- This here is a test paper from a fierce competition.
It is from the Harvard-MIT Math Tournament.
Now just known as HMMT,
it is an annual competition for high school students,
with current Harvard and MIT students
writing the questions and running the event.
It is intended to encourage an interest in math,
but they don't go easy on anyone.
It is very prestigious,
and attracts about 1,000 students from around the world.
I tried some of the questions,
and let me just say they are very tough.
It is an extension beyond what is taught in schools.
So sometimes to solve these,
you need far-flung theorems, tricks and techniques,
many of which I've never encountered before.
And no, you don't get a calculator.
Each year, there is a November tournament held at Harvard,
and then a February or March tournament held at MIT.
The November one is supposed to be easier,
but it's not easy at all.
It's just less focused on mathematical proofs
than the February one.
Let me show you what some of the questions look like.
This is the November General Round paper,
and there are 10 questions.
It looks like they get harder towards the end.
Although the actual points given for each question
is determined by a scoring algorithm,
which takes into account
how many people got that question correct.
It starts off not so bad.
The first question is about four different-colored dominoes:
red-red, red-blue, blue-red, and blue-blue.
We're asked to find the number of ways
to arrange the dominoes in a row, end to end,
such that adjacent ends have the same color.
The answers provided on the website
say that the answer for this one is 4.
From a little bit of trial and error,
and extrapolating all the possibilities.
Question 2 is testing your ability
to manipulate an algebraic expression.
The answers show that if you realize
it's useful to work with the quotient rules,
you can work out the possibilities for the answer.
But then, in my eyes, they get a lot more difficult.
Let's go down to number 7,
and I'll read it out to you in full.
We're told that there's this function,
which is defined by this expression here
if x is not in the range between
minus cube root 2 and 0,
and 0 otherwise.
So when x is in that range, it's 0.
The sum of all real numbers x
for which f 10 of x is equal to 1
can be written like this:
a + b times root c over d,
where a, b, c, d are integers.
d is positive, c is square-free.
And the greatest common denominator of a, b, and d is 1.
Find 1000 a + 100 b + 10 c + d
In brackets, they say here that
f n of x is the function f of x iterated n times.
I couldn't keep up with their solution
that found this different expression for f of x,
implying something else.
And then they reverse-engineer the quadratic formula.
This is the rest of the solution here.
And keep in mind, for this paper of 10 questions,
you have 50 minutes.
And the idea of working through this in the time limit,
I just thought,
I just can't. I just can't.
You can't what?
I can't. I can't even.
You can't even what?
I just can't even.
Moving on to question 8,
we have an interesting-sounding question
about an ant walking around on a circle.
Then question 9, about a cyclic convex pentagon,
and question 10, which is suspiciously similar to number 7.
It says that real numbers x, y, and z
satisfy these three expressions.
The largest possible value of xyz
is a + b root c over d,
where a, b, c, d are integers.
Find 1000 a + 100 b + 10 c + d
That sounds very similar to the question
that we had in number 7,
except this time, instead of being given a function,
we have these three algebraic expressions.
Now I was inspired to take a look at this paper
by my friend PK, who runs the channel "Dr. PK Math,"
and he's been solving some of these problems lately.
I spoke to him about this exam in particular,
and he's kindly uploaded his fully-worked solution
of number 10.
And after seeing how he did it,
I can actually understand it.
PK has also uploaded his solution for question number 7.
So I'll link his videos down in the description,
and you can check them out
if you wanna get a more complete view
of how the solutions are done.
The first thing PK does
is to define a variable k
to be the largest possible value of xyz.
He then takes a look at this first expression,
and factors out the x.
So you have x times 1 + y + yz is equal to 1.
Then take a look at the second expression.
We could replace xyz with k,
and then take k over to that side.
So rewriting this, we'd have y + yz
is equal to 2 minus k.
This left-hand side is pretty similar
to what we have inside this bracket here,
just with an additional 1.
So if we wanted to know 1 + y + yz,
well, that would be equal to 2 minus k, plus 1,
which would be 3 minus k.
Then this would tell us that
x times 3 minus k is equal to 1.
And we can can rearrange to get that
x is equal to 1 over 3 minus k.
This is one of the three important equations
that we'll need to solve the problem.
We can do very similar things to find y and z,
just by changing
which expression we start that process with.
Going through it,
you'd get these as the expressions for x, y, and z.
Then if you multiplied them all together,
you'd get xyz, also known as k, to be this expression.
This can be multiplied out and rearranged
to give you a quartic for k.
And if you managed to find the roots
of this quartic equation
through a combination of guesswork, factorizing,
division, and the quadratic formula,
you'd find that they are 1, 4,
and 5 plus or minus square root 17 over 2.
This last root is in the form
that they told us about in the question.
This a + b root c over d,
and k is the largest when we take the positive of this.
So we can find our a to be 5, our b to be 1,
c, 17; and d is 2.
Adding that all up in the way they want
would give you the answer,
which is written down here as 5,272.
They do it a different way in the solutions,
which doesn't involve solving a quartic,
but in my eyes,
their first step takes a much greater leap of imagination
to think of.
Anyway, this was just the General Round,
and there are plenty more rounds waiting for us.
Here's the Theme Round.
And this will be another 10 questions in 50 minutes.
I'm not sure what the theme was here.
Maybe it was to be as mind-bending as possible,
because all the questions start with this phrase:
"Let n be the answer to this problem."
Good luck with that
if you can't find the answers to the problems.
Question 5 says, let n be the answer to this problem.
The polynomial x to the n + a x squared + bx + c
has real coefficients and exactly k real roots.
Find the sum of the possible values of k.
And then there's a Team Round.
On here, we've got questions such as number 5:
How many ways are there to place 31 knights
in the cells of an 8 by 8 unit grid,
so that no two can attack one another.
Note that a knight attacks another knight
if the distance between the centers of their cells
is exactly root 5.
Then there's something called the Guts Round.
And this seems probably the most fun of the rounds.
You're working in teams,
and it's one of those competitions
where the team sends a runner up
to go and get the questions.
And they come in these groups of three,
and you don't get the next set of questions
until you finish these ones.
These are of varying difficulty,
and the first ones are pretty easy,
but I like number 3.
It says, suppose all the letters that spell "Harvard"
multiplied together,
is equal to the letters of "MIT" multiplied together,
equal to h times m times m times t.
And all of those are equal to 100.
Then find the letters multiplied together
that make up "rad trivia."
It's a question of trying to make
some combination of these letters up here
to be left with these ones here.
And the way it can be done is to do
"Harvard" times "MIT" times "MIT"
divided by "HMMT."
That's 100, 100, 100 divided by 100.
So it's 100 squared, which is 10,000.
And if you cross out all the letters
that appear on the top and the bottom,
you get "rad trivia."
Now let's take a look at the spring tournament.
I believe it's usually held in February,
but last year was held in March, online.
Instead of a General Round,
the individual questions here are split into three topics.
We have have the Geometry Round,
with questions such as,
Two circles with radii 71 and 100 are externally tangent.
Compute the largest possible area of a right triangle
whose vertices are each on at least one of the circles.
Next is the Combinatorics Round,
with questions such as:
A light pulse starts at a corner of a reflective square.
It bounces around inside the square,
reflecting off the square's perimeter n times
before ending in a different corner.
The path of the light pulse, when traced,
divides the square into exactly 2,021 regions.
Compute the smallest possible value of n.
And the last individual round for spring
is the Algebra and Number Theory Round,
where you'll find:
Compute the sum of all positive integers n
for which the expression
n + 7 over the square root of n minus 1
is an integer.
It starts off somewhat comprehensible,
but it devolves into very complicated questions
that I'm not even confident
I'll be able to read out correctly.
And then here's the Team Round for the spring tournament.
The big difference here to the November tournament
is that these are proof type questions.
For example, show this or show that,
determine with proof the value of this expression,
and to see what proof answers
are kind of expected to look like,
let's take a look at the solutions provided online.
For each question,
there tends to be multiple possible solutions.
I was starting to think that
whatever high school students are able to ace these
must be extremely clever.
Wikipedia has a list of the winners from each year,
and taking a look at it,
I noticed that it wasn't uncommon
for the individual champion
to win multiple years in a row.
For example, Luke Robitaille has won
in 2021, 2020 and 2018.
So I looked him up,
and I found he actually has a YouTube channel
where he solves math problems.
Shall I say, much more advanced
than the ones I solve on my channel.
I also looked up a lot of the other names,
and a big trend I saw
was that a lot of them have gone on to win
sometimes multiple gold medals
at the International Math Olympiad.
So this competition attracts people
who are already performing at the very top level.
I found that most of the more recent names
were currently grad students in mathematics,
either at MIT or Harvard or other prestigious unis,
but then scrolling all the way back to the first few years,
this guy Ricky won over 20 years ago.
So I wondered what he was up to now.
And if this is the same person,
then he's currently an assistant professor
working in research mathematics.
And that wasn't uncommon either.
A few names among the list seem to be working in maths.
And I guess after all,
the point of this competition
was to encourage interest in math
and to show what higher math looks like
to high school students.
It doesn't surprise me
that the winners who had really immersed themself in this
want to do more than just be interested in math,
but really wanna turn it into their life and their career.
Thanks for watching this video,
and I encourage you to go and check out Dr. PK Math.
He's got heaps of exam type videos on his channel,
including the two solutions to questions
from this tournament.
Thanks to my patrons for making these videos possible,
and a special shoutout
to today's patron cat of the day, Tom.
(cat meows)