The UK's Hardest Mathematics Exam 

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A bit of history: the Cambridge entrance exam was abolished in 1986. STEP was introduced in 1987 in a number of subjects though it now only remains for mathematics. STEP I was not typically used for entry to Cambridge Maths (STEP I was based on the content of the A-level maths syllabus, and by design was graded one grade higher than STEP II and STEP III - by which I mean that an 1 in STEP I roughly translates to a 2 in STEP II or II etc). Since the adoption a number of years ago of the requirement of Further Maths A-level (or an international equivalent) for entry to the Maths tripos there was no reason to set STEP I as part of any offer. STEP papers II and III are very comparable in level of difficulty and marking. Directors of Studies get plots of candidates performance on STEP II and STEP III and these show no sign of a systematic difference in marks attained in either paper. Whether the average performance is higher in paper II or paper III will vary from year-to-year. Technically STEP is not an entrance exam. It can be taken by students who have not applied to Cambridge (and have no intention of doing so). Also, a good performance in the exam does not guarantee admittance to the course for those who have applied. If a candidate has taken STEP a year early their results will be available to Directors of Studies and Admissions Tutors. However, if they do not perform well at interview a college may decide not to make them offer (which would have to be unconditional, or nearly so since they already have their STEP results). Currently, such students are automatically put into the winter pool (there is no pool for mature applications in the March admissions round). The pool is the mechanism whereby colleges that are over-subscribed with candidates can make them available to other colleges looking for more applicants. Applicants that are compulsory-pooled seem to be less successful than those put in the pool because admissions tutors think they would be of interest to other colleges.

Noticed a couple of minor typos that I’m correcting here :) In 1b, I picked up a negative when calculating the derivative. This was then cancelled later when simplifying. 16:55 - when equating the coefficients akbk. It should be 4 + 8 + 10 - 20 = 2. Everything else remains correct. 33:46 - when solving for the perpendicular here, we should have -R_b * tan(gamma). So there should be a negative instead of a positive. I did this in the next line, just missed the negative to begin with. Everything else remains correct. :)

I did my math degree about 20 years ago and did well, earning a 1st class degree from Imperial. I then worked in investment banking as a commodity trader but my actual theoretical mathematics from undergrad days have forgotten so much. No idea how my young brain coped with all of that. Problem with mathematics if you are not using it frequently then you forget. Just like with the gym. The good thing, like the gym with muscle memory, the knowledge is still in the brain and just requires active recall with some refresh

I'd say the 2nd round of the British Mathematical Olympiad (BMO2) is harder. Also, you should do a video on the International Mathematical Olympiad :)

Very good explanations. So glad I opted to study biomedical engineering instead!

Hi Ellie. Thanks for the video which cast me back to 1993 when I took STEP II and III. Very challenging exams and I only just scraped through to get a place on the Maths Tripos. I had to work damn hard during the degree too, realising within the first couple of weeks that my dream of doing maths research would certainly be out of my reach. I did manage though to get a good result in my final year even if it was beyond what my average ability deserved. Just goes to show that exams are generally not a great indicator of talent or understanding! Just a few thoughts on the questions covered in the video... (1) In Q.1 I did the trig rearranging before making the substitutions with the benefit of making it easier to guess the correct substitution for the later integrals. For the first, multiplying top and bottom of the integrand by cos(x) yields x.cos(x)/(x.sin(x)+cos(x)). Also noting that du=x.cos(x).dx, this shows why the given substitution makes sense. For the second integral, this time multiplying top and bottom by sin(x), one gets a denominator of x.cos(x)-sin(x) making it obvious what the correct substitution should be. (2) For that second integral, you missed a minus sign in du/dx ie one should get: du = - x.sin(x).dx As a result the final answer should also have a minus sign out front. (3) For a bit of fun, an alternative approach to the induction step in the first part of Q.8 is to use matrices: Let v[n]^t = (a[n] b[n]) be the transpose of column vector v[n] and M be the 2x2 matrix with rows (1 2) and (2 5). Then the recursion gives: v[n+1] = M.v[n] Let q[n] = a[n]^2 + 2a[n]b[n] - b[n]^2 = a[n](a[n]+b[n]) + b[n](a[n]-b[n]) = v[n]^t.N.v[n] where N is the 2x2 matrix with rows (1 1) and (1 -1). q[n+1] = v[n+1]^t.N.v[n+1] = v[n]^t.M^t.N.M.v[n] It is easy to show that M^t.N.M = N and so: q[n+1] = v[n]^t.N.v[n] = q[n]

Three hours for (at least) six questions, that’s 30 min per question. As someone who took part in various maths competitions (national olympiads, IMO qualifiers, Middle European Math Olympiad) that is reassuringly little time. You know it gets hard at 90 mins per problem…

I don't know what category these exams have to fall under but if you want to also look at competition papers, BMO2 is the hardest paper for UK.

I studied math in Bonn, which is the leading math faculty in germany and also internationally recognized quite a lot. but you can still enroll with the worst possible highschool degree possible . i think this is because at bonn nobody gives a shit whether you get a job (except professor). many super talented math graduates end up unemployed or just do simple data science, accounting jobs.

Hi! There’s an argument to say that STEP II is harder than STEP III since the content examined in the second paper is less so they increase the difficulty of the questions to compensate

Question 8 has a nice way via X^2-2Y^2=1 having solutions X=a(n)+ b(n), Y=b(n) by Pell

At 17:00 in question #2, I believe the expansion of the middle term will yield 4(a(k))^2 + 18a(k)b(k) + 20(b(k))^2, which for the ab term does come to 4+18-20=2.

Please, at 10:26 "multiply UP and Donw for 1/cos^4" . How did you find that term? Thank you.

Matriculated in 1975 - got a conditional offer for the Maths tripos based on my Maths A-Level result in Lower Sixth form, predicted grades, and interview; not asked to take the entrance exam at all. Based on your video and subsequently having a look at a couple of the STEP papers I would say they are definitely some way above Maths A-Level (at least, as I remember it), and span the challenges between what, at the time, were the *_two_* further maths A-Levels that I took (Pure Maths and Applied Maths - yes, I took three Maths A-Levels, all WJEC!!) and the S-Level Maths paper. The subject matter of 'S' levels was identical to the 'A 'level syllabus, but the questions considerably harder. 'S' levels principally aimed to test candidates confronted with unfamiliar situations and show their reasoning. Special papers were last set in 2001 and then superseded by the Advanced Extension Awards and to some extent by Sixth Term Examination Papers. A-Level physics completed the set. Based on my undoubted preference for applied maths and physics I switched to Natural Sciences before going up, second best decision I ever made. I love maths, but I learned I didn't have what it would take to succeed at it professionally.

I thought step was hard, then I saw what the actual Cambridge maths exam papers looked like (then I left immediately. Absolutely no way I was putting myself through that).

I occasionally do a STEP paper for fun and to keep my hand in. I'm very glad I didn't have to do anything similar aged 18 and applying to university (A-levels were nerve-racking enough).

I'd say the hardest maths exams for students aged 17-18 was the mock exam I had to take for CSYS Maths Paper II (Pure Maths) in Scotland in 1998. The top mark was 51%. I got 34%. Our teacher said, "I already know you can do the easy questions, you already know you can do the easy questions, so there's no point in me testing you on them". The actual final exam (generally considered the hardest high school exam in Scotland at the time) felt easy by comparison. I got an A in the final.

Congratulations on your genius Ellie, this mechanics question is insane 😅😅, a long time ago I solved all the questions in a high school level mathematics book, about finite induction, they don't even come close to the complexity of this question, they were very direct. It's always a pleasure to admire your intelligence Ellie. 😊

I had to do STEP II this year for my Imperial JMC offer, this years paper was so much better than past years and one of the questions on there was a classic example of invariance that I had done many times while preparing for the BMO. I think I did better than on past papers but too bad I missed my IB grades offer so no imperial rip

Some really nice problems here. I could probably manage the proof by induction and could see myself getting in a mess with the mechanics, care is required to get the balance of forces correct, and I don't have the trig identities stored in my head. It is enjoyable watching workthoughs of difficult problems which I wouldn't know how to start but when an expert proceeds through it all slots into place.

Hello, I really enjoyed this video and your walkthroughs on some of the hardest math tests! Just a quick question- at 7:56, wouldn't du/dx be -x*sin(x), causing the answer to have an additional negative sign? Thank you

Please do the math D of ENS exam to get into ens Ulm, its difficulty is legendary in france

Ah the good old STEP examinations, I sat STEP 1, and yeah they were beyond crazy!!! Great video through Ellie))

I just knew it was STEP from recognising the questions in the thumbnail, I didnt do very well at STEP lol Anyway, Im gonna be doing part III in a couple months, and I did watch your videos on guidance for part III so thank you Ellie!

Tbh, 23:40 prove of c_n convergence is required. And limit of 1/b_n^2 is not enough to assume c_n is convergent)

How are you considering an entrance exam which requires a level knowledge only to be harder than any 3rd/4th year uni exam ??

Madam, True one (thing that won't have substructure) - Model Nearing completion on 4 foot rock

How you DO everythings in Best way

your videos are so fun Ellie :)

I feel good to see your videos 💖 ellie

I expected this video to be bad because of how much I hate step, but your explanations are surprisingly easy to understand, thank you 👏

Had to do STEP 2 for Imperial, just praying for even a 3 😅

Nice video, I believe there is one harder math paper for high-school students and that is the International Math Olympiad, this year it took place in Bath, and the problems were really nice!

I am a bit confused here. In the mechanics problem, shouldn’t the coefficient of friction be static? Then, the force of friction is no longer simply proportional to the normal force. Instead, it should be whatever is necessary for static equilibrium. Then it suffices to write out the dynamical equations and set the forces and torques to 0. I got tan(alpha)=-tan(beta). Did I misunderstand the question? Why should we use the kinetic definition of friction here? Also, I am obstinate enough to spend two hours on doing questions from as opposed studying for my exams. Alas, it’s all good fun. Thanks Ellie for the wonderful video as always.

What a very bright young woman !

I swear U THIS the BEST MATH RU-vid CHANNEL

For q8, why did you introduce’k’? Why not just compute for n+1.

Please attempt the HSC Mathematics Extension 2 paper next! It's Australia's hardest mathematics exam.

I wonder who will spot a small mistake in question 8.

omg i just sat both step papers last month and results are coming out soon, hoping for a 1,1 🙏

Great work and nice explanation Ellie, much appreciated! I’m wondering if I can get a copy of the whole exam pdf?

I want video on best Oxford math textbook plz

Hi at 34:20 I think when resolving perpendicular it should be negative R_b * tan(gamma) but in the next line it’s correct when you equate Wsin(alpha)cos(alpha). Also do you have any insight into how these questions are even made? I always wonder how they make the questions so elegant and cancel down to a nice answer.

Can you try an exam of the university of Bonn from germany (Peter Scholze is researching there)? It's one of the hardest universitys for math in europe.

Please provide pdf to download papers you are solving

How do you spot these kinds of substitution 9:34

Hi Ellie, thanks for your video. Please can I ask what you were using that allowed you to circle things on the screen in a way that the inked circle fades after you drew it?

Good work Sleightholm.

That is so easy. They should make the candidate find their own substitution

Thank you

Not in the UK? No A-levels in Scotland

Where can we send you exams that we have found on the internet?

Great explanations. I used to love doing these in school, but people often asked me what was the use of it, i.e. where were they used? I have to say I spent four years at university (natural science) and never did come across the need for these kinds of problem-solving. Could you tell me where they are used then I can shut up, at least in my mind, the dissenters!

GREAT! Liked and subscribed ❤

Purfect thumbnail ba d way🔥🤟

Is there a video on books recommendations?

You should try ISI entrance test(UGA+UGB) for undergrad students from India which is held every year for the prestigious Indian statistical institutes

CMI entrance exam is harder than this :) maybe check once and make a video...

This is great!

Hi Ellie, If I remember rightly, STEP is aimed at the top 5% of students entering for Maths A level, so it's more or less "baked in" that 95% of even A level students are going to fail it.

I studied Mathematics at University in my Engineering Degree.I feel sorry for students who take Physics as their Degree.In Physics you have Quantum Mechanics,Etc.,where the Mathematics is out of this World.

Math is my world

try ENS ulm mp france

sin, cos, tan, cot = essentially a for loop function, that when say calculation miles per hour and distance over time, a square root of 4+3+2+1. Pythagorean theorem. You want to travel 1 mile at 30 miles hr. Not from starting at a resting point, but if the traffic light runs from red to green, and the first batch of cars start from zero velocity at rest and increase their speed. The light must stay green , for the remainder of traffic catching up behind them from a distance, relativity theory outside observer. The next set of cars would be traveling at speeds of 30 miles hour, how many seconds will it take those cars or one of them to reach one mile exactly.

I did these a few weeks ago, results day in < a month 😅

Its close too foobar, 4*4,16*16*256*256 matrix tensor determinant quantum gravity newtons law of body. Friction, innertia. Its that like googke maps on cellphone screen size and rotating map in different direction not parallel to actual road, at center pivot rotation. The riad lines will cut in different directions border by 4 sides of screen devices. Thats foobar, that computer keyboard, but without algorithm for decrypting that, them mind stays locked in a 2 dimensional frame, not a spherical holistic view like that of birds in flight. Stone age.

Very GIFTED☺

Were your parents academics?

I am just going to say it, has it been hard in your profession? Have you ever lost friends or colleagues as as result of math?

Can someone in late 30s eligible to study Maths in cambridge

Beauty with brain amazing and incredible 🙏❤

What degree course is that exam?

Praying I haven’t failed step this year 🙏

Mind. Blown. I think what makes this so hard is spotting the trick to unlock the question. Great explanations and thank you so much!

Bro in my countrie that's exam is just testing Ur self if I can go to exam national of Morocco

i have to take this and the PAT and the ESAT ourghhh

smart! am 14 years old and I wish to be like you

It's the easiest exam just taught badly like robots counting in binary so boring. My maths students excell

Does it pain you that not everyone understands mathematics? I think it is amazing too. I don't understand it that much.

i can see why

Solve ISI math question paper

There is a problem with twitter link

Hey Ellie try ISI Examination paper of India

Do Romania , we have some hard ones

fun x maths ❤

Day 78462 of me requesting you to take a look at X/ENS Maths D, hardest maths exam in France, Alain Connes (fields medalist) sat it in 1966 and got a 0/20. Its part of a series of entrance exams to get into the best engineering school in France (X or Polytechnique) and Ens Ulm (the university with more Fields medalists in its alumni than every single european country and every single american university!), definitely worth a look.

I enjoy UR BRITISH ACCENT too😘

do you solve them without looking into solutions

Wonderful

Indian boy 🇮🇳

Is this like an undergraduate admission course because this stuff was way too easy. I did all those questions in my head. These questions are basic JEE question done by Per University students in India.

normal exam at asia 😀

You so clever and pretty 🤩

indian exam is more harder than uk

First things first she is cute ❤

👍👍👍

Hey sis I am able to solve 4 problem in maths 😊

Really weird.

Let's talk about UPSC math 😅🥶