Solving an IMO Problem in 10 Minutes!! | International Mathematical Olympiad 1992 Problem 1

#IMO #NumberTheory #MathOlympiad

Here is the solution to IMO 1992 Problem 1!!

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I share Maths problems and Maths topics from well-known contests, exams and also from viewers around the world. Apart from sharing solutions to these problems, I also share my intuitions and first thoughts when I tried to solve these problems.

[DaveyJonesLocka]

Regularly kills IMO questions, incorrectly multiplies to get 2 instead of 4 😆

I’m not making fun or criticizing. I think it’s beneficial for students to see that even the greatest mathematicians will make casual errors if they are not careful.

Please keep up the excellent work. I wish I had such instruction in my mathematical competition days as a high school and college student.

[pedrojose]

Good morning!
I did it in a similar way, but quite differnt, as well.
Let k(a, b, c) = (abc-1)/(a-1)(b-1)(c-1) So for a, kmax needs that b=a+1 and c=a+2 (for paritity I coul be more restrictive; b=a+2 and c= a+4, but we would not be able to simplify)
let k*(a, b, c) = abc/((a-1)(b-1)(c-1)
2<= K(a, b, c) <k*(a, b, c), then k*(a, a+1, a+2)>2 ==> a < 4. So a=2 or a=3.
For parity a, b, c are all even and k is odd or a, b, c are all odd and k can be also odd or even.
If a=2
K(2, b, c) is maximum for some b if c=b+1 as k is odd k(2, b, b+1)>=3; then K*(2, b, b+1) >3, so b <5. as b>a and b is even b=4.
kmax(2, 4, c)=k(2, 4, 5)= 10/3. As k needs to be in integer and k is odd and k>1, k=3, so c=8. Then (2, 4, 8) is a solution
Doing the same for a=3
We find that b = 5, k=2 and then c= 15. and (3, 5, 15) is the other solution. Only two solutions.
Note that in both solutions c=ab and the first solucion is a geometric progression (GP).
I liked that you added a term to then factor, with a prime number as a product. And we find b and c at once. I have to restict first for b, then for k and aftewards, solve for c.

[Tommy_]

EXPLAINING a solution to a problem in 10 minutes is not the same as SOLVING the problem in 10 minutes.

[kiranasmaiu]

The amazing solution, it's easy to understand, you're my favorite math channel, I always watched your video and it's helped me improve my math, and help improve my critical thinking... I enjoy your video, thank you very much 🥺 please don't get tired of making videos like this

[niranjan]

You make it so easy even I can understand it. It feels like an AIME question now.

[ignaciobenjamingarridoboba]

amazing method of solving this ecuation. Please try more of these kind of problems with naturals or integers.

[franciscook]

Much the same but slightly differently ...
natural numbers s.t. 1<a<b<c
and P=(a-1)(b-1)(c-1) | abc-1
or k(a-1)(b-1)(c-1) = abc-1 ...(1)
...(1) =>
k<abc/( (a-1)(b-1)(c-1) ) ...(2)
note that for any natural number n>1, we have n/(n-1)>(n+1)/n (because n²>n²-1)
and a≥2, b≥3, c≥4 so max value of ...(2) is for min, a, b, c
from ...(2)
k<24/(6)=4
k≤3
k=1 => ac+bc+ab-a-b-c=0 but ab > a and bc > b and ac > c, so k≠1
looking at a, if a≥4

≤ (4/3)(5/4)(6/5)-1/P < 2 so no solution for a≥4
if a=3
k≤(3/2)(4/3)(5/4)-1/P < 2.5
so k=2
first consider a=2
k≤(2/1)(3/2)(4/3)-1/P < 4
so k=2 or 3
if k=3,
3(2-1)(b-1)(c-1)=2bc-1
3bc-2bc-3b-3c+3+1=0
bc-3b-3c+4=0 add 5 and factorise
(b-3)(c-3)=5 factor pairs 1, 5
solutions a, b, c = 2, 4, 8
if k=2
2(2-1)(b-1)(c-1)=2bc-1
-2b-2c+3=0
2b+2c=3 has no solutions for b, c > a=2
consider a=3 and k=2
2(3-1)(b-1)(c-1)=3bc-1
4bc-4b-4c+4-3bc+1=0 add 11 and factorise
(b-4)(c-4)=11 factor pairs 1, 11
solutions a, b, c = 3, 5, 15

[danielcrespo]

Excellent content. Can you tell me please which software do u use? Is it like an app running on ipad?

[dr.danburritoman]

This way to solve it is fairly good, IMO😉

[adnanahmad]

Really awesome! Must watch videos
Can you pls tell us your secret for solving questions with such an effective strategy I have my maths olympiad coming up.

[Stelios]

That was simple even as a Problem 1. However, thank you for the nice presentation!

[omairsiddique]

At 5:09, shouldn't it be (4/3)(5/4)(6/5). You wrote 6/3 instead of 6/5

[excited]

is the checking part is important at olymp, i mean we should write checking part on our answer sheet?

[divyanshtripathi]

Good video bro, I want to ask you from where you learnt that when to use which approach.?

[dr.rahulgupta]

Thanks for a simple and nice presentation. DrRahul Rohtak Haryana India

[emilsriram]

Says, "And this is equal to two." Contemplates life for two seconds after he contradicts every rule learnes in elemantary school. "No, I'll just stick to regular mathematics for this video."

[gastoncastillo]

Hi, you are really excellent. May I ask where you learnt all of these? And also maybe how you recognize when to use some approach or another? Thank you very much for your content

[michaelblankenau]

Wish I was smart . But we can't have everything .