Hi everyone and welcome back to another exam unboxing video today I'm going to be talking about an exam which I would consider to be extremely difficult If you were going to rank exams on a tier list, this one would be right near the top So this is the Putnam competition which is a math competition, a math exam for American undergrad Students and as a way to demonstrate its you know difficulty you can look at what the scores usually are So this exam even though it has 12 questions in total each worth 10 points So that's a total of 120 points. The median score is usually between 0 & 1 and That's despite the fact that students who sit this exam are usually specializing in mathematics So that means at least half of the students often get zero marks on this exam So that indicates how hard it is and I'll show you the questions in a second at first just a little bit more Context this exam was you could say founded in 1927 in memory of William Lowell Putnam by his wife Elizabeth and I guess it's gone on to challenge and maybe haunt undergrad students ever since then they give out cash prizes to the top students ranging from 250 dollars to $2,500 and if you're one of the top 5 students Then you can be named a Putnam fellow and you might get a scholarship to a prestigious university Like Harvard or you might get poached by a company looking for a very talented students Because the names of the top performing students are published as well as mailed out all the participating universities sometimes these top performing students go on to achieve great things in math and science of the previous Putnam fellows two of them have gone on to win a Nobel Prize in Physics including Richard Feynman and three of them have gone on to win a Fields Medal in mathematics in 2018 the top three performing teams were from Harvard MIT and UCLA, so let's have a little look at the questions. They had to solve last year So this is the entire exam here Now if you want your own copy to follow along with there will be a link in the description where you can download this exam But let's first zoom in a little and take a closer look. Now, you might first notice We've got questions A1 to A6 in this first column and then we've got B1 to B6 down here So when you're sitting this exam you actually get two three-hour sessions to work on it with a lunch break in between so I guess you'll be working on all the A Questions and then doing the B questions after lunch Now that makes it seem like you'll have a lot of time to work on this exam but in reality I think it's so difficult that even if you can answer one or maybe half a question for each session You'll actually still be doing okay. So each question is worth 10 points And of course you get 10 if you have a full and complete Solution with the correct proof, although you could get 9 points for a proof That's nearly there And one point for the beginnings of the correct solution that are on the right track Of course you'd get 0 for any working That's kind of on the wrong track. And also this exam is really about trying to prove the statements You're making and even if you say get the right answer if the proof that you're trying to use to show it is incorrect Then you're not going to be getting any points for that. Fully solving just two questions correctly Should get you a halfway decent mark and in the past a score of around 40 was enough to get into the top percentile. From what I can see online there have only been four perfect scores in the history of the competition these problems cover all kinds of Undergraduate mathematics, they cover things like group theory set theory graph theory Number theory, all the theories of mathematics So the concepts should be no higher than undergrad But the way that they're applied is very clever and often the solutions to some of the later problems They're kind of things you would never think of trying so very clever Questions based on some of the concepts that math majors I guess would be learning about. People that would do well in this competition Are probably people with Experience doing math competitions. Maybe that is the International math Olympiad Which you might participate in in high school It's not necessarily just going to be the top student in math class who would ace this but rather someone who Understands sort of the tricks of competition math and what it means to write a proof in a short span of time That whoever is marking the exam is able to understand and follow, you know, and it seems quite complete I think it is at best a tangential skill set To I guess actual mathematics research or being a great mathematician I know I said that some people who have done really well on this exam have gone on to win the Fields Medal But that doesn't mean that everyone who does well on this exam will be a great Researcher and it certainly doesn't mean that if you don't do well at this, you've got no future in math I think it really does cater to people with competition experience or that you know Really enjoy these sort of high pressure environments of solving math throughout an exam situation Now I do have a copy of some solutions here and again in the description There'll be a link to the source of these solutions They're not the official solutions, but they go some way towards showing you what an answer to a problem here would look like now, of course if you're studying for this exam yourself and Trying to work through the problems, you should take a bit of time to try and solve things yourself before looking at the solution But for the sake of this video, we'll have a little bit of a flick through. So just perusing the solutions We can I guess see some of the types of math that come into the solutions How dense some of these sort of proofs are that you're working on Our solutions use skills from analysis from geometry even - we've got matrices and linear algebra going on here We've got some trig identities even coming in Yeah, definitely we've got some complex numbers I can see and Some really really heavy Math notation and yeah kind of intense things going on. It's it's very intimidating I would say this exam from what I've seen online about the exam There are a number of kind of tricks that come into a few of the problems and understanding these Concepts could make the problems look a little less intimidating for you. So I've got the pigeonhole principle the cauchy-schwarz inequality for real numbers and the AM-GM Inequality, so as a math student who's done Maybe some analysis you might have heard of some of these principles To just state the first one in simple terms. According to Wikipedia the pigeonhole principle States that if n items are put into n Containers with n being bigger than n then at least one container must contain more than one item It seems actually incredibly obvious They're just, you know saying if you've got ten pigeons that you want to put into nine holes One of those holes has to have more than one pigeon in it Many things in math are like this, they seem obvious but to have a principle and a way to apply this To numbers in general or to use as part of a proof can make some of these problems apparently a little easier for you So let's try and comprehend the solution to one of these problems I'm going to work through A1 but actually what I'm going to do because it's a lot neater is to step you through What the solutions here say about A1 so let's have a read of it it's asking us to find all ordered pairs a and b of Positive integers for which this is true that 1 over a plus 1 over b is equal to 3 over 2018 I'm choosing this problem to do because it should be the easiest they should get progressively harder as we go down this column so We've got our best chance of trying to understand this one here Now the first step in the solutions is actually taking the form of our initial formula equation and Rewriting it in a factorized way now I have checked this it is just taking a lot of Algebraic manipulation to get from this form to this one it's slightly non-intuitive to even think of doing it this way along the way you needed to add on a factor of 2018 squared to both sides you had to multiply everything by minus 3 But you'd eventually get to this form here and that is actually pretty great going on from there the Solution and tells us about the fact that each of the factors. So these groups of brackets is congruent to 1 mod 3 now this language means that if you do the number mod 3 I guess you'll get a value of 1 so It's kind of like if you were dividing by 3 you'd have a remainder of 1 so the number 4 mod 3 Well 3 goes into 4 once and there's one left over so 4 would be congruent to 1 mod 3 so would 7 because 3 goes into it twice to give you 6 and then there's one left over so would 2018 squared that's also the same also minus 2018 would be congruent to 1 mod 3, that's because three would go evenly into minus 2019 but in this case we are one More than that, so we're going to have one left over. Hope that kind of makes sense. But that's what that language means So because these two factors have this property I guess we also want that same property in the factors on the right hand side if we were to find the factors of this 2018 squared so there are six positive factors of 2018 squared that are congruent to 1 mod 3 and they are 1 2 squared so for what I mentioned before 1009 Four times 1009 and that is and that makes sense because both of those individually follow the property And then we've got four times 1009 squared so these are the six factors Which I guess have the property that we like so that would lead to the six possible pairs for a and being the first one Being an a value of 673 and a B value of one three five eight one one four. I Hope you're not a calculator to help you with that But I'm not sure and then there's a little statement here as for negative factors The ones that are congruent to 1 mod 3 are minus two minus two times one thousand and nine minus two times one thousand and nine squared however, all of these lead to pairs where a Or b are less than zero and the question stated that we wanted positive integers for a and b so They are ruled out. So actually that is I guess our solution there. I don't know if it's necessarily the best proof or just one way of finding the pair's because even though Like we've spoken about this property of congruence And the modulus it's not totally clear to me that because these factors follow that property that You know, these factors would have to as well. Maybe it makes sense, but I don't really see much of a proof of that But at least that's an example of probably what is the smallest solution in this answer book the next question after this is for A2 Like we might as well read it so calculate the determinant of this matrix M Now this question here has a bit more of a long solution. I don't know it's statements about determinants in general looking at all the indices you can even see like one other matrices here and it continues all the way on to this matrix to doesn't stop until There so Yeah, I'm not even really gonna try to to walk through every step of that because it'll probably take us a long while I guess Technically if you follow the rule that they should increase in difficulty as you go along This one B6 should be the most difficult question. It actually looks Suspiciously similar to A1 in the sense that they love using 2018 as a number I guess because it is the 2018 exam You know We're being asked let s be the set of sequences of length 2018 whose terms are in the set 1 2 3 4 5 6 10 and sum two three eight six zero Prove that the cardinality of this is at most this here Well, I might be able to flounder around for a while But probably wouldn't get anywhere if we take a look at the solution for six We can just take a look through it there So hopefully maybe something in there could make sense to you We've got it continuing on to the last page as well some remarks and we've even got an example Of course, you can't use code in the exam but some these questions, you know, they're very difficult math questions, but if you were able to write some scripts Do some coding you could actually probably solve them a lot easier So I think in some sense they turn into medium problems in terms of computer science But very difficult when you're just there trying to prove them with math So I hope that was kind of insightful for you as usual with these exam videos I do I don't want anyone to be too scared off or depressed just by looking at some of these random questions and solutions because of course if you're not in a class that's teaching these or you're not actively preparing for and studying for this exam Then of course, I don't think you're going to do very well like you need to do some very targeted study at these questions Otherwise a lot of the terms and just the way they're done will be foreign to you if you have Participated in this exam yourself before let us know in the comments how you found it and what your opinion of it is Because that would be a nice Insider's perspective and a huge. Congratulations If anyone who's watching this happened to score more than zero points because yeah, it's certainly very difficult Thanks for watching and I'll see you next time
