UKMT senior challenge

Hi there, I’m exploring the drfrostmaths resources for ukmt. I wanted to ask which topics are the most important studying from, geometry, number theory, algebra, combinatorics, graph sketching reasoning and limits, probability and statistics, functional equations, logic, pigeonhole principle and topics in computer science. Thank you.

Reply 1

mqb2766

Original post by lukeanimin

For the smc, the main thing is to practice a few papers. The most important topics are probably the ones you find difficult/get questions wrong, Though smc is geometry heavy and algebra is important, so maybe have a look through those. Things like logic, functional equations, pigeonhole occur infrequently on the smc.

(edited 8 months ago)

Reply 2

epii+1=0

I have answered a very similar question but focused on Olympiads so I'll just copy up my answer to that answer

"I would recommend doing BMO1 past papers because the past Maclaurin Olympiads are much easier then what they have been recently giving. Also the style of papers they give have changed since previous papers had greater emphasis on "finding lengths / angles" in geometry. Now the papers are much more BMO like so getting good at proof is much more important.

Moreover the arguably most important part is to get comfortable with writing FULL solutions. This means you need to write a structured argument WITH WORDS (not just numbers and algebra) explaining what your doing. Also pattern spotting is NOT enough. While finding patterns is important PROVING ANY ASERTIONS is what will increase your score. This may mean learning proof by induction will be fairly useful (Maclaurin Olympiad 2022 Q4 could be solved very neatly with an "inductiony" argument).

Remember this, the way that the papers are marked is following a 3-/7+ marking strategy (or for some lower level Olympiads 4-/7+). This means that a full solution with proof of assertions will give you 7+ marks (out of 10) with marks deducted for minor omissions. Otherwise just insights or unproven claims will give you at most 3 marks. From UKMT's own words "markers often ask themselves if they were able to fix a candidate's error with a 1 minute interview then they deserve 7+ marks otherwise they will get 3-." This all means that SOLVING PROBLEMS FULLY is really important rather than taking a bite out of each one.

You may want to consider buying one of the UKMT handbooks on a topic you are weak in (the one I received as a book prize for the Hamilton Olympiad last year, namely "Introduction to Algebra (Jeremy D King)" has been pretty useful in that its full of unreleased problems with good solutions) although these books tend to be more advanced than Maclaurin Olympiad level.

Just be familiar with the 4 topics which tend to be in Olympiads: ALGEBRA (identities, systems of equations, sequences, series, inequalities, functional equations), NUMBER THEORY (primes, Diophantine equations, modular arithmetic, divisibility and maybe if your curious theorems like Fermat's little theorem / Fermat-Euler and techniques like LTE lemma), GEOMETRY (GCSE circle theorems, Ptolemy's theorem, intersecting secant theorem / tangent secant theorem, etc..) and COMBINATORICS (choose and permute functions, pigeonhole principal, slot filling, recurrence relations from small cases (the last technique is very powerful))

The above list isn't conclusive by any means but it definitely goes beyond what you'll see at the Maclaurin Olympiad but being overprepared often gives you insight and confidence on easier problems. At the end of the day building up good mathematical intuition and familiarity with a wide range of problems might be the most useful

Some useful resources:
UKMT BMOs: https://bmos.ukmt.org.uk/home/bmo.shtml
DrFrost Riemann Zeta club PowerPoints (this is REALLYYYYYY GOOD): https://www.drfrostmaths.com/resources/page.php?id=6
Art of Problem solving (AoPS) links:
https://artofproblemsolving.com/wiki...y/Introduction
https://artofproblemsolving.com/wiki...a/Introduction
https://artofproblemsolving.com/wiki...y/Introduction
https://artofproblemsolving.com/wiki...s/Introduction
https://approach0.xyz/search/

Don't forget just enjoy the problems don't get too stressed."

The above answer I gave in response to a question focused on the Maclaurin Olympiad but more focused on your question, the most important parts are GEOMETRY, NUMBER THEORY, ALGEBRA (functional equations are more for Olympiads rather than challenges) and COMBINATORICS but remember for challenges it is multiple choice so problems are enumeration problems rather than proof problems so concepts like the pigeon hole principal are more useful for proofs in Olympiads. Graph sketching and probability is more useful for Oxbridge entrance exams / interview question preparation.

Other tips for the Senior Maths Challenge
- Firstly consider timing. You have 90mins for 25 questions. If your goal is to get 25/25 I would recommend spending at most 15mins on the first 10 questions, around 30mins on the next 10 questions and the rest of the time on the last 5 questions with some left over to check. If your goal is 20/25 which is a good aim if your goal is the Kangaroo follow-on round then don't ignore the last 5 questions but if you are clueless on them after 1-2 mins then just leave them and focus on the first 20.

- Next being a multiple choice exam even if you don't "solve" a problem you can remove clearly wrong answers and make a better guess or even deduce the correct answer. Remeber you gain 4 points for a correct answer and lose 1 for a wrong answer so don't guess if you have no idea on a lot of questions but if there are 1 or 2 questions you can't do then guessing them may be a decent ideas since your gamble is at most losing 2 points.

- DO PAST PAPERS!!!!! Recently UKMT have added a paywall to all the papers which sucks (but you can use web archives to access old versions of the website and access papers)