from the IMO shortlist...
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- Опубліковано 28 вер 2024
- We solve a nice problem involving the floor function. This problem was shortlisted for the 1996 International Mathematics Olympiad.
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According to Wolfram Alpha, if you change this problem from "floor" to "ceiling", there are only two solutions: (1,2) or (2,1). Thought that was interesting.
It took about 30 minutes to solve when all guesses as to which direction to head were correct (there were about 4 or 5 of these). In a realistic setting, you could end up going down all manner of rabbit holes that lead to dead ends, or worse going in circles for hours with dozens of pages of scrap paper full of increasingly complicated and inscrutable equations.
Loving these, keep them coming and 100k is around the corner.
take a shot every time he said "let's bring this to the top and keep going"
I was afraid of doing this equation with the floor function. But finally I jumped to it with the solution in this video and... it was great ! Thanks a lot for making possible the discovery of all thoses maths fields...
i love this man ,he helped me so much ,i hope you post more videos like this .I am writing olympiad jr in a month and i hope watching your videos will make me better !!!!! love the way how you teach
One of our friends assigned this problem to us once as POTD, he is also a genius like u, probably all u PRO MATH PPL ARE *CONNECTED*
@@chhabisarkar9057 -_-
yea he is a GAWD
@@chhabisarkar9057 No I didnt
@@chhabisarkar9057 -_-
Yes he is god _/\_
excellent work, here to learn from a master
As another commenter posted, there is a small hiccup at 28:48 where it is argued that m(b^2-2)>= 2(b^2-2). This is true only if b>1.
Good heavens, this was brutal.
This problem reminds me of the Legend of Q6.
When you said that m(b^2-2) is greater or equal to 2(b^2-2) that actually was not true for b=1. That doesn't change the solution, because you had to consider b=1 case separately anyways, but still that was a small inaccuracy.
My head hurts.
Very nice
This frustrated the shit out of me. 1. I never once saw the floor function and 2. This seems extremely detailed for a problem that would be on an exam. Would it take a kid 36 min to solve this?
Well that's probably why it was shortlisted and didn't make it to the final exam
The IMO is quite hard! Anyone who has a chance to take it is definitely familiar with the floor function, and yes I'm sure some of them would have solved it relatively quickly :)
IMO is a 6 question exam where you have more than an hour per question. Alot of time can be spent doing tedious calculations and gaining intuitions. Michael Penn does a good job briefly explaining the motivation for steps in his speech. So mind you, this video would be considered a final product and is quite quick :)
It's also a contest question, right? Not a random test question.
Woah how did I catch this so early
subscribed!
Good teacher
Barney stinson
These are rigged problems..
THE FLOOR IS LAVA
Small error at 16:39. The numerator of the second term in the right floor function should be b^2(m+1) + m^2 . Just missing a plus sign in there. Boy, that confused me for a bit!
Oh, you fixed it at 17:29! I should have known. Nice, you noticed at a taxicab number
Extremely underrated math creator; I love your work, keep it up.
what a video, I could go from the beginning till the end of the video without any doubts and you corrected all little mistakes throughout the way. good job dr Penn, keep it up. I suggest that you try some problems with the ceiling function
Truee
Michael actually makes you wanna become good at maths
@@lucagirelli5223 yeah dr michael is a great person !
35:58
Loading 94.8%... No homework today but I hope you’re having a great time. Stay strong.
Awwww. The puzzles are fun tho
@@gamedepths4792 Yeah... it’s just I didnt have time to have a good pile of homeworks.
On top of that, people like these homeworks but I don’t want to give that much because I’m worried Michael would run out of interesting problems.
Anyway, I’ll think about a new way to give homeworks at 100K subs.
@@goodplacetostop2973 Homework Giveaway!!!!!Pretty innovative
Loved the video, but man I didn't sign up for this many inequalities this early in the morning lol
Is it odd that my first instinct to solving these question is: try 1 first then... break my head?
What patience is needed to work this out? You are amazing.
For the inequality on 26:05 we could use basic induction on m to show that the inequality is greater than 0 with m, b ≥ 2. Leaving us with the only case (b, m)=(1, 1), which is already covered by (a, b) = (b^2 + 1, b^2) up-to permutations with b ≥ 1 and b ∈ N.
=) how about a live video when you get the viewers to try to solve the problem?
Your videos are one of the few things that can distract me from this bad covid situation. So thank you man
The inequality at 28:54 does not hold if b=1 because you will get -m > -2 or m < 2 which contradicts our choice of m
Right
I wonder why this was rejected - too much work needed?
This was worth imo prob , why shortlist?
Because you rejected it :yeet:
@@chhabisarkar9057 -_- xD
I wonder how can you become so talented at math, what is your thinking process when you solve such hard problems ?
Granted, fortunately or unfortunately, some is down to innate talent. Alot is also down to practice and dedication. Do 30 minutedsof math a day for 10 years, 15, 20, and you’ll get really good.
It's just practice. Do a lot of questions (must have variety too) and after sometime you just get Intuition for the answers.
2a=2+a^2 means a is even eg a=2k which after substituting and dividing gives. 2k=1+2*k^2 as 1 isn't even there are no whole number solutions.
Because 2^2 is smaller than 4×2, there are no real number solution
You are a God of Patience. Really amazed.
Don't describe someone or something by "God".
16:45 Sneaky error correction😉
Wonderful Mr Penn 🤝🤝🤝
There are two problems with the working, both while assuming that m ≥ 2.
The first is that m(b² -2) - (b² + 1) ≥ 2(b² -2) - (b² + 1) is only true if b is also ≥ 2, which should have been assumed beforehand by checking for solutions where b = 1 prior to this. Plugging b = 1 in the original equation would have yielded a² = 2a, which fits in the original solution. This should have been stated before performing the inequality, not after.
The second is that there was no need to completely remove the ⌊m²/2⌋ since ⌊m²/2⌋ ≥ 2. So for me, the inequality would have gone as follows:
2m - 2 + 2 ≤ 2(m - 1) + ⌊m²/2⌋ = ⌊[4(m + 1) + m²]/(8 + 2m)⌋ ≤ (4m + 4 + m²)/(2m + 8)
2m ≤ (4m + 4 + m²)/(2m + 8)
2m(2m + 8) ≤ 4m + 4 + m²
4m² + 16m ≤ 4m + 4 + m²
3m² + 12m - 4 ≤ 0
m ∈ [-2 - 4/√3, -2 + 4/√3]
30:23, if b=2 then from m(b^2-2)-(b^2+1)
At 21:19 when he found out that b was greater than or equal to 1 i couldnt help but be like, duh, the question told you b was a natural number
he found that for m=1 we don't have any restrictions for b
it doesnt mean "b is one specific integer greater than 1" but "for any b greater than 1"
@@anastasissfyrides2919 oh thanks for explaining
@@kostaspapadopoulos1480 oh thank you.
Dr penn , since you love floor functions so much , here's another floor function problem from the Indian national maths olympiad 2014 problem 2
Let n be a natural (positive integer) number , prove that floor(n/1) + floor(n/2) + floor(n/3) + floor(n/4) + .... + Floor (n/n) + floor(√n) is even .
Waah induction ka best question
@@samyakmahapatra9154 waah gaawd 🙏🙏 maine waise try nahi kiya tbh, dekh ne me accha laga de diya xD
@giraffemathcoder wow man 👍
@@samyakmahapatra9154 can we use induction in floor or ceiling fxñ???
@@siulibasak3804 hmm induction looks tough on this one , i tried it but failed lol
I love your videos you are one of my favorite math teachers
I feel like there should have been a nicer way to do this one
Hi, is it possible to share our personal solutions with you Dr. Penn? If yes then how can we?
What accurs when you finish math
This problem is painful
👍👍👍👍👍
Sweet
Hello. I am loyal follower of your channel Mr from Morocco.
Great solution as always! However, it is too long and I think that in a maths contest there must be a quick nice solution. You have solved the equation in at least 36 minutes given that you already knew what you will be talking about in the video.
Keep providing us with the great work 😄
That's pretty hard
I thought so too!! Another commenter said it was too easy...
Dear Mr. Penn, i am at 21:41 and want to ask you a question.
You are solving the problem in a straightforward manner, in an ordered way. But, in reality, when faced with these kind of problems, are there backs and forth, twists and turns, comings and goings, before finding the right way?
5:35 (a-b)^2 is STRICTLY bigger than 0
First he checked the case a=b and got no solution so a≠b⇒a-b≠0⇒(a-b)²>0
very nice 😁🤘🏻👏🏻
yet again, hq stuff :)
I have got to that solution right by the moment of the introduction. I mentally have figured out that it is 2. I have jumped to the end of the video and "voila" natural number 2. The rest in between seemed to me quite the display of how people love to complicate themselves for the math art' sake when math itself would feel so embarrassed. With all the due respect for the flow of thinking an the demonstration endeavored afterwards to get to that natural number 2, for sure. I am no math person, however, in the display of the formula, I have intuitively gotten to the number 2 as the result. Intuition. As simply as intuition it can be. Thank you, professor for the challenge.
the floor function Dr.Penn's fav
the floor function Dr.Penn's fav
8:45 ... could you elaborate a little ? that was a bit fast. I get it, but it was a bit 'slick' compared to all the other arithmetic you add in most videos. I think you just saying that you subtracted one detracted from the original fact that a>b>1 is all.
I see you remembered at 10:15 :-)
If you have an expression similar to xy + nx + ny, you can add a constant to factor it. In this case, if you just add n^2, the expression becomes xy + nx + ny + n^2, which can be factored as (x + n)(y + n).
Just rearranging. If it looks weird its because he flipped the inequality and thus changed the direction.
This is a good and profoundly exploring solution of this problem. However, I think it is very long way for exam time and I think you just could replace the b=a×k while k is any rational or irrational number which plays essential coefficient role between a and b and according to that solving this problem would be shorter. I am not sure would it work exactly or not, I just realized something from first glance. Keep posting, your channel is really enjoyable🙂👍🏻
Sir pls do a giveaway when u reach 100k subs!!😁