Loops in a Sequence | International Mathematical Olympiad 2017 Problem 1
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- Опубліковано 16 жов 2024
- #Math #IMO #Sequence
In this video we are going to solve Problem 1 in IMO 2017.
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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.
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Great problem, although I think you could have emphasized the key insight better. When a_0 = 1 mod 3, you will eventually reach a number of the form (3k+2)^2, which sends you back into the 2 mod 3 infinite sequence. On the other hand, when a_0 = 0 mod 3, you will eventually reach the square of a multiple of 3, and (possibly after many iterations of taking square roots) you'll reach the square of 3, 6, or 9, which sends you into the loop. The way you presented the proof does not make it intuitively clear what will happen if you start out at a very large mutliple of 3, e.g. 3,000,000.
Challenge: Do a Q6 question
Fact: 4 =2^2
I remember for some reason most of the participants forgot this fact during the contest and took much time until they noticed it😂
fact: 2×4×1×1=2+4+1+1
@@timetraveller2818 hello
@@timetraveller2818 i know it already 😆😆😆😆
@@timetraveller2818 you can do this for any composite with enough 1 terms.
You can use many different bounds. I for example used 3^(2n)
Very clear presentation until you changed from red to blue pen! How do you come up with the boundaries in the blue cases? Why should a_n be between these values and if it isn't, so what? It is not explained unfortunately.
The only previous hypothesis (at the end of red pen) was that a_n mod 3 = 1 (which we are trying to disprove). Also, the case of a_n mod 3 = 0 should also be mentioned (although this is easily disproven).
really nice solution, very clear and concise
hi, i have a question:
do you think that a large amount of mathematical talent is needed to solve these kinds of problems? or can hard work and practice alone be enough?
great video by the way.
Hi. I'm not the person you are referring to, but anyways. I have some experience on math olympiads and I'm at a high enough level to answer, I think. You could probably make it to a pretty high level with average talent, not as high as the imo though,but a respectable level. But you have to work a lot harder than your peers. I think that all talent in maths stems from creativity, and ,in all honesty, you can approach olympiads by learning tricks and applying them , which requires some creativity, but not a lot. Then you have the other approach, to derive some tricks yourself. This is the most fun and requires the most talent as well. Someone without talent , taking the first approach can certainly solve some high level problems, dont know how high though. Although, unfortunately there is a limit, the sooner someone realizes that the better.
@@exarchoustathees7623 thanks for the reply, that was insightful.
Question was not for me, but I want to answer. I think that you need hard working. Talent can just make your progress faster, but without hard working, nothing can help.
@@exarchoustathees7623 You control your destiny. IMO medalists aren’t magicians, they are kids who have spent thousands and thousands of hours over many many years to solve these problems.
@@JohnSmith-vq8ho I know some IMO medalists, hopefully I will be one as well. (Not old enough yet, but I got the 2nd place in my country U16, and a medal from the junior balkan olympiad). I know many people as well who have studied insane hours, up to 5 times more than me) and didn’t even make it in the national team. I have some experience in the field, and I have heard stories of people studying crazy and they couldn’t even get a medal in the national Olympiad. Hard work + talent beats only hard work no matter how hard that work is. I'm not saying that to unmotivate people, I'm just saying what I think and what I have noticed in my experience.
Could you make ISL A3,N3 up Plz?
The P1 actually observe enough longly and easily solve it.
I think there are more funny question!
BTW the video is great! Thank you!
very nice
Part 2.1. Sequence eventually reaches (3k+2)^2 ?? Don't you mean sequence eventually reaches (3k+2)^2 or (3k+4)^2?
No, because (3k+2)^2 is ≡ 1 mod 3 and, since the sequence goes up in 3s until it reaches a perfect square, it must reach that value first, as the next available perfect square.
Nice
if a0=3k+2 it cannot be a perfect square. and a0+3r cannot be a perfect square. we have to see what happens if a0=3k.
Hi, great problem ,
Challenge: do anything from IOI
my teacher let me to do this problem, i do it in 3 hours. Can't understand how imo-er let it be the easiest problem ... :(
well, at least you can solve it in 4.5 hours... if you participated that year you'll at least get an honourable mention
@@ericzhu6620 as if i could :( . But i'm not good in geometry and the last combination problem in day 1 is very hard, there are only 2 competetors could do it. I think i cant get any prize if i join that competition :/
@@ericzhu6620 by the way, i love math very much, i just want to get a prize in my country's math olympiad, which call VMO :D
@@hoanglaikhanh1383 If the VMO refers to the Vietnamese Math Olympiad, then I got to say it has one of my favorite number theory problems of all time!
@@ShefsofProblemSolving Yes,it's Vietnamese Math Olp. Can i know that problem :D
3k or 3k+1
No just 3k
3k+1 eventually becomes 3k+2
@@emoore06905 It's easy to see the answer isn't JUST multiples of 3, try, for example, 7:
7 -> 10 -> 13 -> 16 -> 4 -> 7
Nevermind I forgot that 4 is a square lmao -- rewatched the video.
I solved it without mod😎
Induction goes brr...
smh dont u remember we failed induction
@@prithujsarkar2010 But now I have succeeded!
First comment
Very unclear presentation, nearly no additional value than purely reading the text of solution. Waisted time.
Hi remember me
My god evem the question itself is convoluted and hard to understand..doesnt everyone agree?