I only got problem 3. And I'm ashamed to say I only got it because I know log_10(2) to 5 digits (it's .30103, it's a palindrome so it's easy to remember).

Very good video, with a lot of interesting problems. A small aside, but10^100 has 101 digits, specifically, one 1 and one hundred 0s. In general, for any natural number n, 10^n has n+1 digits

@EngMorvan

3 жыл бұрын

Correct, but as the same mistake happens in the denominator, the final answer is 210. I checked it with a calculator. 😁

@k_meleon

3 жыл бұрын

honestly I just think that doing a quick estimation of log_10(125^100) is better because in my head I calculated 100*log(125) and I estimated log(125) to be approximately 2.1 so I got the exact answer with a pretty easy method.

@dlevi67

3 жыл бұрын

@@k_meleon Or also note that 125^100 = (5^3)^100 = 5^300. Knowing that log (2) is very close to 0.3, it follows that log 5 is very close to 0.7, so 5^300 ~= (10^0.7)^300 = 10^210.

@Antonio-wh8lh

3 жыл бұрын

@@EngMorvan My calculator can’t handle numbers greater than 10^100

The graphical style reminded me so much of 3b1b... Still, very interesting idea for a video!

@leGEEK84

3 жыл бұрын

I think they use the same python library to make the animations : manim that have been created by 3b1b

The first one is so interesting, when I look at the solution it feels almost obvious, but I would never think of that solution myself

This is the kind of math I dislike. No ideas, only tricks.

@abdulmasaiev9024

3 жыл бұрын

Actually, it's arguably the exact opposite. There's only ideas here, stark and bare, stripped of all fluff. These solutions are rather elegant, too. Tossing these at high schoolers though? Ehhhhhh, I don't know about that. They're very good olympiad problems.

Yea yess, I remember they were called Coffin Problems, I saw them during my jee preparation, and I tried them. I was able to do a few but not many.

this is very high quality, im surprised it doesnt have too many views

@h7x4

3 жыл бұрын

Give it some time, it's just a day old

For problems in the style of problem 2, one can arrive at decent rational bounds of logarithms by using its monotonicity and spamming the property log_b(a) = 1/k log_b(a^k). In the video, log_2(3) = 1/2 log_2(9) > 1/2 log_2(8) = 3/2 = 1/2 log_3(27) > 1/2 log_3(25) = log_3(5). If we want more accurate bounds for log_2(3), we could proceed as follows log_2(3) = 1/7 log_2 (2157) Noting that 2048 1/7 log_2 (2048) 11/7 We could keep increasing the denominator, with the difference between lower and upper bounds being 1/k but this gets challenging for hand calculations as the numbers get bigger. e.g. log_2(3) = 1/20 log_2(3486784401) 2^31 = 2147483648 1.55 = 31/20 So this method isn't really scalable (other methods would be preferred here).

question 2 at 3:26 is unfinished. good video though thanks

@MetaMaths

3 жыл бұрын

Yes ! Apologies.

Nice video!! Really hard problems :(. In the last one I tried putting 5^3 then multiplying and dividing by 2^300 then taking log base 10, then our answer is 300(1 - log2) I tried to estimate log2. I just used log 2 ~ 1/4 (rough estimation) and I got that the answer is approximately 225 digits. I also tried to use pitagorean triples in the first problem but I was not able to get six such points

We did some of these problems at our Calculus lessons, very hard ones indeed, sometimes they are refered to as "Jewish" problems.

@khalidwalid658

3 жыл бұрын

Jewish!!!! Why? What does that mean?

@the_allucinator

3 жыл бұрын

@@khalidwalid658, watch the video again. It can be summarised as a problem given to those under-equipped for the purpose of rejecting them subtly. Target: Jewish Highschooler with no background on calculus Problem given: Something that requires a substantial amount of calculus. Tactic: Unfair and unreasonable difficulty of test to deliberately fail a student.

@Maxence1402a

3 жыл бұрын

@@the_allucinator Well, nowadays they succeed more often than natives, so it's a win I guess?

I tried to do the third one with another method, but I had to approximate a logarithm. Basically, if you apply log base 10 to a number and get the minor integer of it, you get its number of digits it has minus one. So, log(10,125^100) =100*log(10,125) =100*[log(5,125)/log(5,10)] =300/[1+log(5,2)] This is where I found the hard part of the problem. I had to approximate log base 5 of 2. I approximated it to be between 1/4 and 1/2, could've done it better but I didn't really wanted to, I chose the number exactly between both, so (8/3). Plugging this in: 300/[1+8/3] =300*8/11 The minor integer of this number is 216, adding the one I got 217. Pretty close, hehehe. But this is me after finishing a Master's Degree on Physics, no way a high schooler can do this.

I dont know why am i here, i study law.

@arjunnaik4079

3 жыл бұрын

Hahaha me too but love mellowness of these mathematical aficionados

@user-og9nl5mt1b

3 жыл бұрын

Because u like maths

Problem 2 is easily solved with convexity. Log is concave, so is log o log. Comparing log2(3) to log3(5) is like comparing log(3)² to log(2)×log(5). Taking the log of this, it is equivalent to compare log(log 3) to (log(log 2) + log(log 5))/2. Since 3 is inside [2,5], and log o log is concave, log(log(3)) > (log(log 2) + log(log 5))/2. QED.

For the first problem, can someone explain to me why the distance from B, to the reflection of B' across the perpendicular bisector of AC, is a rational length?

@PavanKumar-xv1hg

3 жыл бұрын

you can think of it as a doubling of length BO to BOB' wrt line AC, SO 2.4*2=4.8

@gresach

3 жыл бұрын

Hi Eric Wu: good question. I guess this is why MetaMaths suggested pausing for thought. If O is the origin, then the coordinates are A = (16/5,0), B+- = (0,+-12/5), C = (9/5,0) & D+- = (-7/5,+-12/5). But all 6 points lie on the same circle, so B+D- is a diameter and has length 5. If we compute it the distance from B+ to D- is sqrt(7^2+24^2)/5 = sqrt(25^2)/5 = 5, which is a different pythagorean triad! The same works for any pythagorean triad. A triad has sides p^2-q^2, 2pq, p^2+q^2, and the larger triad uses (up to a constant) pq & p^2-q^2 to get two generators for the larger triangle as P & Q, with |P^2-Q^2|, 2PQ, P^2+R^2. It's not a coincidence that AC = B+D-. Essentially you can regard B+D- as a hypoteneuse of a four Pythagorean triangles, with other vertices A, C and their reflections, A' & C' in the perpendicular bisector of B+D-.

@EastBurningRed

3 жыл бұрын

The simplest answer is that the midpoint of the hypotenuse of a right triangle is also the circumcenter of the triangle, namely the distance between it and any of the three vertices is the same. With that, we know the distance you asked for is exactly the same length as the hypotenuse.

That's nice

I am in high school, they all kinda seemed possible but it wasn't obvious how to do them (including the calculus ones)

@trolleycartwheel9409

3 жыл бұрын

That's every test you'll take in college if you haven't done any preparation/studying.

Could you make videos on olympiad type problems? :)

The real devil with those problems is not only that they are hard and solutions are difficult to find, but also that once you know the solution it often seems somewhat... obvious? Like the one you have shown, with 6 that need to have rational distances between each pair. It seems almost obvious to use symmetric transformations to find those points, but only once you know how it's done.

The last two problems of simular kind were in my highschool finals

Glad to see you went viral, very interesting and great presentation. You deserve more subs! Would love to see this channel grow :) here @ 5.7k

The number of digits of n is the log base 10 of n (integer part) plus one. Therefore intg(log_10(125^100))+1 = intg(100*log_10(125))+1 = intg(100*ln(125)/ln(10))+1 = 210

@Mmmm1ch43l

3 жыл бұрын

How would you calculate this without a calculator though?

@j9dz2sf

3 жыл бұрын

@@Mmmm1ch43l : like him, I need a calculator indeed.

@Mmmm1ch43l

3 жыл бұрын

@@j9dz2sf No, you can do what he does without a calculator fairly easily

Problem 1 . took me 1 minute Yes i don't have exact prove but my intuition says you just need to draw circle around a point with integer radius. Draw all possible circles with integer radius Then at that circle you put another point. Then you draw circle with integer radius from it. all possible circles with integer radius. Then you see where those both circles intersect. There would be infinite amount of intersections. Then you take one of those intersections and draw circles from it with integer raduis. Remember you can only keep drawing circles from the point where ALL of the previous circles have intersected And you just keep doing it. Let me actually physically try it so if it is possible

@pravinrao3669

3 жыл бұрын

yeah this overconfidence is going to ruin my life. Why do i keep oscillating between destructive overconfidence

10¹⁰⁰ has 101 digits but otherwise, nice

We still have such problems at olympiads. And you're kinda have to win an olympiad to get free education at top universities. Obviously these problems are hard but otherwise everyone would've been able to solve them and there would be not enough place at unis. I'm not justifying discrimination against Jews that took place in USSR, but hard creative math problems were always a thing in Russia and thanks to this system I was able to get free education at one of the best universities in Europe.

The first I definitly wouldnt have gotten; the second one I think is not difficult at all, in particular if they back then didnt have pocket calculators they would have known the approximations for sqrt of 2 and 3 for sure (thats how i solved it); the third one can be done.

The log one wasn't too bad with calculus

1. I don't think so, lol...probably an issue with diagonals of the polygon formed by said points...I mean...3 would be possible, a triangle with 3 integer-sides...but the moment you have 4, lol, I don't think so...

in question 1 . BB' is still not a integer , its 4.8, so ...

@santiagoporto6391

3 жыл бұрын

You can scale everything up. In this case 2.4= 12/5 so you multiply every length by 5 and you get all integer distances :)

@xplorewithme2567

3 жыл бұрын

@@santiagoporto6391 oh !! , thanks buddy

I made the second and the third but the First One Is impossible for me

Problem 1 ,is the distance rational or integer?

@Arthoma1810

3 жыл бұрын

Can scale by some factor to make integer

@JohnDlugosz

3 жыл бұрын

Rational is good enough. When you're done, find a common denomonator and scale up the whole thing by that factor.

Problem 1 seems difficult after seeing the solution; but it’s not actually hard if one has done a lot of combinatorial geo...

@valerianmp

3 жыл бұрын

But remember that during its time, you need to solve this orally in front of someone who probably thinks that you are trash

That's so unfair. lol

2nd one everyone does at school in russia lol

Well a lot of them seem to be in the math olympiad style so I’d say any kid with a substational amount of math Olympiad training would solve them (easily, they’re aime lvl lol)

I can’t visualize. Math boggles my eyes.

Your first actually good video. Please, make similar videos, revive your mathematical problem solving insights. Much needed, thanks.

I just came here whilst not understanding anything just because the video is good 😩

My answer to the first problem was to simply have all the 6 points overlap. 0 is an integer, None of the lines are colinear. lol

@yosid1702

3 жыл бұрын

arent they all colinear tho?

They all look like Olympiad-level problems, and the Soviet Bloc pioneered Math Olympiads. They seem reasonable to me. Only those with skill should be allowed in the university under Socialism. They could reapply as many times as they've wanted, couldn't they?

Well, if the students were given the required techniques, maybe they would be able to solve them in a short time. There are similar questions in Chinese Gaokao exams for maths, however all students are taking papers of similar difficulties and they are normally given the required skills to do them. In this case, I feel it is okay to separate students into different groups. After all, not everyone is interested in maths enough to be good at it or talented at maths naturally. This way you can still have people with a good background of mathematics which is required for their chosen field of studies and not worry too much about deeper problems that occur. However, I think there is a much more concerning issue present. The maths education nowadays teaches students how to repeatedly apply same calculation techniques without actually passing on ideas, some do not even care about the ideas. I feel if we can teach maths better, there might be more people able to do these sort of nonstandard questions.

@julianfogel5635

3 жыл бұрын

You missed the point. These impossibly hard problems were not given to all students, just the ones that they wanted to keep out of university because they had a different ethnicity.

@edwardsong5199

3 жыл бұрын

@@julianfogel5635 Yeah I get that and I think that should not happen. I was just trying to say, even though these are used to discriminate people, they can still be used in somewhat harmless way. Sorry maybe it was not a good comment for this video 😁.

These types of problems are imagination killers because they have a rote learning procedure embedded in them. Usually, the people who solve these kinds of problems are not able to become mathematicians or scientists in any other fields. They are not dumb, they are smart. But being smart and being intuitive are two different ideas. These problems are just a waste of time.

@omarinstituteoftechnologys5480

3 жыл бұрын

@1412 Its not uncommon, thats true. But it would be better if people focus more on useful problems that would be more beneficial for the world as a whole.

@omarinstituteoftechnologys5480

3 жыл бұрын

@1412 Yeah people have freedom, they can use their time as they wish. All i am saying is, there is hype in the society about mathematics being a hard and dry subject, the main problem is these problems. Mathematics for people is no more a subject to learn about the nature of quantities, structures, spaces and data. It is mostly for most people a noisy ego satusfying subject now, were intelligent (the so called) tend to show off their rote learned skills. When this rotten society of mathematics gets healed again, world will enjoy learning again about the entities in math, which were useful and delightful.

@omarinstituteoftechnologys5480

3 жыл бұрын

@1412 Not minority, no no, those type of people are the majority. The ignorant ones.

@edwardsong5199

3 жыл бұрын

@@omarinstituteoftechnologys5480 I have read your comments with 1412. I have to say, I agree with you that there are people who pursue a career in science for the sake of recognition and fame rather than for the pleasure of doing the works. I am actually really disgusted by those who make science and maths sounds so abstract and mysterious just to make themselves feel important when then talk down to the general public. However, I really do not feel the science community or the mathematics community are filled with majority of these people. I mean I am a very rookie researcher in a maths department. Though I do not know all the people in my community but at least I feel people are being very nice to me and when I ask a question or trying to explain something, they really listen to things I have to say very patiently. They do not brag about their achievements and do not look down me for not being able to see connections fast enough. This is also what I discovered doing my postgrad, a lot of my friends are not really fast with this sort of questions in the video, but they have good geometric or algebraic intuitions. I do not think it is the maths of these problems that cause people to dislike maths, but rather those who arrogantly think they understand maths just by doing some fast calculations. I feel the people you described are those whom normally do not actually produce any real results, or they may have produced one result and becoming so arrogant thinking they have achieved so much. The problems in the video are good problems to think about, I mean the 2nd one and the 3rd one look like something you would encounter doing analysis, it would be good exercise for error estimation. The first one maybe good for geometric intuition but I'm not so sure. However, I do not think good at these problems is anything worth bragging about or they should be used as a tool for discrimination, it can be obtained by repeated training of using the same techniques and it is not really what mathematics is. I feel the problem is probably how we teach maths as a subject and how people see the prizes and recognition as the rewards of working in these areas. A lot of the calculation techniques are fed to students without any context or the intuition behind doing such calculations. I would be so bored if someone ask me to do 100 integrals involving repeatedly applying the same calculation techniques back and forth. School should really be teaching mathematical thinking rather than the abstract formula without context. I mean just think about the stupid internet question about the answer of a simple arithmetic expression. I saw people commenting " I was taught to do it in this order in the 90s" or "I was taught something different in the 70s". In reality the answer to that question is really not important. There is no context whatsoever and the expression was not well-defined. It is really hard to change how people perceive scientist as intellectual authority and I think that is what most people who pursue fame is going for, the image of being smarter than everyone else. I think the important thing is to realise they are humans too and their thoughts are not as mysterious as what many would have portrayed them, and they are definitely not authority on all matters, in fact most are probably very restricted to their fields of studies. I feel what you said about healing is already happening, many people are putting an effort into making maths transparent and accessible to wider public, and many people are starting to appreciate maths not for its abstraction and difficulties but for its beauty and usefulness.

@omarinstituteoftechnologys5480

3 жыл бұрын

@@edwardsong5199 I agree, The problem with people today is that they have forgotton to appreaciate the beauty and usefulness of mathematics. All they do is rotten procedures. This usually does not happen on undergraduate, graduate or post graduate level. All this nonsensical activities are happening on high school or advanced high school levels. This the root of the causes, because on those levels the math teachers do not have that feel for the subject as their seniors in the fields do (but exceptions are always there).

Problems 2 & 3 aren't difficult questions in Gaokao tbh.

I think it would be hard to tell whether or not these were actually discriminatory type questions aimed at enabling the university to reject certain unwanted individuals, without looking at the other batch of questions that the "other" students got and comparing their difficulties. So unless we could do that, we could very easily assume that the uni simply had high standards for entry which is very acceptable. I haven't myself looked at the sources in the paper but I'd be interested to see if all or just some of the questions given to prospective students were like this.

@Cyberspine

3 жыл бұрын

The fact that they gave different questions based on ethnicity should tell you all you need to know.

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