Challenging Mathematics Questions

…for age 16+

The following questions come from various university Admissions Tests but will be useful for any students really wishing to challenge themselves in Mathematics, particularly those students going on to study Mathematics beyond school.

Advanced Problems in Mathematics

Advanced Problems in Mathematics, Stephen Siklos. Step Paper I 1993

You can download a free copy of  Stephen Siklos’ Advanced Problems in Mathematics:Preparing for University. Whilst Advanced Problems in Mathematics has been written to support students taking STEP examinations, it is excellent preparation for any undergraduate Mathematics course. One aim of the book is to bridge the gap between school anduniversity.


Advanced Problems in Mathematics Problem Index

Following each question, you will find a discussion and a full solution. The clear Contents page lists all 75 problems. Each problem has a title which is a direct link to the problem.

STEP (Sixth Term Examination Paper) Mathematics is a well-established mathematics examination designed to test candidates on questions that are similar in style to undergraduate mathematics. The specification is available on the Cambridge Assessment website.

You will find free STEP and AEA (Advanced Extension Award) solutions on MEI’s website. The STEP question papers are all available on the Cambridge Assessment website, note the STEP resources include a searchable database showing the topic of each question.

The AEA qualification from Pearson is based on the A Level specification and designed for the top 10% of students to help differentiate between the most able candidates. Note that AEA papers and mark schemes can be found here.

Underground Mathematics has STEP questions within their Review Questions. Each question comes with a fully worked solution.
Underground Maths STEP

Underground Maths STEP example

STEP CUSee also, from Cambridge University, their STEP Support Programme. From the home page, access the resources, you will see STEP Support Programme Foundation modules, STEP 2 modules and STEP 3 modules. Any Mathematics students aiming for the highest grades, whether they are sitting university Admissions tests or not can benefit from studying for example the Foundation modules which provide a structured introduction to solving STEP problems.

From Nrich, Prepare for University.
Nrich University prep


To challenge your students, MAT,  STEP and AEA questions are all excellent. Dr Jamie Frost has created such a useful resource with his STEP, MAT and AEA questions all aligned to new A Level chapters. This document is 156 pages of categorised questions (brief answers are given). Also available is a pdf file of just the STEP questions.

For mark schemes see:

Further support for the Mathematics Admissions Test is available on Dr Frost’s site. The resources include topic based worksheets also a problem solving booklet.; the booklet has 151 problems which should certainly keep any students preparing for Mathematics university interviews busy!

Underground Mathematics Review MenuNote the Underground Mathematics Review Questions include Oxford Mathematics Admissions Test questions and full solutions.

TMUA is a newer admissions test only one question is available on the Underground Maths site, however, papers from 2016 onwards and mark schemes here and there is much overlap between the specifications for the TMUA and other tests such as the Oxford MAT, so these questions should provide useful resources for students taking the TMUA examination. Interestingly, Durham University states that “Those students already registered for MAT may substitute those results in place of our own test, if they do not wish to take both.”

Warwick University advise taking one of MAT, TMUA or STEP. As Warwick state, these questions deepen students’ understanding of the A-Level material.

Oxford Admissions Test

As already mentioned, TUMA papers and mark schemes are available from Cambridge Assessment and I would highly recommend the presentation introducing the test, from Julian Gilbey. As suggested – try the questions first (pdf file) before watching the presentation.

Talking to Julian Gilbey, he recommends for the TMUA, the importance of working through the Extended specification notes on the website, to learn about the logic side. (See Test Specifications for the specification and enhanced specification.) He also stresses that the more Maths you can do, the more you work on stretching problems and think hard about maths the better you will get at maths. Examples he mentions for resources are any questions on the Underground Maths website, (not just the review questions already mentioned here), UKMT and olympiad problems, STEP problems (probably just STEP I initially).

“And essentially your ability to ‘think mathematically’ and to solve mathematical problems is all that these tests are testing”


For further sources try UKMT Senior Maths Challenge Questions.


Dr Frost

…and also see MadAsMaths with its many papers and solutions increasing in difficulty.
c1qn Madas


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