Easy. a=2021, c=2020 and b=0.

@aidan-ator7844

Жыл бұрын

You missed half the solutions genius

@TheEmanoeljr

Жыл бұрын

​@@aidan-ator7844whatever. Its correct! Uhuu!

@aidan-ator7844

Жыл бұрын

@@TheEmanoeljr yes but you only have the intuitive solution. There are others.

@ferrel9715

Жыл бұрын

​@@aidan-ator7844Yeah... So you can admit he has good intuitions. Lol...

@aidan-ator7844

Жыл бұрын

@@ferrel9715 without a doubt. Intuition is only one part of thinking that functions best in conjunction with others.

I liked the fact that someone with a nice voice and clear handwriting can provide audio-visuals for an instructional video. I have neither.

@satvikakshintala8030

Ай бұрын

Bro but this isn’t a valid question and neither her answer a valid one Since there are 3 variable you can have multiple answers for eg A = 2021 B = 0 C = 2020

Somewhere, someone's discovered an amazing problem featuring the number 2374 and is just waiting to reach that year to release it

@donelkorantengbrown9439

2 ай бұрын

I don't think that they would still be alive

This is a special case of a much more general problem: ab + c = A (1) a + bc = A + 1 (2) (2)-(1) gives b=1-1/(a-c). Let's replace the variable c by c'=a-c and work with c' from now on (still have 3 independent variables, but more convenient, c can be recovered by c=a-c'.). So b=1-1/c' (3) Substituting this into (1) gives a(1-1/c')+a-c'=A, which yields a=(A+c')c'/(2c'-1) (4). Given an arbitrary A, one therefore only needs to specify c' to get a general solution of a, b and c. For integer solutions, c' must also be an integer, and from (3), b can only be an integer when c'=1 or -1. For c'=1: it follows that a=A+1, c=A, b=0, all integer as long as A is integer. For c'=-1: it follows that a=(A-1)/3, b=2, c=a-c'=(A+2)/3. In order for a and c to be integer for c'=-1, A has to be a multiple of 3 plus 1, i.e., A=3n+1 with arbitrary integer n, which then gives a=n, and c=n+1. The original problem is when n=673. Obviously, there are infinitely many 'problems' with the same condition provided that A=3n+1 with arbitrary integer n. Without restricting to integers, (3) and (4) constitute the general solution for arbitrary A.

@dorgamahmad6033

11 ай бұрын

Also 2×0.5=1 and 0.5×2=1

@andrewcheung7538

11 ай бұрын

The real answer should be a curve in 3 D diagram... [n, f(n), f2(n)], should you think that you haven't given out a proper answer..

@hoagy_ytfc

10 ай бұрын

@@dorgamahmad6033 Except that the problem stated in the first few seconds of the video was "find the integer solutions".

@adivoma7

10 ай бұрын

Basically (1-b)(a-c) = 1 is true only when both (1-b) and (a-c) are 1 or both are -1. These are the only integral solutions. The system having infinite solution is being eliminated by the fact a, b, c are integers. This has only 2 sets of solutions.

@ctsirkass

10 ай бұрын

@@dorgamahmad6033 we are looking only for INTEGER solution. Read the question 3 times before solving (classic teacher's advice from the elementary school)

Look at 4:03. This only works if youre working with integers. The single-step assumption that xy = 1 only has two answers is only valid in integers. Counter-example: x = 1/2 and y = 2. This is also 1. (In this case x = (1-b) and y = (a-c)).

@ShawnPitman

10 ай бұрын

I'm an idiot. The problem statement says "integer solutions".

@atrib_

10 ай бұрын

The video did not stress this point, which is crucial. 3 variables, 2 equations leads to infinite(?) solutions. Even with the additional constraint of integer values, we got 2 solutions

@muhammadabujabal9387

10 ай бұрын

​@ShawnPitman But each part was the combination of 2 numbers , so you will get only integers if you add or subtract integers, I think it's the only step that seems need looking the rest is simple algebra could be done in 1 min

@hickiwawa

9 ай бұрын

Same here. The thumbnail left that out. I was going to solve before watching the video, only to quickly notice there are infinite solutions.

@lipers87

9 ай бұрын

yes, if u name "ab + c = 2020" as X and "a + bc = 2021" as Y, them we can do Y - X we get: (b - 1) • (c - a) = 1. So b can be 0 or 2, but c - a can be anything like (2, 1); (3, 2); (-1, 0); (-2, -3); ...

A lot of commenters are boss stating the simple 0 solution or observing that there are infinite solutions, until they finally read the instructions! Integers only! And there is more than one solution.

Interesting that many people can't read the first four words in red.

@Mr_AbdulRehman

9 ай бұрын

Indeed. It's painful, too painful reading comments.

I solved stuff like this back in highschool. Man life has dragged me down. I need to take math classes again.

@kathrynstemler6331

9 ай бұрын

Right? I remember a time when I could try to get my head around this but I’ve forgotten so much.

a = 2021 b=0 c= 2020

Easy eq1>eq2 by 1 so 2nd equation has 1 extra b c is therefore a+1 b is 2 (one extra plus 1 in row 2) 2020/3 =673 Remainder is 674

Nice use of factorization concepts, well done

In the last step, the two term just need to be reciprocals of eachother and if you get an integer for all values for example (1-b) = 1/2 it is a solution

What is the brand of the pen, I love how thin the lines are.

There is a third solution too. (1-b)=1/(a-c)

Thanks! Delightful presentation of a clever little problem.

That's nowhere near to a maths Olympiad question

I watched just to make sure that my answer was right. I saw the problem while scrolling here on youtube and took only a few moments before I saw the solution. All of the extra steps were entirely unnecessary, but probably can help folks who aren't able to easily see the answers to math problems like this one.

I think I got it!!! Once I opened the video and saw the find all integer solutions I got two answers that both work. I still haven't watched but feel a nice sense of accomplishment!

The RHS equals 1 can be expanded as you did,but also can be expanded as multiplication of I and 1/I, which gives infinite number of solutions

@Mike-rx5uu

5 ай бұрын

If a, b, and c are all integers (given in the problem statement), there is no way to generate a fraction of the form you're suggesting with 1-b or a-c.

Wonderful. this is how simple things can be best seen.. Thanks a mill

2 equation and 3 variable so put 1 variable 0. So only one way we can easily satisfy equations is put b=0 then a=2021 and c=2020

My generic solution for problems with integer solutions: convert the problem to A * B = a small constant. As A, B are integer, we can find all the possible values of (A, B). So, this problem gives a(1 - b) + c(b - 1) = 2021 - 2020 = 1, so (a - c = 1 and 1 - b = 1) or (a - c = -1 and 1 - b = -1)

If u see , there are infinie solutions As 1 can be written 2* 1/2 and infinitely many more 5:41

Thanks for the solution and a great explanation.😊

Love all your explanations. They are so clear and easy to understand

@QUABLEDISTOCFICKLEPO

Жыл бұрын

Not to me.

@popliceanumihai9653

11 ай бұрын

@@QUABLEDISTOCFICKLEPO 😂

@PreservationEnthusiast

10 ай бұрын

Not clear.... she says a into b when she means a x b. a into b is b/a

@christianherrmann

10 ай бұрын

She explained every little step, but not the most crucial one, why none of the factors can be a fraction and hence only can be equal to +/- 1. (because a, b, c are to be integer, the factors (1-b) and also (a-c) are always integer.)

@gardenjoy5223

10 ай бұрын

@@christianherrmann For someone without foreknowledge, she forgot several steps! She did not explain why -(ab + c) is the same as -ab -c. Then she fails to explain why a - ab is the same as a(1-b). Explain those steps in between, and we are good to go. Without: nothing goes. To me this constitutes a bad teacher!

Thank you very much, your solution is clear and simple.

@QUABLEDISTOCFICKLEPO

Жыл бұрын

It's a clear as mud to me.

@sobolzeev

10 ай бұрын

​@@QUABLEDISTOCFICKLEPOYou would prefer it even slower? Or do you need an explanation of an idea of replacing numbers with letters, so called algebra?

@QUABLEDISTOCFICKLEPO

10 ай бұрын

If I couldn't understand it, the fault is not mine. I won't waste time by looking at it again.If I said that it was unclear, it was.

@sobolzeev

10 ай бұрын

@@QUABLEDISTOCFICKLEPO If you hear a dumb sound when a book hits your head, it is not necessary the book's failure.

@QUABLEDISTOCFICKLEPO

10 ай бұрын

Fortunately, I didn't have any "teachers" like that when I was in school. If I had, I never would have learned fractions..

Side note: I gave both Bing Chat and Google Bard this problem. While Bing Chat gave a great step by step, it got the wrong answers. Google Bard got the same answers as the video. Bing Chat: a = 2019.49 or -1919.49 b = 0.51 or 3940.49 c = 2019.98 or 22.02 The first set of answers seemed to be a rounding error. But the second set was completely off. My comment has nothing to do with the video, I just find it interesting how far off these AI are still. Google Bard got this one right but I've had times where Bing Chat gets it right and Google Bard gets it wrong as well. I've found if I ask both the same question, I'll either get a good answer or funny one.

@xxxBradTxxx

8 ай бұрын

I can wait until chat bots can do math properly. I kinda wish OpenAI and Microsoft would block those questions for now until they figure out how to make them accurate.

@santiagoferrari1973

6 ай бұрын

Ask Grok.

@fongwinson5017

5 ай бұрын

So far they are Language Models with some math capability. "Intelligent" in some areas only. SImilarily there should be very powerful math AI that could not have a nice "chat" with you.

Three unknowns cannot be deduced from two equations... There are an infinite number of solutions

@jimbrooks1452

5 ай бұрын

You are correct if there are no restrictions. But, as I tell my students, "read the problem." The problem restricts the solutions to *integers.*

Impressionnant !!

I have seen this classic Olympiad problem so it is easy for me.

@the-mathwizard

6 ай бұрын

Indeed, math is easy if we already seen it before

How come this video appear in my suggestion, it looks like magic to me

Thank you. Elegant.

I brute forced 0's and 1's and found a solution quick. But I'm not taking the test under pressure, I wouldn't have been able to do this in school.

not watched the video but the intuition that comes to mind here is that b scales either a or c with very similar results (2020 vs 2021), so a and c must be very close, hence write c in terms of a by substituting d := c - a (so d will be small), and yeah subtracting the top equation from the bottom then gives d(b - 1) = 1, so (c - a)(b - 1) = 1, then it's easy cos they're integers dividing 1

Subtracting the two equations work here. I wonder how difficult versions of this problem would look like: I am imagining some function of the first equation + some function of the other equation to give some hint in the general case.

@mig_21bison

10 ай бұрын

What is the use of these equations...??? Where they are used??? What is the practical application????Please answer

@sobolzeev

10 ай бұрын

Hm, what is the use of, say, anime pictures? Whom do they depict? What is their application? Please, answer?

@neonblack211

9 ай бұрын

​@@mig_21bisonif you want to consider science, engineering, physics, absolutely all over the place

That was really amazing.

He took 8 mins to explain this

Good answer...it is factual... Based on multiplication principle- 0 multplied by any number is 0

Just like in video, instead subtracting on equation from the other - you can also add one equation to other and get a simplified product for (b+1)(a+c) =4041. It’s pretty much a very straightforward problem

@GauravSingh-gd1yj

9 ай бұрын

This equation is not at all straightforward,the easiest soln was given in the video

@dudedujmovic6562

7 ай бұрын

@@GauravSingh-gd1yj Agree, the easiest is shown. It is a nice problem for basic algebra astute.

As a 7th grade asian we can be sure with you that this math equation is a piece of cake

An easier step at the 2a+c/a+2c part would be just adding them. 3a+3c=4041 a+c=1337 Then you take the a-c=-1 2a=1336 a=673, then c=674 But that I guess it depends on the education system or on your mood. Nice solution.

@gardenjoy5223

10 ай бұрын

? Where did you leave b?

@xyntas

9 ай бұрын

@@gardenjoy5223tell me you didn't watch the video without saying you didn't watch the video

@tcz1757

9 ай бұрын

What about the a = 2021, b = 0, c = 2022 solution?

@gardenjoy5223

9 ай бұрын

@@xyntas Actually I did watch it. What happened to the b? Was I distracted at that bit? Can you give me the time, where one can leave b out to find the answers?

@danieltatar7575

9 ай бұрын

@@gardenjoy5223 6:32

So great , smooth voice , quiet and logic I enjoy

Thank you

This reminds me linear algebra

In case II, we already have c=a+1. Putting in either equation yields 3a + 1 = 2020. Good job otherwise. 👍

b(c - a) - (c - a) = 1 = (c - a)(b - 1): 1) b - 1 = 1 and c - a = 1, so b = 2 and a = 673 and c = 674; or 2), b - 1 = -1 and c - a = -1, so b = 0, and a = 2021 and c = 2020.

@crcurran

10 ай бұрын

This is what I thought just glancing at it. Seems to make sense to me that b = 0, a = 2021 and c = 2020.

Beautiful.

Through inspection a = 2021, b = 0, c = 2020 lol

Clear solution

2:42 signs work in multiplication? How?

I had problems falling asleep. This video was my cure.

I got to 3:37 myself but I had a different train of thought. I thought to myself what do I multiply by each other to get one. I realised that x * 1/x always is one. I got stuck on that and watched the video. I forgot that integer means that it could not be a fraction. English is not my native language although I should have realised it since in a programming I know an integer is a round number from- 2^15 and + 2^15.

@Dannychii

5 ай бұрын

Thanks for the explanation, I was confused at that too :D German here 😄

I tried solving this with substitution before watching the video and got stumped. Nice use of factorials.

I’d have said a = 2000, b = 1 c = 20… Because I’m dumb and just pop out the first answer that comes to mind, give it zero thought, then never think about it again, because I’m not a maths professor or in a maths class, so it’s not like I need to do any equations ever, just like 99.999% of everyone else in the world. The remainder probably do physics or work at NASA or something so they need more maths skills than basic addition, subtraction and potentially basic multiplication or division… But we all have phones with calculators on them for a reason, pretty much for doing DIY, maybe doing refunds at a customer service job, working out how much each person needs to pay at a restaurant, and pretty much nothing else.

love it.

Side note, neat idea with the name on a pen

it was exactly my answer spending just 30s thinking

What grade is this math problem for? In my country, I encountered this problem when I was in 9th grade

love these

Wow very impressive.

ab + c = 2020 and a + bc = 2021 => *(c-a)(b-1) = 1.* Since, a, b and c are natural numbers, so the only way the above product( written in *bold font* ) can be 1, is when b-1=1 and c-a=1, which implies b = 2 and c = a+1. Putting b = 2 and c = a+1 in ab + c = 2020, we get a = 673 and c = 674.

@eddie31415

10 ай бұрын

not natural numbers, but integers

@ctsirkass

10 ай бұрын

We are talking about integers, not natural numbers, so you can get a product of 1 by (-1)*(-1) so you missed one solution.

Diofantic problem. Very interesting.

The calculation in the video is missing a solution. At 3:52 we cannot assume that a x b = 1 then a=b=+-1 like in the video, there are infinite cases that two numbers multiply each other can equal to 1, for instance a x 1/a = 1 as well. According to my calculations there are two set of solutions, one just like in the video: a = 2021 ; b = 0 ; c = 2020 And the other set is: a = 673 ; b = 2 ; c = 674 You can test my solutions by replacing it to the given equation in the beginning of the video. Ty!

@the-mathwizard

6 ай бұрын

Watch the question my brother, it states only integer solutions are allowed

Well, you do not need any calculations for a logic solution: just observe that b cancels a in first equation and also cancels c in second, so if b equals 0 then c is 2020 and a is 2021. 5 second solution.

Easy. c=0, a = 2021, b=2020/2021. With two equations and three variables, we are going to have an infinite number of solutions. We can usually pick one variable to be anything we want. (Edit: I didn’t realize, until after I wrote this that the thumbnail is different from the actual problem, which is misleading. The thumbnail does not say integer solutions.)

Sooo... We have few informations. What I found was simple: b belongs to the set of reals, and it can be any real value, with the exception of 0 and 1. For negative b, a>c. For positive b, a

@ctsirkass

10 ай бұрын

Yeah, you need to pay attention to the question. My suggestion would be to slowly read 3 times and pay attention to each word separately before solving. (hint: we are looking for integer solutions only)

Hi, Did you look at the variation where you add the two equations and get (1+b)*(a+c) = 1*3*3*449. So you could possibly get multiple solutions in addition to what you provided. Curious what kind of solutions are generally acceptable? Thanks!

@user-uo3ko1yt6y

11 ай бұрын

The answer will be the same, the extra solution set provided by your method involves functions.

@donniebao

11 ай бұрын

you get the same answers, but it just takes longer. You can set b equal to all the potential values and have 2 solutions 2 equations that you solve normally, but for the solution values of b = -450, -10, -4, -2, 8, 448, you will get non-integer solutions for a and c. The only values of b that give you integer solutions for a and c are, b = 0 and 2.

@AlfredoTifi

11 ай бұрын

I solved for natural integers (b+1)(a+c) = 4041 = 3•3•449 (excluding factor 1, which would yield b = 0). I have got 4 equations for 4041=(8+1)(a+c) = (448+1)(a+c) = (2+1) (a+c) = (1346+1) (a+c) with b>0, of which only the third, with b = 2 gave integer a and c (673 and 674).

Its good it says integer solution because with two equations and 3 variables you can get infinite real solutions, lol. a = 1125 b = ⅘ c = 1120 For example. Very nice, I forgot to factor that way, and so I trailed and errored to get (a-c)(1-b).

awesome !

So interesting, thanks 👌✨️

I am embarrassed to say I spend 15 min going in circles until I realized what I need to do and it was solved in 5 min.

Why are you only using integer solutions?

0.5*2 equals to 1 as well

In my augustus opinion, add up all the stuff and put its variable b in evidence.

This was great!

Thank you very much

The product of 1 can be obtained by 1/2 × 2 or 1/3 x 3. There are infinite solutions to that problem.. You have 3 unknowns and 2 equations. You can't have a definite solution for that!!

@andykyllo6856

Ай бұрын

The directions state integer solutions only.

The easiest is you choose one variable to be zero like a = 0 Then you eliminate quite a lot. c = 2020 and b = 2021/2020. So multiple solutions Not a fair Math problem but resourcefull one.

@tawfikahmed.2526

5 ай бұрын

This is decimal solution not integer solution for b

lol i can't do all of that weird stuff but I did solve it in my head by just making b 0 cause it was the only one multiplying in both and working from there lol.

awesome

My solution also seems to work a=0 not b, b=2021/2020, c=2020. Waht is wrong with that ?

Besides the 0 solution. C=674 A=673 B=2

1-b=1/2, a-c=2.

This is so simple.

Nice solution

Solving systems of equations is so satisfying. Plug and chug, baby

A pretty clear explanation except that it is not proven at 3:45 that 1 * 1 and (-1) * (-1) are indeed the only products of two integers that yield 1.

easier to solve using a system of equations. we express A, and then substitute it into the first equation..., and then it’s obvious

I was not able to follow the solution. But the first one was obvious to me looking at it. 😂

There is an infinite solutions, as you can express any variable as dependent of the other two, as that is a two equation system with 3 incognities

@jige1225

10 ай бұрын

There is an additional constraint that a, b, c are integers

abc (nailed it)

I need to reteach myself math fr.

Use it in a realistic world problem

i have to count to 20 by taking my socks off and i was able to do solution 1 in my head :)

Man… it’s been too long since college.

Ab+c = (a + bc) - 1 Now solve it... There are probably more solutons... or more easy c =2020 - ab Now replace this c in second equation with (2020 - ab). a + (b(2020-ab))= 2021 a + (2020b -ab^2)=2021 Now we get: 2020b-ab^2 = 2021-a and c = 2020-ab Here on we choose one variable and multiple solutions present ourselves... One of them: b = 1 we get 2020-a=2021-a 2020=(2021-a)a a^2-2021a+2020=0 a=1 c=2019

@tawfikahmed.2526

5 ай бұрын

Wrong

Поскольку ни француским, ни ангийским не владею, то нюанс про целые числа я упустил. Только когда он начал разбираться с "а" и "с" я понял в чем дело. А потом еще увидел и "Іnteger" в условии.

a= 673; b=2; c=674 and a=2021; b=0; c=2020

how does this solve in this universe? 10 for a=1 to 1999:for b=a+1 to 2000 z1=2020-a*b:z2=(2021-a)/b:if z1=z2 then stop next b:next a

I loved this.