to solve the problem using modular algebra. Take modulo 3: mod(3^x,3)+mod(x,3)=mod(30,3) [mod(3,3)]^x+mod(x,3)=0 0+mod(x,3)=0 --> mod(x,3)=0 It means that x=3k where k is any integer. Plugging it back to the given equation: 3^(3k)+3k=30 Divide by 3: 3^(3k-1)+k=10 =3²+1 Comparing both sides we get k=1 Thus x=3. As check: LHS=3^x+x=3³+3 =27+3 =30 equals to RHS

What a dumd solution...it could have been solved in 3 steps...he made a mess of it.

Невозможно смотреть такие глупые решения

x = 3. Because If x > 3 => 3^x + x > 3^3 + 3 If x < 3 => 3^x + x < 3^3 + 3 Simple answer.

It's vvvvsimple. It's = 30. So u put number x 1, 2 , 3. Then u find when x=3. Both sides equal. It takes not even 1 min

i just can not understand from where u got 3 and 27 ? why is not 1 and 29 or 2 and 28 etc

I look3d 2 Seconds and Said 3 ist the answer but dont expect an explanation

I mean the answer is obvious if you think about it hard enough, still that was good

I solved it in 3 seconds yet I can't pass the most basic highschool exam

Define f:R -> R by f(x) = 3^x + x - 30. Note that f is differentiable (and thus, continuous) on (-inf, inf) and f(3) = 0. Note that f '(x) = 3^x ln 3 + 1 > 0 on (-inf, inf) since 3^x > 0 on (-inf, inf). Suppose by contradiction there exists a in R such that f(a) = 0 and a != 3. If a < 3, then by the Mean Value Theorem there exists c in (a, 3) such that f '(c) = [f(3) - f(a)] / [3 - a] = 0, but this is a contradiction since f '(x) > 0 on (-inf, inf). We can use the MVT to show that a > 3 also leads to a contradiction. Therefore, f has one root at x = 3.

f(x)=3^x g(x)=-x+30 f(x) monotonously increasing, g(x) it decreases monotonously, but knows if there is only one solution. In this case, it is obvious x=3

X=3 , I needed 8 seconds for the answer.

Very simply one can assume the value of x as 3 by using common sense.

I have no idea what's going on since I'm still on junior high school but very cool

Задачу проще решить графический за 3 минуты

Who else just looked at that and intuition immediately knew that x=3

I knew it was 3 right off the bat since 3 to the power of 3 is 27 amd + 3 is 30

I said 3 in 2seconds. I think it was more simpler than u :)

Mesmo substituindo o 3 por X, mentalmente a resposta é X= 3.

Well , we can just do this 30 is not a big number So we can try 1 or 2 or 3 to get the answere

It is very simple just assume X=3 and work out.