A Big Secret in Solving Number Theory Problems | Turkish Junior Mathematical Olympiad 2012 P1

#NumberTheory #MathOlympiad #ProblemSolving

Here is a BIG secret in solving Number Theory problems! What do you know about Number Theory? Always ask yourself, have you made the most of your knowledge?

———

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.

[SaidVSMath]

this made me so giddy, I’ve always struggled with congruence and mods but watching you prove the only value of p was so awesome!!

[macho_a_little]

This channel helps me find my location to the number theory and teach my students who prepare various mathematical competitions. thanks a lots. i hope your channel to spread out all over the world!

[cmdcs]

One of the best channels for actually *explaining* the methods and thoughts. I hope you return to Youtube one day

[klarkunskap]

What a great channel! Cant belive your channel is this small, but I guarantie you if you continue like this your channel are going to be huge.

[yoyokojo]

I really enjoy your video style, very thorough explanation!

[tonyha]

Wow, a hard problem for a Junior MO! Great solution and insightfully using mod, Thanks!

[blankino-]

Why do we multiply by 4 when completing the square?
Really great video by the way, it helped a lot to hear your own first thoughts about the problem. Looking forward to further problem solving videos!

[СВЭП-иф]

My solution: x(x -3y) = p(12-py^2) => x mod p = 0 or x - 3y mod p = 0. First case, x = kp, where k belongs Z. Plug in to get p(k^2 + y^2) = 3 (3 + ky). k^2 + y^2 can't equal to 3, so p = 3. x - 3y = kp also leads to same result. Then it follows this solution

[mukaddastaj]

You really make it look so easy and fun! Thank you so much for this)

[emrahkaragoz]

My name is Emrah from Turkey. You are great .I have learned so much things from you.I am not teacher or student but I enjoy so much your teaching videos.
Thank you so much.

[AlephThree]

These are very good. Keep going with the output!

[sadececansu]

Türk bayrağı görmek sevindirdi TÜRKİYE'DEN selamlar...
Thanks for your video....

[SatyamKumar-yksz]

Please make video on strategy to solve number theory problem based on imo Or Putnam level olympiad.

[pedrojose]

Good Morning! I waked up this morning and I saw an easy solution, not with many value, cause I saw, only after doing it once, and so backwards. But it is intersting.
x^2-3xy + (py)^2= 12p (i) It is easy to see that x^2 + (yp)^2 = 0 mod3 and then x=yp=0 mod3.
Then p=0 mod3, also. p=3.
p=3 and (i) x^2-3xy + 9y^2 = 36 (ii). And as x^2 -6xy + 9y^2= (x-3y)^2 *, we have a perfect square and also (x-3y)^2=0 mod3 ; because x^2-3xy + 9y^2 =0 mod3 and (x-3y)^2-(x^2-3xy + 9y^2)= -3xy=0mod3.
So we have (iii) (x - 3y)^2 = (6 + 3k)^2 for k integer and k > -3.
(iii)-(ii) -3xy = 36k + 9k^2 : for k>=1 -3xy >=45; but no way becauseit implies in (ii) that x^2+ (3y)^2 <= -9
So, as I had found in the first solution xy>=0 and then k have to be less than 1.
Three options k=0; k=-1 and k=-2.
k=0 -3xy=0 ==> x=0 (y=-2 or y=2) or y=0 (x=-6 or x=6). Then four solutions (0, -2, 3); (0, 2, 3);(-6, 0, 3);(6, 0, 3)
k=-1 -3xy=-27; no integer solutions
k=-2 -3xy=-36 ==>xy=12 and solving by factoring 12 or solving a biquadritic equation we have y=-2 (x=-6) or y=2 (x=6) and we have two more solutions: (-6, -2, 3); (6, 2, 3)
I hope you enjoy it. I would really enjoy it, if it was my first solution. Best regards.
* It was missing that from Bézout x-3y=w it has integer solutions for w integer, since (1, 3)=1

[TechToppers]

Oh, can you get some Geometry problems rolling? I want to try them too.
Great vid btw.

[yashvardhan]

I solved it exactly like u did
First I thought that maybe We can factor by difference of squares after completing the squares but immediately realised that it won’t work .Then after some time I finally saw that rhs was divisible by 3 if we move the 3xy term
Then I reduced modulo 3 and eventually solved it like u . It took me 40 min though.
Ur choice of questions is great.

[michaeleissenberg]

Hello! I don’t understand why at 0:53, you added those 9! Can someone explain to me what happens in your mind so you think about this please?

[SYZY-]

please make a video for the jbmo 2024 problem 3 number theory

[kqpgyy]

Delight! You don't stop on arithmetics! If I don't get it, I stop the video and get it. 🌷

[thayanithirk]

Sir please provide the solution for Turkish junior national maths Olympiad 2013 question number 2