Math Olympiad 3^m-2^m=65 😊| Math Olympiad Problems | Algebra
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Решается в уме, за время меньшее чем взять карандаш и бумагу.
That is not a generic solution but a guess-fitting situation. Not correct mathematics
81 - 16 = 65. m=4. This can be solved orally in 30 secs.
También lo hice así.
Math it's not oral. It requires proof
Trial and error method is applicable.@@paulkersey2424
Factors of 65 can also be 65 and 1.
... and does that give you an answer that solves the problem? (In a way that a Math Olympiad student would be able to solve)?
@@professorbsamazinglaboratory If the student used factors of 1 and 65, it will lead to an equation where 3 to the power of m = 1089. m = ln 1089/ln 3. As a teacher, I am not going to mark the student wrong.
This does not prove there are no other solutions. It assumes that 3^(m/2) - 2^(m/2) and the sum are both integers. This narrows the search space for the roots, so, in theory, some roots may not even be considered. This cannot be called a solution.
@chinchang5117 You wouldn't mark the student wrong if they came up with an incorrect solution? Why? Genuinely curious.
@@wrc1210 Because m = ln 1089/ln 3 is a correct solution!!
In fact an infinitely amount of Real numbers can give a product of 65. So the conclusion to calculate with the prime numbers 5 und 13 ist only an arbitrary choice.
True.
Exactly. Interestingly, though, if you choose some other factors of 65 other than 13 and 5, and then calculate m for x and y separately, you'll come up w 2 different values of m. This is obviously not a solution to the problem since we are looking for a single value for m. But if you plug those two different values back into the original equation, you do end up w 65. For example, if you choose 65 and 1, and then calculate m for x, you get m=2×(ln 33/ln 3)=6.3653... Calculating m for y you get m=2×(ln 32/ln 2)=10 Plugging those back into the equation you do get 3^6.3653 - 2^10 = 65 Not really profound or anything since it just shows there are an infinite number of solutions to 3^p - 2^q = 65 But I think it does kind help illustrate the arbitrariness of picking 13 and 5, which just happen to be the one combination out of infinitely many that does give the same m for both x and y.
It took 10 minutes to solve... if I had tried trial and error, it would have taken 1 minute. 😂😂😂
10 seconds
Why you take only integer factors?
great to learn new problem
All effort to teach is much appreciated
This is not a solution, just an assumption. Prob.: 65=1*65 65=a*b is only true if a and b are not 1 and 65 and are elements of whole numbers. It's not a solution anyway. This is just a guess.
A much easier approach would be to just think of powers of numbers. Powers of 2: 1, 2, 4, 8,16, 32, etc. Powers of 3: 1, 3, 9, 27, 81, etc. 3 ^4 = 81, 2 ^4 =16 81-16 = 65 therefore m =4.
Brute force is hardly elegant though.
@@CrYou575 It is not a brute force, it is a logical approach to the problem.
Wow what is easy way
It is good way to explain
Intuitively 4
Сравнить степени 2 и 3 до 4-й, это 16 и 81. 81- 16 = 65.
Thanks,, very good
Very well explained
First You must prove that m is even. otherwise 3^(m/2) or 2^(m/2) may not be integer. (Proof) applying mod3 , 0 - (-1)^m = -1, which means m is even. Second Since x + y > 0, x - y > 0 and x + y > x - y You must try 65 x 1 case. (actually there is no solution)
Neat! Thx a lot.
You can start with a number that makes 3^m > 65. First guess is m=4 and just satisfies the equation
65=65х1 too 😀
4 быстрее находится прямым подбором степеней. А вот доказать, что решение единственное чуть сложнее. Через производную сложно, а через математическую индукцию для n и n + 1 в самый раз.
Подобрать быстрее. Сразу же ясно, что число маленькое
Это всё так, но в математике нет понятия ПОДБОР. Здесь показан метод, может быть не самый простой.
Wow!!!
There's a solution to prove that is unique... look that the function is increasing (take the first derivative)...
M=4 by trial and error
Took me about 20 seconds by T & E.
@@silverhammer7779 same
👍👏👏👏
Excelente 👍
Из монотонности следует максимум один корень, подбор. Ну это разминка средней школы России. Я представляю лицо автора, если он увидит реальную олимпиадную задачу))
А было в условии, что m - натуральное число?
@@v.volynskiy Пересмотрите видео, ответ найдете. Почему я должен отвечать на странные вопросы?
@@v.volynskiy А, если корень не подберется? Что в этом случае делать? Так и надо формулировать вопрос, а не строить из себя не пойми кого. Очевидно, что тогда надо решать. Только у меня вопрос встречный- идея монотонности опровергается вами? Нет...Корень тоже нет. Мой коммент был написал под конкретный пример, а не на все случаи жизни. Если вы этого не понимаете, то не стоит смотреть математические видео...правда не стоит
@@user-wi8iq3hn3k помилуйте, ничего вы не должны. С удивлением вижу от вас ещё текст, но очень переживаю, что побеспокоил, так что читать не буду.
@@v.volynskiy Мне не понравился контекст вопроса, поэтому ответ жёсткий. Надо думать прежде чем пытаться унизить незнакомого человека.
65= 3^4 - 2^4 3^m - 2^m = 3^4 - 2^4 Então m=4 Vamos ver o vídeo !!!
Кто сказал, что 65 можно представить лишь как (5*13)? С таким же успехом можно записать 65 как (2, 5*26). И ещё миллионом способов. Считаю приведённую методику нерациональной и вредной. Потому что формирует ложные навыки...
👏👏👏👌👌👌
Thanks 🙏
Eu sabia essa mas era com o X.
Ma'am, I think we can easily do this by trial and error...
You also better explain that + means addition, and what is addition, and '-' means subtraction, and what is subtraction. It will complete your detailed explanation in even more detail.
Согласна, писанины лишней много.
perfect video for olympiad
Thanks
مول الحانوت لمسرهط : بحال هاد الفاكتورة ممكنة ولكن ب n ماشي ب m!🍋👻😁
Your explanation was excellent ; you are a good Teacher who demonstrates every step of the equation ……….Thanks for sharing !
besides the solution is wrong
Will guess and check be accepted in any major maths test?😅
you can set up table and considering m from 0 to 4,5,6... you can choose m=4 is correct answer.
Yea, for small numbers you can use brute force. But what do you do if it says million or billion on the right side of the equation? The result may not always be an integer.
m=4 3⁴ - 2⁴ = 81-16 =65
4... 5seconds
con i logaritmi 3 secondi.....
M yerine 1,2,3,4,5... Koyarak cok kolay bulunuyor. Cevap 4
Deneyerek m=4 olduğunu bulması çok kolay bir soru.
Al ojo: m = 4
4:10 - where did you have taken number 4???
you didn't explained why (x+y)(x-y)=65x1 doesn't work
Because … u get x = 33 and y = 32…. Which means 3^m/2 = 33 and 3^m = 33^2…take log on both sides… m = 2log33/log3 = 3.6797….. We repeat it with 2^m/2 = 32 and this is simple cuz 32 is just 2^5, so 2^m/2 = 2^5…. Taking logs on both sides…. M/2(log2) = 5(log2)….that means m/2 = 5 and m = 10….as u can see M from 3^ m/2 and and 2^m/2 are not the same if u take 65 x 1 factorisation…. Which is why u don’t take it…. Now it is perfect fine to take 65 x 1 factorisation when u define x and y as 3^m/2 and 2^n/2 (m not equal to n) but given this question wants to satisfy the condition of both 2 and 3 having the same number, we take the factorisation that satisfies the given condition
@@RoronoaDPuneethand x+y=65.
Also x-y=65.
@RoronoaDPuneeth Yeah, everything you said is true, but the point is how do we know to go right for 13 and 5 as the factors to use as opposed to any of the infinitely many other factors of 65 over the Real numbers, like 2 x 32.5 or 26 x 2.5 or pi x 65/pi? This was never explained in the video. So in the end, the solution was just guessed at and not "solved" algebraically. And, I'd add, it was guessed at after doing a whole lot of algebra to put it in a form that was harder to guess at than the original.
M 4
M=4. Благодарности не нужно 😂
Mam your soooooo talanted. I love maths
3^m - 2^m, where it's go m, why it's come x and y. Why all maths problem choosing x, y, m, z, ....., is there any particular reason, this type of maths exactly where it's use any explanation
I solved this in 10 sec. My answer: 3^x-2^x is inreasing function. 3^4-2^4=81-16=65=>m=4.THE END. Cheers from Poland.
Need to do this quick transformation 3^m = 2^m + 65 (3/2)^m = 1 + 65/2^m From which follows that rhs is monotonically increasing, while the lhs - decreasing. This at least proves that the number of solutions cannot be more than one.
Now use the same technique [edit: as in the video] to solve: 3^m - 2^m = 2059 Hint: the prime factorization of 2059 is: 29 × 71 Show your work :)
m=7, just by guessing under 1 minute.
@@fajarwp8148 Did you miss the "use the same technique" bit? Yes, it's very easy to solve with trial and error. But how easy is it to solve using the technique from this video? In other words, is the technique from the video really a generic solution?
@@wrc1210 sorry i missed that, but honestly i can't use the same technique if i were in an olympiad. Not efficient.
@@wrc1210She demonstrated the solution of a constructed Diophantine equation of the chosen form. This can only be solved in the way described for certain sums p*q of the form (3^n+2^n)*(3^n-2^n) with m=2*n. Therefore, m must be even and p and q must both be even or odd. For n
4
Perfect solution
No it’s not
Даже решать нечего. m=4, 81-16
Is it really maths Olympiad question?? Stop calling any random question as Olympiad level
-13 x -5 = 65 so , there is an error
But 65 = 65x1, has the problem been ompletely solved?
Practically it will not produce any desired result.
Consider 2 factors of 65 whose sum is13 and difference is 5😊
Also, since there is absolutely no reason to constrain ourselves to integer factors of 65 for that step, why not: 65 = 32.5 x 2 65 = 26 x 2.5 65 = pi x 65/pi etc, etc. In the end, picking 13 and 5 was just a guess.
Това е един частен случай само за цели числа.
Não pode tentar resolver um problema complexo com uma solução simples. Foi muito forçado!
10 saniyelik soru
Et le cas 65×1. 1×65
This is not exactly solution just time pass😂😂😂
Buqədər uzatmaqın nə mənası var?
3^4 - 2^4 = 65
Hint: 81 - 16 = 65
65=78х0,8333333... х=39,416666... где в условии задачи сказано, что решение надо искать в целых числах? А если по условию не 65, а, например, 66. В этом видео не решение задачи, а удачно угаданный ответ! Hello from Russia!
Она не говорила, что нужно найти все решения, а искала только одно. Поэтому она молчаливо предположила, что это диофантово уравнение. Вы бы знали это, если бы... Привет из Билефельда.
3⁴ - 2⁴ = 65
81-16=65
Pl use a bettervpen
m=4
Se pasó todos los principios de las matemáticas por el *. 4 x 3 = 6 x 2 ==> 4 = 6 y 3 = 2 jajajajajaja Horrible forma de resolver este ejercicio.
M=4
गझकोरि । स्टूडेंट को तंग करना ।
Use logarithms
This is not good mathematics. You should have stated at the beginning m > 0 and an integer. If m is a decimal number there are million alternatives.
😂😂
Why you did not use ln way
Is there a way to use logarithms to solve it? I'd love to see it. I fooled around with it a little bit, but couldn't do it. The problem is that you've got two exponential terms with different bases. So it's very difficult. Not saying it's impossible but I don't know how, but I would love to learn if it is possible.
@wrc1210 no I did not try I'm just asking can we do it or no?
@@mostafamahmoud5882 Yeah, I'm not sure. I wish I knew too!
If i think, we can solve it in another way.
Now, to solve to solve it the other way round, question is 3^m-2^m=65. We can also take that m=4, m=2+2, and m=2×2. If we take m=2×2, we get 3^(2×2)-2^(2×2)=65. By using the laws of the exponents, we get (3^2)^2-(2^2)^2=65. 9^2-4^2=65. (9+4)(9-4)=65. 13×5=65. 65=65.
I think 3^m much > 65 Start put m=4 iin equation can get the answer
Элементарно делается путём анализа секунд за 10 . Разумеется, если не идиот
А что, глаза не хватает, чтоб почти мгновенно вычесть из 81 шестнадцать? Да, секунды 3 надо, чтоб понять, кто такой м. Еще секунды три надо, чтоб повозмущаться, куда делся х. Но остальное то время нафига тратить?
what a olympiad!!!! Is olympiad a foolish game?
глупее логики не слышал. Столько воды. Пятиклассник в уме методом подстановки чисел решит за 5 минут.
For basic school in India or Balkan 😅
I solved this in 6 seconds. Your way is too much bla bla bla
Put m = 4 You are all set
4
3^4 - 2^4 = 65
m = 4
M=4
4
m = 4
M=4
m = 4