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