France - Math Olympiad Question | An Algebraic Expression | You should be able to solve this!

Maths Olympiads are held all around the world to recognise students who excel in maths. The test is offered at many grade levels and provides them with numerous possibilities to win certifications, awards, and even scholarships for higher studies.

[brianbutton]

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

[eliatgnu]

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.

[ShawnPitman]

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)).

[muskyoxes]

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

[mikesmovingimages]

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.

[Crom]

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

[masterthnag]

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

[balkansenjoyer]

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

[Moharidy]

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

[M.Melkonyan]

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

[yogeshchaure]

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

[proman]

That's nowhere near to a maths Olympiad question

[lamttl]

Nice use of factorization concepts, well done

[avalagum]

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)

[AhirZamanSairi]

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

[arteffectsshivam]

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

[richardleveson]

Thanks! Delightful presentation of a clever little problem.

[rnseby]

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.

[jasonsternburgh]

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.

[alanklajnsek]

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